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ISSN 0002-9920 (print)
ISSN 1088-9477 (online)
Volume 71, Number 8
of the American Mathematical Society
The cover design is based on
imagery from “Topological Photonics:
A Mathematical Perspective,” page 994.
September 2024
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Cover Credit: The image used in the cover design appears in “Topological Photonics:
A Mathematical Perspective,” p. 994, and is courtesy of Ross Parker.
A Word from... Stephan Ramon Garcia ..................... 992
Feature: Topological Photonics: A Mathematical
Perspective .................................................................. 994
Ross Parker and Alejandro Aceves
Feature: Clusters and Weaves................................... 1004
Mikhail Gorsky and José Simental
Feature: Multiscale Modeling of Viscoelastic
Fluids .........................................................................1015
Paula A. Vasquez
Feature: Codimension One Foliations on
Projective Manifolds ................................................ 1025
Jorge Vitório Pereira
Early Career: Building Community and Keeping
Momentum............................................................... 1032
Finding and Creating Community in Your
Department .......................................................... 1032
Alan Chang and Rachel Greenfeld
Transitioning as an Early-Career
Mathematician ..................................................... 1034
Rosemarie Bongers
Do Mathematics Every Day ................................. 1037
Daniel J. Thompson
Productivity and Time Management in
Research ............................................................... 1040
Steven Senger
How Does Your Daily Life Change When You
Become the Graduate Coordinator? ................... 1042
Chun-Kit Lai
Dear Early Career ................................................. 1043
Memorial Tribute: Eli Goodman (1933–2021)
and Ricky Pollack (1935–2018) .............................. 1044
János Pach, Micha Sharir, Noga Alon,
and Andreas Holmsen
Book Review: The Mathematician ........................... 1054
Reviewed by Thomas Garrity
Education: Freeman Hrabowski: Advocate for
Mathematics and STEM Visionary .......................... 1056
Christian Anderson
What is... a Parking Function? ................................. 1062
J. Carlos Martínez Mori
Bookshelf .................................................................. 1067
Communication: Label Bias: A Pervasive and
Invisibilized Problem............................................... 1069
Yunyi Li, Maria De-Arteaga, and Maytal Saar-Tsechansky
Communication: Double-Anonymous Peer
Review in Mathematics: Implementation for
American Mathematical Society Journals ............... 1079
Dan Abramovich, Henry Cohn, David Futer,
and Robert Harington
FROM THE
AMS SECRETARY
Calls for Nominations & Applications ...................... 1084
JPBM Communications Award ........................... 1084
Joan and Joseph Birman Fellowship for
Women Scholars .................................................. 1084
Centennial Research Fellowship ......................... 1084
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State of the AMS: Executive Director Report .......... 1086
2024 Election ............................................................. 1089
Election Information ........................................... 1089
Candidate Biographies ........................................ 1091
2025 Election ..............................................................110 9
Call for Suggestions ..............................................110 9
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BEGIN Career Resources ....................inside back cover
INVITATIONS
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Washington Update: The AMS Marks Twenty
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Karen Saxe
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New Books Offered by the AMS ............................... 111 8
Meetings & Conferences of the AMS ........................ 11 2 5
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The American Mathematical Society is committed to promoting and facilitating equity, diversity and inclusion throughout the mathematical sciences. For
its own long-term prosperity as well as that of the public at large, our discipline must connect with and appropriately incorporate all sectors of society. We
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A WORD FROM. . .
Stephan Ramon Garcia
The opinions expressed here are not necessarily those of the Notices or the AMS.
National Hispanic Heritage Month (September 15—
October 15) has long been recognized by the Notices with
an annual special issue. For the past several years, I have
had the honor of writing the “A Word from.. .” introduc-
tions for these wonderful issues. My 2023 piece, written to-
ward the beginning of that calendar year, stated that “stun-
ning new developments in articial intelligence are chang-
ing the world around us.” Far from being prophetic, these
words perhaps undersold the extent to which articial in-
telligence, and its uses and misuses, have changed every-
thing.
The 2024 election season is just beginning to hit its
stride as I write these words, with the major parties having
just identied their presumptive nominees. Bad actors are
ooding the internet with falsehoods nearly indistinguish-
able from reality. With AI-generated text, images, sound,
and video polluting the public discourse, the existence of
simple facts can no longer be taken for granted. Objective
truth seems an old-fashioned notion these days.
Closer to home, AI has forced instructors at all levels
to confront how we embrace, or defend ourselves against,
the AI revolution. While our colleagues in the humanities
and social sciences are already grappling with computer-
generated essays written at a disturbingly high level, gener-
ative AI has been somewhat slower to affect the advanced
mathematics curriculum. While Wolfram Alpha and sim-
ilar websites have long been able to answer standard al-
gebra and calculus problems, complete with step-by-step
derivations, it is now only a matter of time before sophisti-
cated AI algorithms can convincingly answer proof-based
questions in upper-level courses. What will we do then?
Stephan Ramon Garcia is W. M. Keck Distinguished Service Professor and
Chair of the Department of Mathematics and Statistics at Pomona College. He
served as an associate editor of the Notices from 2019–2021. His email address
is stephan.garcia@pomona.edu.
For permission to reprint this article, please contact:
reprint-permission@ams.org.
DOI: https://doi.org/10.1090/noti3005
How will we adapt to the new landscape? Will we embrace
the challenge or shrink from it?
The ip side of all this is that the AI revolution is par-
tially the byproduct of advances in the mathematical sci-
ences. After all, computers are fundamentally mathemati-
cal in nature: when it comes down to it, they are machines
for performing mathematical computations.
It’s hard to tell where things will stand a year from now,
save that AI will have progressed in novel and unexpected
directions. How will the AI revolution help or hurt our
communities? Will algorithmic bias, which we have been
warned about for some years now, combine with AI algo-
rithms to disadvantage some at the expense of others?
In this time of uncertainty, there is one thing that is cer-
tain: this year’s Hispanic Heritage issue contains some ex-
citing articles that should interest mathematicians of all
stripes.
The issue kicks off with four terric feature articles. First,
Ross Parker and Alejandro Aceves provide a mathemati-
cal perspective on topological photonics. Then Mikhail
Gorsky and Jos´e Simental introduce us to cluster algebras
and weaves. Next Paula Vasquez tells us about multiscale
modeling of viscoelastic uids. In our last feature article,
Jorge Pereira explains codimension one foliations on pro-
jective manifolds.
In our Math-Education section, Christian Anderson cel-
ebrates Freeman Hrabowski’s mathematical advocacy and
his contributions to diversity in STEM.
Our “What is... ?” column was written by J. Carlos
Mart´ınez Mori, who answers the question: “What is a park-
ing function?”
We also have two important Communications in this
issue. First, Yunyi Li, Maria de-Arteaga, and Maytal
Saar-Tschanskye warn us about label bias. Then Dan
Abramovich, Henry Cohn, David Futer, and Robert Har-
ington catch us up on the implementation of double
anonymous peer review in the mathematics journals of the
AMS.
992 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
Although I highlighted the articles in honor of Hispanic
Heritage Month, there is a lot more exciting content in this
issue as well. Now let’s get on to the 2024 Hispanic Her-
itage Month special issue of the Notices!
Stephan Ramon
Garcia
Credits
Photo of Stephan Ramon Garcia is courtesy of Gizem Karaali.
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SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 993
Topological Photonics:
A Mathematical Perspective
Ross Parker and Alejandro Aceves
1. Introduction
Topological photonics is a framework that follows both
condensed matter physics and topology. It refers to de-
signing the guiding properties of the propagating medium
(e.g., a photonic crystal or a waveguide lattice) in such a
way that the transport of electromagnetic energy is real-
ized in unique, robust, and sometimes unexpected ways.
Consider a simple thought experiment: imagine rst the
two-dimensional wave equation on a square domain, and
assume homogeneous Dirichlet boundary conditions. We
know that the accessible modes extend in periodic form
throughout the whole domain and, in time, waves can
propagate in all directions. This behavior is in response
to the inherent symmetries of the medium. Imagine in-
stead that we engineer the medium in such a way that all
the energy concentrates in the boundary of the medium
and propagates in only one direction. (In the language of
optics, this would be seen as inhibiting back reection and
making the bulk medium act like an insulator).
Typically, in describing a photonic system, we refer to
physical quantities such as frequency, wave vector, polar-
ization, and dispersion. Instead, in the relatively new eld
of topological photonics, the term “topology” refers to a
property of a photonic material that characterizes global
behavior of the wavefunctions on their entire dispersion
map; most importantly, this property takes quantized val-
ues. Think of this as characterizing the “genus” of an ob-
ject, like a doughnut, with the “object” being described
in wave vector space rather than in physical space. There
are analogues in photonics to the topological fact that con-
tinuous deformations will not change the genus of an ob-
ject. As an example, photonic topological insulators that
are designed using articial materials can support topo-
logically nontrivial unidirectional states of light. These
Ross Parker is a research staff member at the IDA Center for Communications
Research, Princeton. His email address is r.parker@ccr-princeton.org.
Alejandro Aceves is a professor in the department of mathematics at Southern
Methodist University. His email address is aaceves@smu.edu.
Communicated by Notices Associate Editor Reza Malek-Madani.
For permission to reprint this article, please contact:
reprint-permission@ams.org.
DOI: https://doi.org/10.1090/noti2998
states are characterized by a particular “genus-like” num-
ber. Since this number is quantized, this unidirectional
property will be robust to perturbations in the underlying
photonic structure.
Photonics research often parallels or aims at explaining
phenomena in other physical contexts. Bose-Einstein con-
densation in condensed matter physics is governed by the
Gross-Pitaevskii equation, which is identical to the nonlin-
ear Schrödinger equation that governs intense laser beam
propagation in a dielectric medium such as air. In the
quantum realm, nontrivial states of two-dimensional mat-
ter (e.g., a periodic lattice of atoms) with broken time-
reversal symmetry can have the property that the bulk is
an insulator while states (modes) exist that carry current
along the sample edges without dissipation. The character-
istic “genus-like” integer is called the Chern number (see
section 3 for an example), which arises out of topological
properties of the material’s band structure (see the discus-
sion in section 2 and section 3 below).
In photonic crystals, a periodic variation of the di-
electric properties of the medium affects photons in the
same manner as solids modulate electrons (with the caveat
that photons are bosons, while electrons are fermions).
The question is whether the topological features are repli-
cated in the analogous photonic system. In two foun-
dational papers by Haldane and Raghu [HR08, RH08],
the authors highlight the photonics analogue to quantum
properties. They demonstrate the ingredients necessary
to create a “one-way waveguide” which exhibits proper-
ties similar to the Quantum Hall Effect. While the model
in [HR08] has not been experimentally realized, it moti-
vated further work by Wang, Chong, Joannopoulos, and
Soljaˇci´c, in which they rst predicted the existence of edge
states in a magneto-optical crystal in the microwave regime
[WCJS08] and then demonstrated these experimentally
[WCJS09]. Experiments by Rechtsman et al. [RZP13]
found topological edge states without the need for an exter-
nal magnetic eld by using a photonic crystal comprising
helical waveguides.
Since then, the eld of topological photonics has ma-
tured and continues to be very active, both in theory and
experiments, as well as in the linear and nonlinear regimes.
While we have briey discussed its origins, it is not our
994 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
Figure 1. Top: schematic of SSH model, unit cell in
dotted box. Coupling constant is within unit cell, and
between unit cells. Bottom: reciprocal lattice, first BZ
shown in red.
purpose to give a detailed history of the eld (for this pur-
pose, we point the interested reader to the review articles
[JS14, LCG22]). Instead, we will focus our discussion
on three prototypical examples: the one-dimensional Su-
Schrieffer-Heeger model, the two-dimensional Haldane
model, and the model of a photonic Floquet topological
insulator from [RZP13]. We hope that this article high-
lights why this is a fertile area for mathematicians to ex-
plore and contribute to with their expertise.
2. SSH Model
The Su-Schrieffer-Heeger (SSH) model [SSH79], is the sim-
plest lattice model that exhibits topological features. It was
devised to describe the electrical conductivity in a doped
polyacetylene polymer chain. The lattice comprises re-
peating, two-node unit cells, where the couplings within
and between unit cells are given by and , respectively
(Figure 1, top). The optical analogue is a linear lattice
of waveguides in which the nearest-neighbor couplings
are staggered (this can be implemented, e.g., by altering
the physical spacings between the bers). Mathematically,
the SSH model can be described by the discrete nonlinear
Schrödinger equation
(1)
where is the th unit cell, and is the strength
of the cubic nonlinearity. (A rigorous mathematical deriva-
tion can be found in [AC22]).
Our analysis follows that of [AOP16, Chapter 1] and
[AC22, Section 8]. The topological features of the opti-
cal SSH model can be understood by studying the linear
model (). As a rst step, we look for plane wave so-
lutions of the form
w
(2)
where is the frequency and is the wavenumber. Equa-
tion (2) is periodic in with period , since w
wfor any integer . The points dene
another linear lattice, which is called the reciprocal lattice
(Figure 1, bottom). The rst Brillouin zone (BZ) is the set
of points closer to the origin than any other point of the
Figure 2. Left: band structure of the SSH model for
(top) and (bottom). Right: circle in the complex
plane traced counterclockwise by for .
reciprocal lattice, which in this case is the interval .
Due to the -periodicity, the BZ is topologically equiva-
lent to the unit circle .
Substituting the ansatz (2) into (1) and simplifying,
we obtain the -dependent eigenvalue problem v
v, where v, and is the Hermitian ma-
trix
Since is -periodic, we only need to consider
, i.e., in the rst BZ. We note that since we are posing
the problem on the full integer lattice, can take any value
in . The eigenvalues of are
(3)
which is the dispersion relation relating
the frequency and the wavenumber . Each eigenvalue
is a continuous function of the wavenumber on the
rst BZ, and is called a band. Since is a matrix,
the SSH model has two bands. All of the bands of the
system form its band structure, which is illustrated in the
left column of Figure 2. (These terms are borrowed from
solid state physics, where the band structure describes the
energy levels that electrons can occupy in a solid).
When , there is a space between the upper and
lower bands, which is known as a band gap. This band gap
has size , where . The band gap closes when
. Since the eigenvalues (3) are unchanged if and
are exchanged, it appears at rst glance that the cases
and are identical, i.e., that the problem is
symmetric about . Interestingly, this is not the case.
For a complete picture, we need to look at the eigenvectors
of as well.
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 995
The normalized eigenvectors corresponding to the
eigenvalues are given by
v
(4)
Since
is a complex number of unit modulus whose
argument is the same as that of , we can write vas
v
Im
Re
(5)
As the wavenumber varies from to over the BZ, the
complex number traces a clockwise circle in the com-
plex plane with center and radius . This circle en-
closes the origin when , but does not when
(Figure 2, right column). The topological invariant is the
winding number of , which is the number of times
travels counterclockwise around the origin. We can
see from Figure 2 that
Ind
(6)
where a winding number of represents a single clock-
wise trip around the origin. (The winding number is un-
dened if ).
The same topological information can be obtained in a
different way by computing a quantity known as the Berry
phase [Ber84] (also known as the Zak phase [Zak89] in
1D). Intuitively, the Berry phase is the phase angle accu-
mulated by a complex vector, e.g., one of the eigenvec-
tors v, around a closed contour in -space. We will
take a brief digression to discuss these concepts, following
[AOP16, Chapter 2] and [Van18, Chapter 3], before com-
puting them for the SSH model.
Let vbe a normalized eigenvector of . For any
wavenumbers and , we dene the relative phase be-
tween vand vby
vv (7)
where is the phase, or argument, of the complex num-
ber . (We are using the Hermitian inner product uv
uv
, where the dagger symbol denotes the con-
jugate transpose; the complex conjugation is placed on the
rst component to be consistent with the Dirac notation
of quantum mechanics). The relative phase satises the
equation vv
vv(8)
It is important to note that the eigenvector vis not
unique. In particular, since it is a unit vector, it is spec-
ied only up to multiplication by a constant unit com-
plex number . The transformation v vis
called a gauge transformation. The relative phase is not
invariant under a gauge transformation, since if we take
vv,vvtransforms to
vvvv
thus .
We wish to dene the change of phase of vin such a
way as to be gauge invariant. To do this, we take a sequence
of points in -space ordered in a loop. We
then dene the discrete Berry phase by
vvvvvv
which is the phase accumulated by varound the loop.
Unlike the relative phases , the Berry phase is gauge
invariant; if we take the gauge transformations v
v, the Berry phase transforms to
, which is equal to , since all of the
cancel. We note that the Berry phase is only unique up
to an integer multiple of unless we take the principal
value of the argument, i.e., restrict to .
We now move from discrete to continuous. In partic-
ular, we wish to compute the phase accumulated by v
along a continuous, closed path. For small , let be
the relative phase accumulated between vand v.
Following (8), satises the equation
vv
vv(9)
Since is small and vis a unit vector, the denominator
in (9) is approximately 1, thus
vv
Expanding both sides in a Taylor series to rst order in
and and simplifying, we nd that is approximately
given by v
v (10)
We dene the Berry connection by
v
vv
v (11)
which is the coefcient of on the RHS of (10). We then
dene the Berry phase to be the integral of Berry connec-
tion around a closed contour :
(12)
As in the discrete case, the Berry connection is not gauge
invariant, while the Berry phase is invariant under gauge
transformations (modulo integer multiples of ).
996 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
Figure 3. Eigenvalues of for (top left), (top
right), and (bottom left). Edge mode eigenvectors
vabfor (bottom right); a(solid blue line) and b
(dotted orange line) are either in-phase (top) or out-of-phase
(bottom). unit cells with Dirichlet boundary conditions.
Returning to the SSH model, we rst compute the Berry
connection using the eigenvector v:
We then compute the Berry phase by integrating the Berry
connection from to . This is a closed contour in the
BZ since the endpoints of correspond to the same
point on the unit circle . The Berry phase is
which is the change in phase of the eigenvector vover
the BZ. Since it is a constant multiple of (6), it conveys the
same information as the winding number.
The fundamental topological difference between the
two asymmetric lattice congurations (and )
becomes evident when we consider a nite lattice. Specif-
ically, we take a lattice comprising waveguides (unit
cells) with Dirichlet boundary conditions at the two ends
(and ). For the linear system, solutions
are standing waves of the form ab, where a
and brepresent the and sublat-
tices of the system. Substituting this ansatz into (1) yields
the eigenvalue problem vv, where vab
and is the off-diagonal block matrix
An intuitive understanding of the difference between
the two cases can be gained by considering the two extreme
congurations, where one of the coupling constants is set
to 0. If and , the lattice comprises inde-
pendent dimers with internal coupling constant . The
eigenvalues of are , each with multiplicity . On the
other hand, if and , the lattice instead com-
prises independent dimers (staggered from the ones
in the previous case) with internal coupling constant , as
well as two unconnected nodes at the ends of the lattice. In
addition to eigenvalues at , each with multiplicity ,
the matrix has two eigenvalues at . The eigenvectors cor-
responding to these zero eigenvalues are and
. These are called edge modes, since they are
localized at the ends of the lattice.
Since is a matrix, its spectrum is a discrete
set of eigenvalues, as opposed to the two continuous
bands of eigenvalues found from the dispersion relation
in the innite lattice case. The eigenvalues of are shown
in Figure 3 for ,, and . The two
asymmetric congurations contain a “gap,” which closes
when . This eigenvalue gap is analogous to the band
gap in the innite lattice case. When , there
are no eigenvalues in this gap, and all of the eigenvectors
are nonlocalized. When , however, there are
two eigenvalues close to (but not exactly at) 0 which lie
within this gap (these eigenvalues approach 0 in the limit
). As in the case where , these eigenvalues
correspond to edge modes, since aand bare localized to
the left and right edges of the lattice, respectively (Figure 3,
bottom right). All of the remaining modes are nonlocal-
ized.
Finally, we briey comment on what occurs when a cu-
bic nonlinearity (is present (see [MS21] for a more
thorough treatment). Standing wave solutions of the form
ab solve the equation vvv, where
v. Numerical continuation exper-
iments show that the edge modes from the linear model
persist for small .
3. Haldane Model
We now turn to a two-dimensional model. We start with
a honeycomb lattice (Figure 4), which is constructed from
a two-site unit cell, with sites labeled and . These unit
cells tile the plane periodically along the two primitive lat-
tice vectors v
and v
to obtain a
hexagonal lattice. We note that the -sites and -sites form
two offset, triangular sublattices. This structure is similar
to that of the material graphene, which is a hexagonal lat-
tice constructed entirely from carbon atoms. The spatial lo-
cation of a unit cell is specied by the vector rnvv,
where n . It is therefore natural to in-
dex the unit cells by the vector n; the locations of the lat-
tice sites nand nin unit cell nare nand n,
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 997
respectively. Each node in the honeycomb lattice is
connected to its three nearest neighbors with coupling
strength , which is a coupling between sublattices.
The directions of the nearest-neighbor (NN) couplings are
given by the vectors
,
, and
, which are depicted in Figure 4.
The resulting linear model can be written as
n
mm
n
mm (13)
where the angle brackets indicate that the sum is taken over
nearest neighbors. The system (13) obeys time-reversal
symmetry, i.e., is invariant under the transformation
,nn
n
n.
The Haldane model [Hal88] adds two more terms to
the honeycomb model. We will see that this results in a
band gap in the spectrum, similar to what occurs in the
SSH model with unequal couplings. First, the Haldane
model has an on-site energy term of magnitude , which
takes opposite signs on the and sublattices. In addi-
tion, there is an imaginary, next-nearest neighbor (NNN)
coupling term with strength . (In the original Haldane
model, this coupling term has the complex strength ;
we take here for simplicity). Each node is cou-
pled to its six next-nearest neighbors, which is a coupling
within sublattices. These couplings are staggered so that
there is no net ux into or out of a single lattice site. The
directions of the next-nearest neighbor couplings are given
by the vectors v,v, and vv(Figure 4).
Using this notation, the linear model for the Haldane lat-
tice is nn
mm
⟪m⟫m
nn
mm
⟪m⟫m (14)
where the double angle brackets indicate that the sum is
taken over next-nearest neighbors. The signs of the NNN
couplings are indicated by the arrows in Figure 4, where
outward and inward pointing arrows denote couplings of
and , respectively. The arrangement of the arrows
in two staggered, counterclockwise triangles ensures no net
ux results from the NNN term. Time-reversal symmetry is
broken when , but is unaffected by the on-site term
.
As in the SSH model, the rst step is to compute the
band structure, which is found by looking for plane wave
solutions to (14) of the form
wkn
n
krn(15)
where k is the wave vector. As in the one-
dimensional case, we restrict ourselves to a bounded re-
Figure 4. Schematic of the Haldane lattice. Rhombus is unit
cell with sites and . Primitive lattice vectors vand v.
Nearest neighbor coupling vectors ,, and .
Next-nearest neighbor coupling vectors ,, and .
gion in k-space, since wkis periodic in k. Specically,
wkr wk, where ris called a reciprocal lattice
vector. To determine the reciprocal lattice vectors, we note
that wkrwkif and only if
krnkrrnkrnrrn
This implies that rr, i.e., rrrvrv
is an integer multiple of . To satisfy this criterion, we
take rv
v
for integers and , where
v
v
The points rdene another lattice, which is called
the reciprocal lattice, and its periodicity is given by the
primitive reciprocal lattice vectors v
and v
. For a two-
dimensional lattice, v
and v
can be computed in terms
of the primitive lattice vectors vand vusing the formulas
v
v
vvv
v
vv
For the honeycomb lattice, v
and v
. The rst BZ is the set of points closer to the
origin than any other point of the reciprocal lattice (out-
lined hexagon in Figure 5, top left). This is the Voronoi
cell around the origin, which is a unit cell of the reciprocal
lattice. Equivalently, the rst BZ is the rhombus spanned
by the reciprocal lattice vectors v
and v
. Since opposite
sides of this rhombus are identied due to periodicity, the
rst BZ has the topology of a torus.
Substituting the ansatz (15) into (14) and simplifying,
we obtain the k-dependent eigenvalue problem kv
kv, where
kk k
k k
k
k k
k
We note that kkand kk. Since
kis periodic in kalong the reciprocal lattice vectors, we
998 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
Figure 5. Top left: reciprocal lattice for Haldane model (blue
dots). Red arrows are primitive reciprocal lattice vectors v
and v
. Outlined hexagon is first BZ, which is equivalent to
rhombus. is the origin of the reciprocal lattice, and are
the two Dirac points, and is the midpoint of the Dirac
points. Band structure of Haldane model for ,(top
right), ,(bottom left), and ,
(bottom right), following piecewise linear path
in BZ. .
only need to compute the eigenvalues of kover the rst
BZ. The eigenvalues of kgive us the two bands k,
where kkk(16)
The corresponding normalized eigenvectors are
vk
kkkkk
k
kk
Plots of the band structure of the Haldane model for sev-
eral parameter congurations are shown in Figure 5. (As is
typically done, e.g., in [HR08, Figure 1], the bands are plot-
ted following a piecewise linear path in the BZ). The eigen-
value khas six minima, which are called Dirac points.
These are located at the corners of the hexagonal BZ (Fig-
ure 5, top left). We label the two Dirac points with
by
and
. The Dirac points
and are not equivalent in the BZ, since they are not re-
lated by translation through reciprocal lattice vectors. (The
remaining Dirac points are equivalent to either or ).
Near the Dirac points, the bands are called Dirac cones,
since, to leading order, they are linear in k. At the Dirac
points,
(17)
It follows from (17) that there is no band gap when
and (Figure 5, top right). For and ,
there is a band gap of size (Figure 5, bottom left).
The Dirac cones at both and touch when . For
xed and , the behavior at the two Dirac points
is no longer symmetric (Figure 5, bottom right). It follows
from (17) that , thus the Dirac cones at can
never touch. The Dirac cones at , however, touch when
, where
. Therefore, there is a band gap
for both
and
, and the band gap closes
when
. The closure of the band gap corresponds to
a topological transition, which we will discuss below.
To understand what is occurring topologically, we will
extend the concepts we discussed in section 2 from one
to two dimensions (for a more rigorous treatment, see
[AOP16] and [Van18]). Let vkbe a normalized eigen-
vector of k. As in the one-dimensional case, we are in-
terested in how the phase of vkchanges along a closed
path in the BZ. For any kand k, we dene the relative
phase between vkand vkby
vkvk (18)
Analogous to equation (11), we dene the Berry connec-
tion Aby
Akvkkvk (19)
where kvkis the Jacobian matrix for vk. The Berry
phase is the integral of the Berry connection around a
closed contour in k-space:
Akk(20)
Using Stokes’s theorem (i.e., Green’s theorem in two di-
mensions), we can write the Berry phase as
k (21)
where is the region in the plane enclosed by , and
k(22)
is called the Berry potential. Evaluating the derivatives in
(22) and using the equivalence of mixed partials, we can
also write the Berry potential as
kIm vkvk(23)
The topological quantity of interest is the integral of the
Berry connection around the boundary of the rst BZ. By
the Chern theorem (see, for example, [Nak90], as well as
the intuitive explanation below), this is an integer multiple
of . We then dene the Chern number of vby
BZ
Akk
BZ k (24)
which is an integer.
We will use a numerical method to compute the Chern
number [FHS05]. We rst choose a mesh size , and then
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 999
discretize the rst BZ (rhombus in Figure 5, top left) using
the points
k
v
v
The discretized grid comprises rhombus-shaped pla-
quettes , with corners k,k,k, and
k. Let
be the relative phase between vk
and vk. Then the discrete Berry phase around the
boundary of the BZ is dened by
(25)
where the subscript denotes the mesh size. Instead
of computing this, which involves the sum of gauge-
dependent relative phases, we will take the sum of the dis-
crete Berry phases around each plaquette [AOP16, Chapter
2.1.3]. The discrete Berry phase around is given by
(26)
where we take the principal value of the argument, i.e., take
. Taking the product of the Berry phases
for all plaquettes,
(27)
Consider an internal edge of the mesh connecting the
points k and k. This edge appears in exactly two
adjacent plaquettes, but in opposite orientations, which
implies that the sum on the RHS of (27) contains the rel-
ative phases
and
, each exactly once. Since the
Hermitian inner product is conjugate-symmetric,
, thus the relative phase contributions from all in-
ternal edges cancel. This implies that the exponents on the
RHS of equations (25) and (27) are the same, i.e.,
(28)
from which it follows that the discrete Berry phase and
the plaquette sum
are equal, modulo an inte-
ger multiple of .
Figure 6. Ribbon of Haldane lattice with armchair edges on
top and bottom, infinite in horizontal direction. Unit cell
enclosed in dotted lines.
Finally, we use the discrete Berry phase around the BZ
to dene the discrete Chern number
(29)
Since the BZ is a torus, opposite boundaries of the BZ are
identied. In particular, this means that opposite external
edges of the mesh are equivalent, e.g., the edge between
kand kis the same as that between kand k.
Since the two members of each pair of equivalent external
edges appear exactly once, but in opposite orientations, in
the sum on the RHS of (27), the relative phase contribu-
tions from all external edges cancel as well. This implies
that the exponent in (28) is 0, so that is an integer mul-
tiple of , and is an integer. We can think of the Chern
number as the limit of as , which provides an
intuitive explanation for why the Chern number is integer
valued.
Returning to the Haldane model, we compute the
Chern numbers of the two bands using the above dis-
cretization with . First, we consider the case when
. When , the Chern numbers of both bands
are 0, and when , the Chern numbers of the upper
and lower bands are and , respectively. When ,
the Chern number of the upper band is
and the Chern number of the lower band has the same
magnitude but opposite sign. The Chern number changes
from 0 to 1 when the band gap closes at
. This tran-
sition from the nontopological to the topological regime
is analogous to what occurs in the SSH model.
The fundamental difference between the nontopologi-
cal and topological states can be most easily seen in a lat-
tice which is nite in at least one dimension. As in the one-
dimensional case, imposition of a boundary will give rise
to edge modes. The simplest example is a ribbon lattice,
1000 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
Figure 7. Band structure for ribbon of Haldane lattice with
armchair edges. Top left: and (nontopological
state). Top right: and (topological state). Blue
and red lines are edge modes which connect the upper and
lower bands. Band gap closes at
. Bottom:
eigenvector abcorresponding to the two edge modes
from top right for .and in all cases.
which is nite in one direction and innite in the other.
There are many possible congurations for the edges of the
ribbon (see, for example, [LFR24]), and we note that the
edge mode behavior and properties depend on this choice.
We will use armchair edges for the top and bottom edges
of the ribbon, which are illustrated in Figure 6. The unit
cell of the ribbon (dotted square in Figure 6) comprises
sites of each type, for a total of sites.
Solutions to the linear system are standing waves of
the form ab, where aand b
represent the and sublattices of the system.
The wavenumber runs over the rst BZ, which is .
Substituting this ansatz into (14), we obtain the eigen-
value problem v v, where v ab ,
is the block matrix
(30)
is the tridiagonal matrix whose rows alternate
between and , and is the pen-
tadiagonal, skew-Hermitian matrix whose rows alternate
between and
.
When
, there is a band gap, but the system
does not possess any edge modes (Figure 7, top left). This
corresponds to a Chern number of 0 for both bands. As
is increased, the band gap closes at
, and then re-
opens for
. A topological transition occurs at
,
which is concurrent with the band gap closure. For
,
the Chern numbers of the two bands are . The resulting
topological state is characterized by the appearance of edge
modes (blue and red lines in Figure 7, top right) which
connect the upper and lower bands. A plot of the associ-
ated eigenvectors for these edge modes (Figure 7, bottom)
shows that they are indeed localized to the bottom and top
edges of the ribbon.
4. Photonic Floquet Topological Insulator
While the Haldane model has not been realized experi-
mentally, it has motivated further theoretical and exper-
imental work in topological photonics. In [RZP13],
Rechtsman et al. performed experiments with a photonic
crystal array of coupled, helical waveguides arranged in a
honeycomb lattice (as in the Haldane model). The helical
waveguides induce temporal modulation of the photonic
crystal, which breaks time-reversal symmetry and leads
to topological states. We start with the following lattice
model, which is a modication of (13) and can be derived
from Maxwell’s equations (see [AC17,AC22]):
n
mArmrnm
n
mArmrnm (31)
where the dot denotes differentiation with respect to ,
A, and the angle brackets indi-
cate that the sum is taken over nearest neighbors. We point
out that in the photonics setting, the paraxial direction
is the “time-like” variable. The parameters and repre-
sent the radius of the helical waveguide and its frequency
of rotation, respectively. The unit cells nnand posi-
tion vectors rnare the same as in the Haldane model, and
rmrnis the displacement vector between two unit cells.
Time-reversal symmetry is broken when , since mak-
ing the transformation ,nn
n
nand
taking complex conjugates takes in (31).
We look for solutions of the form
n
n
krn(32)
where the wave vector kranges over the rst BZ. Unlike the
Haldane model, the system (31) is nonautonomous, i.e.,
it depends on . As a consequence, the functions and
will also depend on . Substituting this ansatz into
(31) and simplifying, we obtain the linear, k-dependent,
nonautonomous ODE
uku (33)
where u,
k k
k
k
Ak(34)
and the are the nearest-neighbor coupling vectors from
Figure 4.
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1001
Figure 8. Floquet bands for (33) for (left) and
(right). Plot shows imaginary part of Floquet exponents over
the first BZ. For both cases, ,.
Since the matrix kis periodic in with period
, it follows from Floquet theory (see, for exam-
ple, [Chi06, Chapter 2.4]) that (33) has solutions of the
form uv, where vv, and is called
a Floquet exponent. To compute the Floquet exponents,
let be the fundamental matrix solution for the sys-
tem, so that uuis the unique solution to (33)
with initial condition u. Let , which is called
the monodromy matrix. The Floquet multipliers are the
unique eigenvalues of the monodromy matrix , and the
Floquet exponents, which are unique modulo , are
related to the Floquet multipliers by . Since the
two columns of the monodromy matrix are the unique
solutions to (33) at with initial conditions
and , respectively, it is straightforward to compute
the monodromy matrix numerically using a standard ODE
solver. Once we have computed , we can then calculate
the Floquet multipliers and exponents numerically.
Figure 8 shows the Floquet band structure of the system,
which is obtained by plotting the imaginary part of the Flo-
quet exponents as the wavenumber kvaries over the rst
BZ (the Floquet exponents are purely imaginary). When
, there is no band gap, which is expected since the
system reduces to the honeycomb model (13). In this case,
the matrix kis constant in , and the Floquet expo-
nents are the eigenvalues of . A band gap opens when
, which can be seen in the right panel of Figure 8.
This band gap is associated with a topological state, i.e.,
a nonzero Chern number. The Chern number for each
Floquet band can be computed using the corresponding
eigenvector of the monodromy matrix and the numerical
method from the previous section. When , the Chern
numbers of both bands are 0. For and frequency
, the Chern numbers of the two bands are and
. The edge modes that are a consequence of the non-
trivial Chern number are both predicted theoretically and
demonstrated experimentally in [RZP13].
5. Conclusions and Future Directions
In this article, our hope is to illustrate by way of examples
the rich mathematics underpinning the study of topologi-
cal photonics. By conning photons with topological pro-
tection, we can generate structures which have applications
in topological lasers, buffers, and other optical elements.
The signature of these states (e.g., edge modes) is the
presence of band gaps in the frequency versus wavenum-
ber dispersion relation. While there have been many ex-
perimental demonstrations of these concepts, fully three-
dimensional topological photonic bandgaps have not
been achieved to date, which is a very promising direc-
tion. Another exciting emerging eld where mathematical
modeling can play an important role is that of quantum
topological photonics. In a quantum setting, the appli-
cation of topological photonics to quantum optics could
help to generate robust quantum light sources and protect
photons from decoherence during photon propagation.
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[LFR24] J. L. Lado and J. Fern´andez-Rossier, Theory of edge
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Ross Parker Alejandro Aceves
Credits
Figures 1–8 are courtesy of Ross Parker.
Photo of Ross Parker is courtesy of Sarah Louise Parker.
Photo of Alejandro Aceves is courtesy of Adriana Aceves.
NEW FROM THE
The Structure of Pro-Lie
Groups
Karl H. Hofmann, Technische Universität, Darmstadt, Germany;
and Tulane University, New Orleans, US; and Sidney A. Morris,
La Trobe University, Bundoora: Australia, and Federation University
Australia, Ballarat, Australia
A topological group is said to be almost connected if the
quotient group of its connected components is compact.
This book exposes a Lie theory of almost connected pro-Lie
groups (and hence of almost connected locally compact
groups) and illuminates the variety of ways in which their
structure theory reduces to that of compact groups on the
one hand and of finite dimensional Lie groups on the
other. It is, therefore, a continuation of the authors’ mono-
graph on the structure of compact groups (1998, 2006,
2014, 2020, 2023) and is an invaluable tool for researchers
in topological groups, Lie theory, harmonic analysis, and
representation theory.
EMS Tracts in Mathematics, Volume 36; 2023; 840 pages;
Hardcover; ISBN: 978-3-98547-048-8; List US$129; AMS
members US$103.20; Order code EMSTM/36
Explore more titles at
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A publication of the European Mathematical Society (EMS).
Distributed within the Americas by
the American Mathematical Society.
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1003
Clusters and Weaves
Mikhail Gorsky and Jos´e Simental
1. Cluster Algebras
Cluster algebras were dened by Sergey Fomin and Andrei
Zelevinsky [9] in 2001, with the goal of providing a com-
binatorial framework for problems related to total positiv-
ity and canonical bases in Lie theory. Since then, various
connections have been developed between the theory of
cluster algebras and many areas of mathematics, includ-
ing representation theory and categorication, integrable
systems, mathematical physics, symplectic and algebraic
geometry, and higher Teichmüller theory, to name just a
few. The goal of this article is to introduce weaves as a tech-
nique to construct cluster algebra structures on the coor-
dinate rings of certain algebraic varieties that appear nat-
urally in the study of Legendrian links and in Lie theory,
and which generalize many of the motivating examples of
cluster algebras.
1.1. Cluster algebras. Cluster algebras are a special type
of commutative algebras that enjoy nice algebraic and
combinatorial properties. In contrast with the commu-
tative algebras we encounter in a rst algebra course, or
even in an algebraic geometry class, cluster algebras are
typically not explicitly given by generators and relations.
Rather, cluster algebras are dened to be subalgebras of a
eld of rational functions specied by a
set of generators , known as the set of cluster vari-
ables. While the set may be innite, it can be expressed
as a union of distinguished nite subsets of the same car-
dinality , which is the transcendence degree of the eld .
Each of these nite subsets is called a cluster. A key prop-
erty of cluster algebras is that, in order to know all of the
cluster variables, it is enough to know a single cluster and
a certain combinatorial datum together with a rule, called
mutation, for how to get all the other such pairs from it.
Mikhail Gorsky is a postdoctoral researcher in the Department of Mathematics
of the University of Vienna. His email address is mikhail.gorskii@univie
.ac.at. His work received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
(grant agreement No. 101001159).
Jos´e Simental is an investigador asociado C at the Instituto de Matem´aticas
of the Universidad Nacional Aut´onoma de M´exico. His email address is
simental@im.unam.mx. His work received support from CONAHCyT Project
CF-2023-G-106.
Communicated by Notices Associate Editor Han-Bom Moon.
For permission to reprint this article, please contact:
reprint-permission@ams.org.
DOI: https://doi.org/10.1090/noti3000
Thus, a cluster algebra is dened by two pieces of data:
1. A nite set , called an initial cluster,
consisting of algebraically independent elements that
form a free generating set of the eld .
2. A complementary piece of data given by a quiver (i.e.,
an oriented graph) without loops or oriented -
cycles, whose vertex set is identied with .
The pair is known as an initial seed, and it
determines a cluster algebra : from the initial seed
all other clusters are obtained as follows. For ,
dene a new element by the equation:
(1)
where by and we mean arrows in the quiver
. The condition that has no loops is imposed so that
does not appear on the right-hand side of (1), while
the condition that has no -cycles is imposed so that
no element divides the right-hand side of (1). Now we
create a new cluster by replacing in the element
by the element , that is, . Note
that this gives new clusters: . Each one of
these new clusters is known as a mutation of the original
cluster .
Now we want to mutate each of the clusters
, and so on. However, we also need to up-
date the quiver in a compatible way every time we mu-
tate. This is given by the procedure of quiver mutation.
Given a vertex of , the mutation is a new quiver
obtained from by the following three-step procedure:
(i) For each pair of arrows , insert a new arrow
.
(ii) Reverse all arrows incident with .
(iii) The previous two steps may have created -cycles. Re-
move a maximal collection of these.
With this in hand, we start with an initial seed
; we create new seeds ,
; from each one of these seeds we create more seeds,
and so on. As an exercise, the reader may prove that muta-
tion is involutive: . By denition, the clus-
ter algebra is the subalgebra of generated by all
clusters that can be obtained from by iterated mutations.
Note that, since mutation is involutive, the choice of an
initial seed is not important: the cluster algebra associated
with an arbitrary seed which can be obtained from by
nitely many mutations coincides with .
1004 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
Variations. Before giving examples of cluster algebras,
let us explain variations of the denition that are com-
monly used in the literature. First, given our initial cluster
, we may declare some elements of it to be unmutable, or
frozen. We are not allowed to mutate at these elements. As
a consequence, these elements will belong to every clus-
ter. To reect this, we declare the corresponding vertices
of to be frozen; we are not allowed to mutate at these. A
quiver with frozen vertices is also known as an ice quiver.
Second, if is a frozen variable, it is common to adjoin
the element
to the cluster algebra . In this case, we
say that the frozen variable is invertible. Below, we indi-
cate frozen variables by putting them in a box; invertible
frozen variables will, in addition, be colored blue.
Example 1.1. Consider the space of -matrices, whose
algebra of polynomial functions is .
One such polynomial function is the determinant
. Note that . Interpret-
ing this as an exchange relation of the form (1), we obtain
that is a cluster algebra with the follow-
ing initial seed:
Note that if we choose to invert the frozen variable , we
obtain the algebra of functions on the group .
1.2. The geometry of cluster algebras. From the deni-
tion, each cluster variable is a rational function in the ini-
tial cluster. The following is one of the key early results in
the theory of cluster algebras.
Theorem 1.2 (The Laurent phenomenon).Let be a clus-
ter algebra, and let be a cluster of . Then, each cluster
variable of can be expressed as a Laurent polynomial in the
variables of .
The Laurent phenomenon was proved by Fomin and
Zelevinsky in [9]. One way to interpret this result is that,
for every cluster , we have
where both algebras are considered inside our xed eld
. Note that, since , we actually have
, so if we adjoin to
the inverse to the element , we have:
(2)
The equation (2) has a neat algebro-geometric interpreta-
tion. To each nitely generated commutative algebra
with no nonzero nilpotent elements, algebraic geometry
associates a space , called an afne algebraic variety,
whose algebra of regular functions to is precisely . For
example, if then con-
sists of the points in where the polynomials
all vanish. As another example, if
then is . The condition that the algebra
is determines uniquely: if we have
an afne algebraic variety whose algebra of regular func-
tions is , then is isomorphic to .
Then, (2) tells us that the set of for which
all are nonzero, which is open in ,
is isomorphic to a torus . Thus, each cluster denes
an open torus , known as a cluster torus.
Example. We go back to Example 1.1:
. We have a cluster . Then,
Theorem 1.2 tells us that the set of invertible )-
matrices with is isomorphic to a torus
. Indeed, one can check that the following map is
an isomorphism:
The union of all the cluster tori is typically properly con-
tained in . The Laurent phenomenon also suggests
to consider the upper cluster algebra
is a cluster
where the intersection is taken inside our xed eld . By
Theorem 1.2, . In some cases, these two algebras
coincide, but this does not always happen. As we will see
next, from a geometric point of view it is more natural to
consider the algebra . Note that, by the same reasoning
as with , we have that every cluster denes a cluster
torus .
1.3. Constructing cluster structures. Given a commuta-
tive algebra , how to decide whether it has an (upper)
cluster algebra structure? Since cluster algebras are subal-
gebras of elds, a necessary condition on to have a cluster
algebra structure is that it is an integral domain. Beyond
that, this seems like a daunting task. We would need to:
1. Construct a set of cluster variables.
2. Partition into clusters.
3. Find a mutation rule to move between the clusters.
The Laurent phenomenon, Theorem 1.2, gives us a geomet-
ric way to move forward. If is to be an (upper) cluster
algebra, then must contain cluster tori. So a rst
task is to nd candidates for these cluster tori. After nding
these tori, we must nd a system of coordinates on them:
these will be the candidates for the cluster variables. When
this is done, the most difcult part is to nd a mutation
rule that allows us to mutate every coordinate of a cluster
torus which is not an invertible element of , for we can
dene the invertible elements to be frozen variables. This
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1005
is not an easy task. However, the following result tells us
that we do not necessarily have to nd all (possibly inn-
itely many!) cluster tori: we only need to nd one of them
and all of its possible one-step mutations.
Lemma 1.3 (The starsh lemma [1]).Let be a nitely
generated -algebra that is a normal domain with fraction eld
. Assume that we are given a seed with
such that:
1. The cluster consists of elements of .
2. The noninvertible (in ) elements of are pairwise co-
prime in .
3. For each noninvertible element , mutation at the
vertex replaces with , and are coprime
in .
Then, the upper cluster algebra associated to is contained
in .
Let us say a few words on the assumptions of the theo-
rem. The assumption that the algebra is normal is made
so as to be able to use an algebraic version of Hartogs’s the-
orem from complex analysis, that says that a function on
a normal algebraic variety that is regular outside of codi-
mension must be regular everywhere. This condition is
satised if, for example, is smooth. The coprime-
ness conditions on the variables are made so as to ensure
that both the cluster and its one-step mutations generate
the fraction eld of . As we will see below, the starsh
lemma is a powerful tool when trying to nd cluster alge-
bra structures on a given commutative algebra.
1.4. Why? Let us mention a few reasons why you would
be interested in constructing a cluster structure on a given
commutative algebra. As mentioned above, one of Fomin
and Zelevinsky’s original motivations was to provide an
algebraic framework for the study of total positivity. The
example is a good illustration of this. Let us
say that a -matrix is totally positive if all of its minors
are positive real numbers. Since
, to check
total positivity it is enough to verify that the elements of
the cluster given in Example 1.1 give positive numbers: we
do not have to check all minors. In general, the algebra of
functions on admits a cluster structure, with
each cluster containing elements. To check total posi-
tivity of a matrix it is sufcient to check that the elements
of a single cluster, i.e., functions, evaluate to positive
numbers, which is more efcient than verifying that all the
minors of the matrix are positive.
More generally, given a cluster algebra one can dene
the totally positive space to be the set of points
such that for every cluster variable
. It is a highly nontrivial result that it is enough to verify
that for all elements in a single cluster, [13]. Thus,
having a cluster structure not only allows us to dene a
notion of positivity, it also provides (many!) efcient tests
for it.
Another motivation of Fomin and Zelevisnky was the
study of (semi)canonical bases in Lie theory. Much later,
several families of bases for quite general (upper) cluster
algebras (resp. ) were proved to exist in [13, 17]; see
[18] for a survey. Each of these is a vector space basis of
(resp. ) including all cluster variables; the basis con-
structed in [13] also has positive structure constants.
In general, having a cluster structure on comes with a
wealth of combinatorial properties that allow to explicitly
study the geometry and topology of the variety .
2. Points in the Projective Line
In this section, we nd cluster structures on varieties de-
ned from congurations of points. Let denote the
complex projective space of dimension , that is, the set of
lines passing through the origin in . We will use homo-
geneous coordinates and denote by the line
passing through the origin and . Thus,
for any , i.e. we can simultane-
ously rescale the coordinates without changing the point
in . We will denote .
Now let and consider the space consisting
of -tuples of elements of ,,
satisfying:
(i) ,.
(ii) for all .
Thus, is a single point. Also,
so that , the space of nonzero
complex numbers. These are both algebraic varieties. Let
us show that is an algebraic variety for all .
Since we are required to have , we must
have with . Rescaling, we have that
for a unique . Now, since
and 1form a basis of , , where
. Since , we have
. Rescaling again, we may assume and, re-
naming , we have . Simi-
larly, is represented by an element in of the form
for a unique . Continu-
ing recursively, we obtain a family of polynomials dened
by , and
(3)
so that . The require-
ment that now becomes , since
. Thus,
1The choice of sign is made so that the matrix
has determinant .
1006 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
We see that is an afne algebraic variety: it is given by
the zeros of a polynomial equation. Note also that
implies that and, conversely, if we
obtain an element of by setting . We thus
obtain an equivalent description of the variety :
which realizes as an open subvariety of . As an
open subvariety of a ,is smooth of dimension
and
The recursion (3) is very similar to the rule (1). To make
this more explicit, let us rewrite (3) as
(4)
So that seems to be a clus-
ter in , with mutations given by
and
. Note that since is in-
vertible, we may consider it frozen. In order to verify that
this is the mutation rule for a cluster structure on ,
we need to verify that there exists a quiver providing
a mutation rule of the form (1) that coincides with (4).
From (4) we can see that the neighbors of must be
and , so that exists and is as follows:
Since is smooth, is normal. We are then in
position to apply the starsh lemma 1.3 to conclude that
the upper cluster algebra is contained in .
On the other hand, the algebra is generated by
and the invertible variable . These all
belong to . We then conclude that
and thus these are all equalities.
By virtue of it being a cluster variety, the variety
must contain a collection of open tori , one for each
cluster . Let us study these cluster tori, starting with
the initial cluster . By denition,
consists of the points such that
for all . Recalling that
and that ,
we obtain:
Let us represent this torus pictorially. First, we represent
the elements of graphically, as follows:
(5)
Here, two points in labeling adjacent regions separated
by a blue edge are required to be different. Since in the
torus we are forced to have for every
, we can represent this torus in a similar way:
(6)
so that now every point is separated
from by a blue edge. Thus, this diagram represents
the points such that for all
, which is precisely the cluster torus .
Let us now examine the mutation
. In the torus , we have
. As before, the non-
vanishing conditions on ’s mean that
, . What does the condition mean?
By denition, represents the line passing through the
origin and a point . The points are dened re-
cursively by ,and .
Thus, , and if and only
if the elements are linearly independent,
that is, if and only if . Thus, is given by
and can be pictorially represented as follows:
(7)
In fact, every cluster torus in can be pictorially
represented by a diagram similar to those in (6) and (7).
Thus, any cluster torus is dened by requiring some pairs
of elements among to be distinct.
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1007
Conversely, every diagram like those in (6) and (7) (that is,
a rooted binary tree with leaves) denes a cluster torus
in . See Figure 1.
Figure 1. Graphical representations of the cluster tori inside
.
In this context, seed mutation has a clear combinatorial
meaning. It corresponds to a local move of the following
form:
(8)
The varieties are some of the simplest varieties
whose coordinate algebra admits a cluster algebra struc-
ture. Moreover, in this case we have a complete under-
standing of the cluster tori, and there are only nitely many
of them. The number of cluster tori in is given by the
Catalan number - an ubiquitous sequence in combi-
natorics. In the next section, we will generalize the variety
to braid varieties. This is a rather large class of vari-
eties that appear naturally in many different mathematical
contexts: geometric representations of Weyl groups, Leg-
endrian link invariants, and homological algebra in a Lie
theoretic context, to give just a few examples.
3. Braid Varieties
We generalize every ingredient appearing in the denition
of .
The ag variety. The projective space naturally gener-
alizes to ag varieties. Let us x a number . A complete
ag in is a sequence of vector subspaces:
with the property that for all . One
way to think about a ag is as an invertible -matrix: if
is an invertible -matrix, its columns
are linearly independent, so the space
is -dimensional and this gives rise to a ag:
In fact, any ag arises from a matrix in this way. Different
matrices may give rise to the same ag. Multiplying the
columns by arbitrary nonzero scalars does not change the
subspaces
. Also, adding scalar multiples of the rst
columns to the -th column does not change the space
. Recalling that elementary column operations can be
expressed by matrix multiplication on the right, we see that
for every invertible upper triangular matrix . In fact, if
we let be the group of invertible -matrices
and its subgroup consisting of upper triangular
matrices, the set of all ags in can be identied with
The space is known as the ag variety, and it is a
projective algebraic variety. It is not afne, but can be glued
from afne varieties in a similar way to how is glued
from many copies of .
To see how generalizes the projective space ,
note that consists of all chains of subspaces
. Since the rst (that is, ) and last () spaces
are always xed, we see that can be identied with the
space of all lines in . This is precisely , i.e., .
Flag varieties appear naturally in representation theory:
the ag variety can be used to geometrically construct
all irreducible representations of the symmetric group .
They also appear naturally in Schubert calculus, which is
the study of incidence problems of (translations of) linear
spaces, and can be interpreted as a study of the cohomol-
ogy of and related varieties.
(Anti)standard ags. Having generalized the projec-
tive space , we now nd counterparts of the points
. Let
be the standard basis.
The standard ag
is the ag dened by
and the antistandard ag
is the ag dened by
When , we have that
, and
. Thus, we see that in this case the standard ag
is precisely , while the antistandard ag is .
Relative position of ags. So far, we have generalized:
It is tempting to generalize by considering all tuples
of consecutively distinct ags that start with
and end
with
. However, this variety would be too large to be
well-behaved, and we will instead study pieces of its certain
decomposition. For example, if , the variety is
, that is not an afne algebraic variety when
.
1008 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
Instead, we observe that if are dif-
ferent ags, then they are forced to be different in the rst
subspace: as and , we have
. Generalizing this, we will specify where two ags
are different.
Denition 3.1. Let be ags, and let
. We say that are in position if ,
but for all .
Let us classify the ags that are in position with respect
to
. Such a ag must satisfy .
Since , there must exist such that
. Since we are requiring
, we must have . Rescaling, we may assume
and thus for a unique .
In fact, with a similar reasoning it is possible to param-
etrize all ags that are in position with respect to a given
ag. In order to do this, we dene a family of matrices
depending on a parameter :
where the nonidentity block is in the -th and -st rows
and columns.2
Lemma 3.2. Let be an invertible matrix. The set
of all ags which are in position with respect to the ag
is
given by
We warn the reader that, if
, it does not follow
that
for all . Rather, what is true
is that for there exists a unique such that
. In other words, the parametrization of
the ags in position with respect to
given by Lemma
3.2 depends on the chosen matrix , and not just on the
ag
.
Braid varieties. We are ready to generalize the varieties
. Instead of one natural number , the variety
will depend on an -tuple, or a word,
that species how the ags change.
Denition 3.3. Let . The
braid variety is the variety of -tuples of ags in
,
, satisfying:
(1)
;
.
(2) For ,
and
are in position .
2The sign is there just so that , which makes some formulas
nicer.
Note that, if and
, then ,
so the varieties do indeed generalize from Sec-
tion 2. We have the following two facts.
(i) If and
, then the varieties and are isomor-
phic provided that .
(ii) If and
, then the varieties and are
isomorphic.
These isomorphisms mimick the relations in the positive
braid monoid. This is the reason that is called a braid va-
riety. We refer the reader to [2,4] for details on this. Just as
from Section 2, is a smooth, afne algebraic vari-
ety. We verify that it is indeed an afne algebraic variety by
showing that it is dened by the vanishing of several poly-
nomials. For this, given an element
, we
choose particular matrices in representing the ags
. Since
, we have that
, where
is the -identity matrix. Since
is in posi-
tion with respect to
, thanks to Lemma 3.2 there exists
a unique such that
Since
is in position with respect to
, there exists a
unique such that
Continuing like this, we obtain a unique element
such that
for each . Now, the condition that
imposes conditions on the matrix
, and we obtain that is iso-
morphic to the collection of tuples such
that if
Note that each entry of the matrix product
is a polynomial in . Thus,
is given by the vanishing of several polynomials, and
so it is indeed an afne algebraic variety. The fact that
is smooth is more complicated this time; in general,
cannot be identied with an open set in . For smooth-
ness of we refer the reader to [7].
Braid varieties generalize a wide class of algebraic va-
rieties appearing in Lie theory, such as positroid and
Richardson varieties [2, 10, 14, 15]. They also have incar-
nations in the study of Legendrian link invariants, such as
the augmentation variety or the moduli space of microlo-
cal rank 1 sheaves associated with certain Legendrian links
[5,12, 20].
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1009
The fact that the varieties admit cluster structures
generalizes as follows.
Theorem 3.4 ([4]).For any such that
the braid variety is nonempty, the coordinate ring
admits a cluster algebra structure.
Remark 3.5.Let us elaborate on the condition . If
the word is not “complicated enough,” then there will be
no space to get from
to
according to the word ,
and thus will be empty. For example, if
and , then
Since we cannot have simultaneously and
, there are no triples such that
the ag associated to is the antistandard
ag, and so . One way to x this is to relax
the condition on the last ag being
, and require that it
is a ag all whose subspaces are given by the vanishing of
some coordinates, and that is as transversal to
as the
word allows. See [4] for details. In what follows, we will
use this more general denition of a braid variety so that
is always nonempty. These varieties still admit a clus-
ter structure, and the construction we will explain applies
to this case as well, with minimal modications.
The construction of an initial seed for the cluster alge-
bra on is considerably more involved than its coun-
terpart for described in Section 2. However, a great
deal of cluster tori in are still readily accessible using
a graphical calculus similar to the one used in Section 2.
The calculus becomes more complicated because we need
to take into account the isomorphisms (i) and (ii). The
key object is a weave, introduced in [6], that we discuss in
detail in the next section.
4. Weaves
In order to motivate weaves, let us take a word
, and picture an element of the braid variety
as follows:
Here, the vertical lines are colored with colors .
The ag in the leftmost region is
, and the ag in the
rightmost one is
. If two regions are separated by a
line of color , then the corresponding ags are in position
, that is, they differ precisely in the -th subspace.
Now suppose we have the following conguration:
where both vertical lines are of the same color . Here, the
ags
and
either differ in precisely the -th subspace,
or they are equal. It is natural to picture these two possi-
bilities using the following diagrams, that we refer to as a
trivalent vertex and a cup, respectively.
Note that the condition
is open, and the comple-
ment
is closed. Demazure weaves, which we will
now dene, are designed to characterize open subsets of a
braid variety, and so they only use trivalent vertices, never
cups. A Demazure weave on is a graph in
a rectangle , such that edges of are colored
and vertices are of four types:
1. Univalent vertices, located only on the top and bot-
tom sides of . The edges adjacent to the vertices on
the top side spell precisely , and the word spelt by the
colors of the edges adjacent to the vertices on the bot-
tom side is so that no more trivalent vertices can be
drawn starting from it, even after applying tetra- and
hexavalent vertices described below.
2. Trivalent vertices, as pictured in Figure 2.
3. Hexavalent vertices, as pictured in Figure 2.
4. Tetravalent vertices, as pictured in Figure 2.
Figure 2. The types of vertices in the interior of the rectangle
of the definition of a weave. Note that the edges adjacent to
a trivalent vertex all have the same color; the edges adjacent
to a tetravalent vertex are of two distant colors; and the edges
adjacent to an hexavalent vertex are of neighboring colors.
See Figure 3 for an example. Given a Demazure weave
on , we consider the space of all congurations of
ags
, one per connected component of , satisfy-
ing the following conditions.
The ag labeling the region bordering the left side of
the rectangle is
.
The ag labeling the region bordering the right side of
the rectangle is
.3
3More precisely, this region should be labeled by the ag whose subspaces are
given by vanishing of coordinates and that is as transversal to
as allows;
see Remark 3.5.
1010 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
If two regions are separated by an edge of color , then
the corresponding ags are in position .
Note that, by denition, for any element of , the
ags labeling the regions bordering the top side of the rec-
tangle give an element of . In fact, these ags prop-
agate, using isomorphisms (i) and (ii), to ll the entire
rectangle in a unique way, i.e., there is an embedding
. Not every collection of ags forming an ele-
ment in will propagate consistently through the entire
diagram to give an element of : the trivalent vertices
impose open conditions on the element of . See Figure
3.
Note that, when , we recover precisely the dia-
grams in Section 2 that give cluster tori in . More
generally, the elements of that propagate to give an
element of form a cluster torus . We remark that
different weaves may determine the same tori: the map
from Demazure weaves to cluster tori is not injective.
Figure 3. A weave on and its flag moduli.
All the flags are determined by and . The
shortest path from to is indicated in orange; the one
from to is indicated in purple.
To get a seed from a cluster torus , we need to specify:
(1) A collection of functions in whose nonvanish-
ing locus is precisely the torus .
(2) An ice quiver, , specifying the mutation rule.
Both constructions are quite technical. We explain
some ideas behind them and refer the reader to [4] for de-
tails. We take the weave in Figure 3 as a motivating exam-
ple. Intuitively, a collection of ags
belongs to if and only if each ag is as
far from the standard ag as possible. This can be made
precise using matrix minors. For example, the ag
is not equal to the standard ag if and only
if the entry of the matrix is nonzero.
Note that , where is a transposition
in the symmetric group. Then, we can rephrase by say-
ing that is not equal to the standard ag if and only
if the minor is nonzero.4Sim-
ilarly, the ag is as far as possible from if the minor
is nonzero. Note
that where , and that the path
joining the regions labeled with and in Figure 3
that crosses the minimal number of edges in the weave
crosses precisely an edge of color followed by an edge of
color . By looking at ags immediately to the right of each
trivalent vertex, we are led to also consider the following
minors:
The torus is given by the nonvanishing of .
However, note that , so that and
are not coprime. Nevertheless, once we factor into ir-
reducibles, there will be exactly one irreducible factor that
does not appear in for . These irreducible factors
are the cluster variables. For a general weave, the cluster
variables will be irreducible factors of certain minors of
partial products in , that measure how far
from each other the corresponding ags are.
To nd the quiver we need to introduce one more
ingredient: Lusztig cycles. Each trivalent vertex has an as-
sociated Lusztig cycle , that is an integer-valued function
on the edges of the weave . This function is completely
determined by requiring that it is in every edge that is
not below the trivalent vertex ; that it is on the southern
edge of ; and that below it satises a tropical version of
Lusztig’s relations between factorization coordinates [16].
We refer the reader to [4] for the most general case of these
rules. When all the values of the Lusztig cycle are equal
to or , the Lusztig cycle propagates downward through
the vertices of according to Figure 4. Figure 5 shows the
weave of Figure 3 with all its Lusztig cycles drawn.
Once the Lusztig cycles are computed, the quiver is
obtained using an intersection pairing between these. In
the setting of [6], where a weave is a combinatorial shadow
of a surface in , Lusztig cycles represent homology cycles
in this surface and the intersection pairing corrresponds
to topological intersection. The intersection pairing can
be computed combinatorially, by adding contributions at
each - and -valent vertex. We again refer the reader to
[4] for the most general case of this. In case all Lusztig cy-
cles have weight or , the contributions are as in Figure 6.
Finally, frozen variables correspond to those trivalent ver-
tices whose Lusztig cycle has a nonzero value at the south-
ern boundary of the weave. The reader is invited to verify
that the quiver associated to the weave in Figures 3 and 5
4Recall that if is an -matrix and are sets of the same size,
then the minor is the determinant of the square submatrix obtained by
considering only the rows indexed by and the columns indexed by of .
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1011
is as follows, where circles indicate mutable vertices and
squares indicate frozen ones.
A more precise version of Theorem 3.4 is:
Theorem 4.1. For any Demazure weave on a word , the
functions on obtained using irreducible factors of ma-
trix minors, together with the quiver obtained using Lusztig
cycles, give a cluster structure on . Moreover, any two
weaves on give rise to the same cluster structure on .
Figure 4. Lusztig propagation rules when all values are or .
An edge colored purple means that the value of the Lusztig
cycle at that edge is ; otherwise the value is .
Figure 5. The weave from Figure 3 with its Lusztig cycles
indicated. There is one Lusztig cycle per trivalent vertex, and
each Lusztig cycle only takes values or .
Special cases of braid varieties include double Bruhat
cells, double Bott-Samelson varieties, open positroid and
Richardson varieties. The case of double Bruhat cells is one
Figure 6. Intersection pairing between Lusztig cycles at - and
-valent vertices.
of the motivating examples of cluster algebras [1], while
the case of positroid varieties is studied by P. Galashin
and T. Lam in [10], and cluster structures on double Bott-
Samelson varieties were studied in detail by D. Weng and L.
Shen in [19]. That open Richardson varieties admit clus-
ter structures was conjectured by B. Leclerc in [15], and
proved using braid varieties and weaves in [4]. Braid va-
rieties provide a unifying context to all the cluster struc-
tures mentioned above. Moreover, even in the cases when
a cluster structure was previously known, weaves usually
give combinatorial access to more clusters than earlier con-
structions; see, e.g., [3].
Finally, we mention some exciting recent developments
related to braid varieties: independent constructions of
cluster structures using a 3D analogue of Postnikov’s
plabic graphs [11] and, for some braids and similar vari-
eties, entirely via symplecto-geometric means [5]; a con-
struction of cluster structures for braid varieties of other
Lie types [4]; a construction of cluster Poisson structures
on braid varieties and the proof of cluster duality for these,
including the existence of -bases [4, 8, 13]; and connec-
tions to Khovanov-Rozansky homology [2].
ACKNOWLEDGMENTS. We are grateful to the ref-
erees, O. Chugreeva, J. de Loera Ch´avez, and R.
Manr´ıquez for their thoughtful suggestions.
References
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links and braid varieties, arXiv:2105.13948.
[3] M. Castronovo, M. Gorsky, J. Simental, and D. Speyer,
Cluster deep loci and mirror symmetry, arXiv:2402.16970.
[4] R. Casals, E. Gorsky, M. Gorsky, I. Le, J. Simental, and L.
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[5] Roger Casals and Daping Weng, Microlocal theory of
Legendrian links and cluster algebras, Geom. Topol. 28
(2024), no. 2, 901–1000, DOI 10.2140/gt.2024.28.901.
MR4718130
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[7] Laura Escobar, Brick manifolds and toric varieties of brick
polytopes, Electron. J. Combin. 23 (2016), no. 2, Paper 2.25,
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of local systems and higher Teichmüller theory, Publ.
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DOI 10.1090/S0894-0347-01-00385-X. MR1887642
[10] Pavel Galashin and Thomas Lam, Positroid varieties and
cluster algebras (English, with English and French sum-
maries), Ann. Sci. ´
Ec. Norm. Sup´er. (4) 56 (2023), no. 3,
859–884, DOI 10.24033/asens.2545. MR4650160
[11] P. Galashin, T. Lam, M. Sherman–Bennett, and D.
Speyer, Braid variety cluster structures, I: 3D plabic graphs,
arXiv:2210.04778.
[12] Honghao Gao, Linhui Shen, and Daping Weng, Aug-
mentations, Fillings, and Clusters, Geom. Funct. Anal. 34
(2024), no. 3, 798–867, DOI 10.1007/s00039-024-00673-
y. MR4743511
[13] Mark Gross, Paul Hacking, Sean Keel, and Maxim Kont-
sevich, Canonical bases for cluster algebras, J. Amer. Math.
Soc. 31 (2018), no. 2, 497–608, DOI 10.1090/jams/890.
MR3758151
[14] Allen Knutson, Thomas Lam, and David E.
Speyer, Positroid varieties: juggling and geometry, Com-
pos. Math. 149 (2013), no. 10, 1710–1752, DOI
10.1112/S0010437X13007240. MR3123307
[15] B. Leclerc, Cluster structures on strata of ag va-
rieties, Adv. Math. 300 (2016), 190–228, DOI
10.1016/j.aim.2016.03.018. MR3534832
[16] G. Lusztig, Total positivity in reductive groups, Lie theory
and geometry, Progr. Math., vol. 123, Birkhäuser Boston,
Boston, MA, 1994, pp. 531–568, DOI 10.1007/978-1-4612-
0261-5_20. MR1327548
[17] Fan Qin, Bases for upper cluster algebras and tropical points,
J. Eur. Math. Soc. (JEMS) 26 (2024), no. 4, 1255–1312,
DOI 10.4171/jems/1308. MR4721032
[18] Fan Qin, Cluster algebras and their bases, Representa-
tions of algebras and related structures, EMS Ser. Congr.
Rep., EMS Press, Berlin, [2023] ©2023, pp. 335–369.
MR4693645
[19] Linhui Shen and Daping Weng, Cluster structures on dou-
ble Bott-Samelson cells, Forum Math. Sigma 9(2021), Paper
No. e66, 89, DOI 10.1017/fms.2021.59. MR4321011
[20] Vivek Shende, David Treumann, and Eric Zaslow, Leg-
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5. MR3608288
Mikhail Gorsky Jos ´e Simental
Credits
All gures are courtesy of the authors.
Photo of Mikhail Gorsky is courtesy of Eugene Gorsky.
Photo of Jos´e Simental is courtesy of Adolfo Arroyo-Rabasa.
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Multiscale Modeling
of Viscoelastic Fluids
Paula A. Vasquez
1. Introduction
Although you may be unfamiliar with the term viscoelas-
ticity, viscoelastic uids (VEFs) are ubiquitous in our daily
lives. The shampoo you used this morning, the salad dress-
ing you ate yesterday, and about every biological uid in
your body; all of them are VEFs. As their name implies,
viscoelastic materials exhibit both viscous and elastic be-
haviors. Viscous behavior is related to a uid’s resistance
to ow. The higher the viscosity, the more the uid resists
motion. Honey, for example, has a high viscosity, while
water has a low viscosity. Conversely, elasticity pertains to
reversible deformations, such as the snapback of a rubber
band.
Under applied deformations, VEFs display an instanta-
neous pure elastic response, followed by a time-dependent
mechanical response and energy dissipation, characteristic
of viscous liquids. The differences from one VEF to an-
other comes from the relative timescales of these elastic
and viscous responses. This “duality” in the behavior of
VEFs plays a critical role in their applications. For exam-
ple, paints can be thin enough to be applied with a brush,
yet thick enough to stay on the wall. And although may-
onnaise appears semisolid in a jar, it can be easily spread
on bread.
The study of VEFs falls within the eld of rheology. Rhe-
ology investigates how materials deform or ow during
and after a load is applied. Measuring rheological proper-
ties is pertinent to all materials, from liquids such as water,
polymers, and protein solutions to semisolids such as gels
and creams and to solid polymers such as resins. Within
rheology, at the most basic level, uids can be divided
into Newtonian and non-Newtonian according to their re-
sponse to ow. From a modeling point of view, all Newto-
nian uids are described by the well-known Navier–Stokes
equations [Bat99]. This set of equations works well on sys-
tems in which the ow does not alter the dynamics of indi-
Paula A. Vasquez is an associate professor of mathematics at the University of
South Carolina. Her email address is paula@math.sc.edu.
Communicated by Notices Associate Editor Reza Malek-Madani.
For permission to reprint this article, please contact:
reprint-permission@ams.org.
DOI: https://doi.org/10.1090/noti3001
vidual constituents. In contrast, for non-Newtonian uids,
applied elds can alter the local microstructure. Hence,
there is no single set of equations that can comprehen-
sively describe all non-Newtonian materials. What these
uids have in common is that their properties emerge from
the collective behavior of many microstructural compo-
nents.
The eld of VEFs offers many opportunities for math-
ematical exploration, particularly in the development of
new constitutive models and numerical techniques. The
absence of a unied equation for VEFs and their wide range
of applications in industrial and biological processes has
resulted in extensive research activity in this eld. How-
ever, there are still many issues that need to be addressed.
On the numerical side, key challenges involve the loss of
accuracy and convergence of numerical methods due to
nonlinearities in the constitutive equations. Another chal-
lenge is the change of type of the partial differential equa-
tions (PDEs), which sometimes leads to a loss of well-
posedness. Additionally, in certain cases, uid ows re-
sult in rapid changes of the solutions in specic regions,
making it necessary to implement adaptive mesh tech-
niques. For reviews on this area, the reader is referred to
[OP02,Keu04,AOP21].
Fractional calculus is another rapidly growing area in
the eld of VEFs. It provides a more detailed understand-
ing of the memory effect through the use of fractional
derivatives [Mai22]. The behavior of VEFs falls on a spec-
trum between that of fully elastic solids and fully viscous
uids. Fractional models provide a unied framework
to understand this entire range by varying the order of
the fractional derivative. This allows fractional models to
achieve comparable accuracy to classical models, but with
fewer parameters, making data tting less complex.
In addition, the eld of VEFs offers numerous oppor-
tunities for the mathematical analysis of existing consti-
tutive equations. To understand the dynamics of VEFs,
it is crucial to evaluate the stability of these systems and
understand how the interplay between viscous and elastic
properties inuences their behavior. However, there is still
much to be explored regarding how different ow condi-
tions affect the solutions of these models. Only a limited
subset of these equations has been thoroughly studied in
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1015
Figure 1. VEFs exist at different scales because their
macroscopic responses depend on the dynamics of their
microstructural components.
terms of their existence and uniqueness [RT21]. Investi-
gating the connections between VEFs constitutive models
and broader concepts in dynamical systems could uncover
previously unexplored complexities.
Perturbation analysis is another valuable eld com-
monly used to gain insight and address unresolved issues
pertaining to VEFs. Singular perturbation methods are es-
pecially well-suited for examining VEFs because they can
effectively manage the wide range of temporal and spatial
scales inherent in the dynamics of these materials. By an-
alyzing extreme cases of high and low elasticity, we can
gain valuable insights into the underlying dynamics, cir-
cumventing the need to solve complex systems of equa-
tions.
This review aims to introduce the reader to basic con-
cepts related to VEFs, focusing mainly on constitutive mod-
eling. We note that each class of VEFs has unique proper-
ties, posing different mathematical challenges. Providing
a complete overview of every class of VEFs and their math-
ematical representations is beyond the scope of this intro-
ductory review. Instead, we discuss some commonalities
and focus on a specic class of VEFs. When suitable, the
reader will be referred to more in-depth reviews and text-
books.
2. Multiscale Modeling of VEFs
From a mathematical perspective, modeling challenges of
VEFs arise from the need to describe the dynamics resulting
from the complex interactions among microscopic con-
stituents and how such interactions dictate material prop-
erties and functions at the macroscale (Fig. 1). In short,
there is a need for continuous communication across mul-
tiple scales of time and space. Ideally, a constitutive equa-
tion for VEFs should provide sufcient insight into the
microscopic changes that lead to a given macroscopic re-
sponse.
This coupling of micro and macro scales is particularly
crucial for some materials. For example, one might like
to study biochemical changes in the mucus network and
their impact on the mucociliary clearance process, or per-
haps investigate how changes in the viscoelastic properties
of the cytoplasm affect cellular function. In these situa-
tions, modeling platforms that consider dynamics at the
microscale are better suited.
To model the microstructure explicitly, we consider a
VEF to be composed of two major components: a solvent
and some dynamic network at the microscale. We consider
the system to be a continuum. This means it can be de-
scribed by classical mechanics and its state under a defor-
mation is determined by the fundamental hydrodynamic
elds of density, momenta, and energy, [Bat99,BCAH87],
Conservation of mass:
Conservation of momentum:
Conservation of energy:
Here represents the material density, the velocity eld,
the Cauchy stress tensor, body forces, the internal
energy per unit mass, the thermal conductivity and the
temperature. And, the material derivative is dened as,
If the ow is incompressible (constant), isothermal
(constant), and in the absence of body forces (),
the conservation equations become,
(1a)
(1b)
The Cauchy stress tensor can be decomposed into
isotropic and extra stress components,
where is the identity tensor. At equilibrium, the isotropic
component is the thermodynamic pressure, , while the
extra stress tensor, , vanishes [Gra18]. Under these con-
siderations, the resulting conservation equations are,
(2a)
(2b)
For a viscous or Newtonian uid, the stress is directly
proportional to the strain rate, , so that
. In this case is known as a viscous stress and
is the uid’s viscosity. We note that in some references,
the strain rate tensor is dened as
, and .
Here we follow the notation proposed in [BCAH87] and
1016 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
use . The resulting conservation of momentum equation
is known as the incompressible Navier–Stokes equation,
(3)
Inspection of (3) shows that differences between materials
will only affect the value of , but not the functional form
of the governing equations. Then, by varying the viscos-
ity constant, the same numerical algorithms can describe
different materials, as long as they behave as Newtonian
uids.
Coming back to (2a)–(2b), in the case of VEFs there
is a need to introduce an extra term in stress, , which
accounts for the contributions from the microstructure,
. In our discussion of (3) we noted that for New-
tonian uids there is a single, general constitutive equa-
tion capable of describing different Newtonian materials,
so that (3) sufces to describe a large class of uids. In
contrast, there is not a single all-encompassing constitu-
tive equation for viscoelastic materials. The details of ,
particularly the constitutive equation that denes it, will
differ depending on the material or the specic ow con-
ditions. Thus, the formulation of constitutive equations
describing different viscoelastic materials is a prolic area
of research [BCAH87,Lar99].
Broadly speaking, we understand relatively well the con-
nection between VEFs and external ow elds: the dynam-
ics of the underlying microstructure directly control the be-
havior of a VEF under these deformation elds. Therefore,
when dealing with constitutive equations for VEFs, chal-
lenges arise at three main levels:
At the level of their derivation, since different ma-
terials require different mathematical descriptions
of their microstructure;
At the level of their numerical simulation, since the
resulting equations are of different class and each
has its unique numerical challenges;
At the level of their mathematical treatment, since
researchers have only established the existence
and uniqueness of the solution of VEF constitu-
tive equations in only a few cases (See for example
[Ren85,LM00, RT21]).
The mathematical representation of the microstructure,
the length scale at which this representation will be ren-
dered, and the numerical methods used to solve the result-
ing constitutive equations will all depend on the particu-
lars of the uid and the chosen mathematical description.
Accordingly, many studies have tackled one aspect or an-
other and even combinations of them. However, a full
description is out of the scope of this review. Moving for-
ward, we focus our attention on one type of models which
originates from kinetic theory and a specic category of
VEFs: polymeric uids.
2.1. Coarse-grained representation of the microstruc-
ture. To better understand how we can develop mathe-
matical models of VEFs based on representations of the
microstructure, here we will focus on polymeric uids.
In these uids, polymer chains compose the microstruc-
ture. The underlying dynamics driving the material’s re-
sponse to deformation result from both individual chain
congurations and interchain dynamics. Among others,
these include coiling and uncoiling processes, hindered
motion due to physical entanglements, hydrodynamic ef-
fects caused by the presence of other molecules, and in
some cases, physical cross-linking between the polymer
chains.
Within the context of kinetic models and polymeric u-
ids, various approaches have developed to describe the
coupling between macroscopic responses and microstruc-
ture dynamics. One family of models is the so-called
bead-spring models. These models use molecular coarse-
graining to describe the behavior of polymer chains repre-
sented as beads connected by massless springs. These mod-
els are based on molecular physics by considering the in-
teraction between individual polymer chains and the sur-
rounding uid. The extra stress arising from the polymer
molecules, i.e., the microstructure, depends strongly on
their spatial conguration, the most important features be-
ing their orientation and their extension.
The simplest of these models considers only two beads
and it is known as the elastic dumbbell model. The cong-
uration of each dumbbell is fully specied by its stretch-
ing and orientation. Although it is widely recognized
that a dumbbell is too simple to be able to describe any
complicated dynamics in polymeric molecules, it is also
well known that stretching and orientation alone sufce
to give a qualitative description of steady-state rheologi-
cal properties and ows with slow characteristic timescales
[BCAH87]. Accordingly, these models had been exten-
sively used to develop “an elementary but broad under-
standing of the relation between macro-molecular mo-
tions and rheological phenomena” [BCAH87].
2.2. Dumbbell models. To model a given microstructure
this class of models uses a coarse-grained approximation
at the mesoscale consisting of noninteracting elastic dumb-
bells. The VEF system will be described by the dynamics
of these dumbbells in a solvent. The solvent is assumed to
be an incompressible Newtonian uid of viscosity . The
conguration of the dumbbell is described by the end-to-
end connector vector and the center-of-mass
vector
.
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1017
To keep it short, this review only discusses homoge-
neous ows. This means that the velocity gradient is as-
sumed to be the same everywhere. We refer the reader to
[BAB91] for a discussion on how to introduce spatial vari-
ations into the dumbbell equations.
To capture the dynamics of each dumbbell, we start
from a balance of forces at the inertia-less limit. That is
we use Newton’s second law, , but assume the
mass is negligible [BCAH87,Ött96],
Drag force
Spring force
Thermal noise (4)
Here is the drag coefcient, is the uid velocity, de-
notes a functional form of the spring force, and is the
thermal energy. denotes a Wiener process where each
component of the vector is a random number drawn
from a normal distribution with zero mean and variance
equal to 1.
Equation (4) shows that changes in the conguration of
the dumbbell, namely its orientation and extension, are
the result of three competing forces. The drag force, im-
posed by the solvent molecules onto the beads, has the
tendency of aligning the dumbbells with the macroscopic
ow. The thermal or Brownian force tends to randomize
their conguration. And, the spring force tends to bring
both beads together, which counteracts the stretching ef-
fects of the drag and thermal forces.
Rearranging (4), gives the following two stochastic dif-
ferential equations (SDEs) describing the evolution of
each bead’s position,
(5a)
(5b)
Since the spatial location of the dumbbells is not relevant
under the homogeneous ow assumption, the only vari-
able of interest is the end-to-end vector, . By subtracting
(5a) from (5b), we obtain the SDE describing the evolu-
tion of ,
(6)
The only remaining task is to establish the form of the
spring law, denoted as , where .
2.2.1. Hookean dumbbells. For Hookean dumbbells
, where is the spring constant. With this,
(6) becomes,
(7)
For convenience, we will make these equations non-
dimensional. To couple these equations with the conser-
vation equations, we will scale the time using the macro-
scopic timescale, . In addition, we introduce the follow-
ing characteristic microscopic time and length scales, re-
spectively,
The nondimensional variables are then given by,
Scaling (7) and dropping the tildes gives,
(8)
where the nondimensional group is the so-
called Deborah number. Since it is the ratio of micro-to-
macro characteristic timescales, this dimensionless group
compares how long it takes for a material to adapt to defor-
mations relative to the process’s characteristic timescale.
Note that, we could have chosen a different macro-
scopic timescale, namely, , where and are, re-
spectively, characteristic macroscopic length and velocity.
Here, the resulting nondimensional group, , is
called the Weissenberg number, and it represents the ratio of
elastic to viscous forces. For many applications
and it is very common to confuse these two nondimen-
sional groups. To better understand the difference be-
tween and , we recommend reading [Poo12].
2.2.2. FENE dumbbells. The linear spring law used in the
Hookean dumbbell model is unphysical, since it allows
the end-to-end vector, , to stretch without limit. One
modication of this law uses nitely extensible nonlinear elas-
tic (FENE) springs laws. FENE-type models are derived by
introducing Warner’s force law [BCAH87],
(9)
where and is the maximum allowed
extension of the dumbbell.
1018 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
Introducing this spring law into (6) and nondimension-
alizing as before gives
(10)
where
.
Finally, we note that, in this dumbbell formulation, we
only considered the most basic dynamics. Still, dumb-
bell models can integrate other physical assumptions. For
instance, certain models account for a nonisotropic drag
[BD83] while others include breaking and reforming dy-
namics [VMC07].
Before we move forward, we note that dumbbell models
are not the only, nor the most prominent, class of mod-
els used to describe polymeric systems. We have chosen
this particular class of models for specic reasons. Among
them, the Hookean dumbbell model, although physically
unrealistic, stands out as the only model in its class with
an exact closure [BCAH87,BAB91]. This allows for a more
coherent explanation of the transition through the differ-
ent length scales that are discussed in this review. We ac-
knowledge that dumbbell models have their limitations,
especially when dealing with complex ows. These draw-
backs primarily stem from their oversimplication of sev-
eral key aspects in polymer physics. For instance, they do
not account for entanglements, excluded volume effects,
nor internal chain dynamics. For a deeper understanding
of more physically relevant models, we highly recommend
the book by Doi and Edwards [DEE88]. For readers inter-
ested in the analytical and numerical solutions to SDEs-
based models, the book by Öttinger is another excellent
starting point [Ött96].
2.3. Fokker–Planck representation. In the previous sec-
tion, we discussed how SDEs can describe the evolution
of the end-to-end vector . This implies that is a sto-
chastic or random process. To fully capture the system’s
dynamics, it is necessary to solve thousands of SDEs. Now,
if instead of “following” individual realizations of this pro-
cess, i.e., solving (8) or (10), we decide to “follow” ensem-
bles of realizations, we would need a different set of equa-
tions. This new set of equations should describe the same
system, but instead of using stochastic variables, it uses de-
terministic variables that uctuate because of stochasticity.
To accomplish this we use the probability density function
(PDF), , which describes the probability of nding
dumbbells with congurations in the interval
at time . Risken’s book [Ris84] excellently explains the
differences between these two representations, which we
summarize in Fig. 2.
Let represent the conguration number den-
sity function, so that the number density of dumbbells
Figure 2. Levels of description of a system using Langevin
and Fokker–Planck Equations. Figure adapted from [Ris84].
with end-to-end vector and center of mass at position
, at time is given by,
In the homogeneous case the spatial dependence can be
neglected, so that .
When dealing with PDFs, one can do an expansion with
similar avor as the Taylor expansion taught in calculus.
This expansion is known as the Kramers–Moyal expansion
[Ris84]. If the expansion is truncated after the second term,
the resulting equation is called a Fokker–Planck equation,
also known as the forward Kolmogorov equation. A brief
summary of how Brownian dynamics can be described
by Langevin and their corresponding Fokker–Planck equa-
tions is given in [MDV20], but for a more comprehensive
treatment, see [Ris84].
The general form of a Fokker–Planck equation on the
variable is
(11)
which corresponds to the Langevin equation,
(12)
Thus, we can use (12) together with (8) or (10) to ob-
tain the Fokker–Planck equations corresponding to the
Hookean and FENE models.
Hookean dumbbells
(13)
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1019
FENE dumbbells
(14)
Just like it was the case for the Langevin equations, we
can relate each term in the Fokker–Planck equations with
their physical counterpart. Since the derivatives are on ,
this means these terms depend on the dumbbell’s cong-
uration. In (13)–(14) the rst term on the right-hand side
describes transport by the macroscopic ow. The second
term relates to the spring force, whose effect is to concen-
trate the distribution function about . The last term
represents diffusion in conguration space; its effect is to
“spread out” the distribution function.
Just as with the previous section, we will avoid going
into details about solution strategies for these Fokker–
Planck equations in order to maintain brevity and accessi-
bility in this review. However, the book by Risken [Ris84]
is a good starting point for those interested in this subject.
2.4. Macroscopic representation. So far, we have de-
scribed two ways in which we can capture the dynamics of
a microstructure comprising chains represented by elastic
dumbbells. The rst representation considers the equation
of motion of individual dumbbells given by SDEs. The sec-
ond representation considers ensembles of dumbbells and
gives the evolution equation for their PDFs as partial differ-
ential equations (PDEs). The next logical step is to discuss
how to couple these dynamics with the macroscopic ow
eld described by the conservation of mass and momen-
tum equations.
As discussed in previous sections, the equations for the
conservation of mass and momentum of an isothermal, in-
compressible, viscoelastic uid, in the absence of external
forces, are given by
(15a)
(15b)
Here we have used the notation to denote the viscosity
of the Newtonian component of the VEF, i.e., the solvent.
We will scale this equations as,
where and are characteristic time and velocity, is the
dumbbells’ number density, and is the thermal energy.
Dropping the tildes, gives
(16a)
(16b)
Here, is the ratio of the solvent to the total vis-
cosity, where is the zero-shear rate viscosity
Figure 3. Components of the stress tensor, where are
stresses resulting from forces in the -direction, acting in the
face of the volume with normal vector in the -direction. In
general, is a symmetric tensor; see proof for symmetry in
Appendix A1 of [OP02].
of the uid, with being the polymer contribu-
tion to the viscosity. And, is the Reynolds
number, which is the ratio of inertial to viscous forces.
Together with a constitutive equation for ,(16) give the
mathematical description, in time and space, of the resulting
ow eld of a VEF. However, these equations are applicable
only when considering the uid as a continuous medium.
This means that we need to nd an expression for that
can “translate” the dynamics at the microscopic and meso-
scopic levels, described in previous sections, to the contin-
uum level.
As dumbbells move about the solvent uid, there is a
drag force imposed on the beads by the uid’s velocity, .
This drag on the beads causes an extra stress on the solvent,
which in turns changes its velocity. This exchange between
the dumbbells and the uid depends on the momentum
transferred between the beads in each dumbbell, which
is modulated by the connector vector . Because of this
dependence on , momentum transfer perpendicular to
the ow exerts additional viscous forces, i.e., resistance to
ow. While momentum transfer parallel to the ow gives
elastic properties to the uid.
In order to understand how single dumbbells con-
tribute to the stress, let’s start by examining the signicance
of each entry in the stress tensor. In three-dimensional
space, the stress tensor is a matrix. If we consider
a piece of uid as a cube, then the entry corresponds
to the stress resulting from a force in the direction im-
posed in the face with a normal vector in the
direction;
see Fig. 3.
The contribution to the stress from a single dumbbell is
then given by Kramer’s relation [BCAH87,Ött96],
where is the spring force, the identity ma-
trix, and indicates the tensor product of vectors. And,
we can connect , at the microscopic or mesoscopic scales,
to the macroscopic stress using ensemble averages. In
the conguration space represented by , the ensemble
1020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
average of any function is given by
(17)
We use (17) to nd the stress resulting from the collective
dynamics all of dumbbells as
where is the contribution from the tension on
the spring with spring law , and capture effects
due to Brownian motion.
For Hookean springs, and we obtain,
(18)
For FENE springs , so
that,
(19)
Here, we used the same nondimensionalization as before.
To nd a constitutive equation for , we start with
the general form of the Fokker–Planck equation for elas-
tic dumbbells,
Recall that in this representation we assume the spring
force is of the form . To nd an expression
for , we multiply (20) throughout by , and inte-
grate over the conguration space. We use the divergence
theorem and the fact that as to obtain
[BCAH87]
If we dene the upper convected derivative as,
we arrive at,
(20)
In the next sections we show how we can use (20) to
formulate constitutive equations for corresponding the
Hookean and FENE dumbbells.
2.4.1. Hookean dumbbells. In this case , so that
Using (18) we obtain the constitutive equation for the ex-
tra stress tensor of Hookean dumbbells,
(21)
This is the well-known Upper Convected Maxwell (UCM)
model [BCAH87]. We should emphasize that the UCM
model provides an exact closure to the Hookean dumbbell
model because (21) can be directly obtained from (8). As
we will see below, this is not the case for the FENE dumb-
bells.
Finally, if instead of considering a constitutive equation
for only , we consider the total extra stress, ,
the constitutive equation for is known as the Oldroyd-B
Model. For a discussion of the mathematical considera-
tions and challenges arising from the description of VEF
using this model, see [RT21].
2.4.2. FENE dumbbells. For FENE dumbbells, (20) gives,
Because of the nonlinear term, it is not possible to obtain
a close-form constitutive equation of for FENE dumb-
bells. Instead, several closures have been suggested to
allow the formulation of macroscopic constitutive equa-
tions. Here we will discuss the so-called Peterlin approx-
imation, which results in the FENE-P model [BCAH87].
Other FENE closures are discussed in [DLY05].
Peterlin proposed a separate average of the numerator
and denominator of the spring law [BCAH87],
(22)
(23)
In this way, instead of restricting the length of individual
dumbbells to be less than , the Peterlin’s approxima-
tion relaxes the restriction where only the average dumb-
bell length needs to be less than the prescribed maximum
extension. Individual dumbbell lengths can thus exceed
as long as the average stays within bounds.
For convenience, we dene a nondimensional congu-
ration tensor, , as
(24)
where is the mean-square end-to-end spring length
at equilibrium (absence of ow) and is the dimen-
sionality [BCAH87]. Note that if we scale the end-to-end
vector as before, we have a nondimensional conformation
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1021
Figure 4. Solutions in simple shear flow for macroscopic
models (solid lines) compared to stochastic simulations
(dashed lines). For simple shear flow, the velocity is
prescribed as . Defining gives the
nondimensional velocity as .(A) Hookean
dumbbell vs. UCM model. (B) FENE dumbbell vs. FENE-P
model.
tensor . The constitutive equation for in
three-dimensions is then found as,
(25a)
(25b)
To elucidate the trade-offs involved in the Peterlin ap-
proximation, in Fig. 4 we show solutions of (8), (10), (21),
and (25) under simple shear ow. We compare the macro-
scopic closures given by the UCM and FENE-P models with
their corresponding stochastic counterparts: the Hookean
and FENE models. Since the UCM model is an exact clo-
sure of the Hookean dumbbell model, solutions to (8)
and (21) will always agree with each other, as shown in
Fig. 4(A). On the other hand, the approximation used in
the FENE-P formulation leads to deviation between solu-
tions of (10) and (25). Fig. 4(B) shows that these dif-
ferences are more noticeable at higher deformation rates,
when dumbbells are close to their maximum extension.
As mentioned above, FENE-P is not the only closure pro-
posed for the FENE model. The mathematical analysis of
various closures for the FENE model remains an active area
of research, involving the exploration of different approxi-
mations and/or higher order moments, e.g., [DLY05].
3. Conclusions
This review aims to introduce the reader to the fundamen-
tal aspects of mathematical modeling of viscoelastic uids
(VEFs). The most important factor being that the underly-
ing microstructure of VEFs is what determines their prop-
erties at the macroscale. This microstructure can comprise
polymer molecules, colloidal particles, emulsion drops,
etc. The common characteristic is that these structures are
larger than the solvent molecules and, as a result, bring
about additional stresses to the system. This results in
two components of the extra stress, one from the New-
tonian component (solvent) and the other from the mi-
crostructure. We discussed three levels of description used
in modeling this microstructure. The rst description
focuses on the evolution of individual dumbbells using
Langevin-type SDEs. The second description is concerned
with the evolution of the PDF of the dumbbell’s congura-
tion through Fokker–Planck equations. The third descrip-
tion provides information at the macroscopic level using
PDEs.
Among the different representations, macroscopic con-
stitutive models offer a higher level of computational fea-
sibility, enabling us to nd solutions in complex ows or
geometries. However, by using these models, one must
make a compromise regarding how accurately we can de-
scribe the underlying molecular physics. Similarly, al-
though computationally expensive, Langevin or Fokker–
Planck descriptions are more amenable to incorporating
additional degrees of freedom. This allows us to develop
models that better capture the intricacies of physical pro-
cesses. Hence, in nding the appropriate level of descrip-
tion from a mathematical standpoint, it is important to
strike a balance between the complexity of molecular in-
formation and the computational costs involved.
References
[AOP21] M. A. Alves, P. J. Oliveira, and F. T. Pinho, Numerical
methods for viscoelastic uid ows, Annual Review of Fluid
Mechanics 53 (2021), 509–541.
[BAB91] Aparna V. Bhave, Robert C. Armstrong, and Robert A.
Brown, Kinetic theory and rheology of dilute, nonhomoge-
neous polymer solutions, The Journal of Chemical Physics 95
(1991), no. 4, 2988–3000.
[Bat99] G. K. Batchelor, An introduction to uid dynam-
ics, Second paperback edition, Cambridge Mathematical
Library, Cambridge University Press, Cambridge, 1999.
MR1744638
[BCAH87] Robert Byron Bird, Charles F. Curtiss, Robert C.
Armstrong, and Ole Hassager, Dynamics of polymeric liquids,
volume 2: Kinetic theory, Wiley, 1987.
[BD83] R. Byron Bird and J. R. DeAguiar, An encapsulted dumb-
bell model for concentrated polymer solutions and melts i. theo-
retical development and constitutive equation, Journal of non-
Newtonian Fluid Mechanics 13 (1983), no. 2, 149–160.
1022 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
[DEE88] Masao Doi, Sam F. Edwards, and Samuel Frederick
Edwards, The theory of polymer dynamics, Vol. 73, Oxford
University Press, 1988.
[DLY05] Qiang Du, Chun Liu, and Peng Yu, FENE dumbbell
model and its several linear and nonlinear closure approxima-
tions, Multiscale Model. Simul. 4(2005), no. 3, 709–731,
DOI 10.1137/040612038. MR2203938
[Gra18] Michael D. Graham, Microhydrodynamics, Brownian
motion, and complex uids, Cambridge Texts in Applied
Mathematics, Cambridge University Press, Cambridge,
2018, DOI 10.1017/9781139175876. MR3837153
[Keu04] Roland Keunings, Micro-macro methods for the multi-
scale simulation of viscoelastic ow using molecular models of
kinetic theory, Rheology Reviews 2004 (2004), 67–98.
[Lar99] Ronald G. Larson, The structure and rheology of complex
uids, Oxford University Press, 1999.
[LM00] P. L. Lions and N. Masmoudi, Global solutions for
some Oldroyd models of non-Newtonian ows, Chinese
Ann. Math. Ser. B 21 (2000), no. 2, 131–146, DOI
10.1142/S0252959900000170. MR1763488
[Mai22] Francesco Mainardi, Fractional calculus and waves in
linear viscoelasticity—an introduction to mathematical models,
World Scientic Publishing Co. Pte. Ltd., Hackensack, NJ,
2022. Second edition [of 2676137], DOI 10.1142/p926.
MR4485803
[MDV20] Andrei Medved, Riley Davis, and Paula A Vasquez,
Understanding uid dynamics from Langevin and Fokker–
Planck equations, Fluids 5(2020), no. 1, 40.
[OP02] R. G. Owens and T. N. Phillips, Computational
rheology, Imperial College Press, London, 2002, DOI
10.1142/9781860949425. MR1906885
[Ött96] Hans Christian Öttinger, Stochastic processes in poly-
meric uids, Springer-Verlag, Berlin, 1996. Tools and
examples for developing simulation algorithms, DOI
10.1007/978-3-642-58290-5. MR1383323
[Poo12] R. J. Poole, The Deborah and Weissenberg numbers,
Rheol. Bull 53 (2012), no. 2, 32–39.
[Ren85] M. Renardy, Existence of slow steady ows of vis-
coelastic uids with differential constitutive equations, Z.
Angew. Math. Mech. 65 (1985), no. 9, 449–451, DOI
10.1002/zamm.19850650919. MR814684
[Ris84] H. Risken, The Fokker-Planck equation, Springer Series
in Synergetics, vol. 18, Springer-Verlag, Berlin, 1984. Meth-
ods of solution and applications, DOI 10.1007/978-3-642-
96807-5. MR749386
[RT21] Michael Renardy and Becca Thomases, A mathemati-
cian’s perspective on the Oldroyd B model: progress and fu-
ture challenges, J. Non-Newton. Fluid Mech. 293 (2021),
Paper No. 104573, 12, DOI 10.1016/j.jnnfm.2021.104573.
MR4266315
[VMC07] Paula A. Vasquez, Gareth H. McKinley, and
L. Pamela Cook, A network scission model for wormlike mi-
cellar solutions: I. model formulation and viscometric ow pre-
dictions, Journal of non-Newtonian Fluid Mechanics 144
(2007), no. 2-3, 122–139.
Paula A. Vasquez
Credits
All gures are courtesy of the author.
Photo of Paula A. Vasquez is courtesy of the University of
South Carolina.
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1023
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the AMS, and private donors.
NSF funding decision to
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Codimension One Foliations
on Projective Manifolds
Jorge Vit´orio Pereira
In simple terms, a smooth foliation on a manifold is
a decomposition of into a disjoint union of immersed
smooth subvarieties. Although the origins of the subject
trace back to the end of the 19th century, the denition of
a smooth foliation only appeared in the mid-20th century,
in the work of Georges Reeb.
Foliations naturally appear in various mathematical dis-
ciplines, subjecting them to different perspectives. For
some, they are topological objects, others emphasize their
dynamical nature. Here, our focus lies on seeing folia-
tions as algebro-geometric objects. In particular, we will
restrict ourselves to the discussion of holomorphic folia-
tions, with an emphasis on codimension one holomor-
phic foliations on projective manifolds.
1. Foliations
A smooth holomorphic foliation of codimension on
a complex manifold is dened by a collection of holo-
morphic submersions , where
the sets form an open covering of , and whenever
the domains of two submersions, say and , have a
nonempty intersection, there exists a holomorphic transi-
tion function satisfying
.
We will use “smooth foliation” rather than “smooth
holomorphic foliation” throughout the text to avoid
wordiness.
A smooth foliation denes on an equivalence rela-
tion, namely the smallest equivalence relation which iden-
ties points on the same connected components of bers
of the submersions . Its equivalence classes are the leaves
of which are immersed submanifolds of .
In practice, a smooth foliation is rarely presented
through a collection of submersions as above. A classical
theorem by Frobenius allows to recover a smooth foliation
by means of the holomorphic vector elds tangent to the
Jorge Vit´orio Pereira is a researcher at IMPA. His email address is jvp@impa
.br.
Communicated by Notices Associate Editor Han-Bom Moon.
For permission to reprint this article, please contact:
reprint-permission@ams.org.
DOI: https://doi.org/10.1090/noti2999
level sets of the submersion , or by means of the holo-
morphic -forms vanishing on the level sets of the . If
is a smooth foliation on then the holomorphic vector
elds tangent to it dene a subbundle of the tangent
bundle of . The Lie bracket of any two local sections of
dened on the same open subset of is also a local
section of . Reciprocally, the Frobenius theorem guar-
antees that any subbundle of with set of local sec-
tions closed under Lie bracket is the tangent bundle of a
(uniquely dened) smooth foliation . When this is the
case, we say that the subbundle is involutive.
Dually, the holomorphic -forms which are zero when
restricted (that is, pulled-back) to the leaves of a smooth
foliation dene a subbundle
of
called the conor-
mal bundle of . When has codimension one, our main
case of interest, any section of
over a open set
where is dened by a submersion
is of the form for some holomorphic function
. It follows that . Reciprocally, a dual formu-
lation of the Frobenius theorem, implies that for any rank
one subbundle of
such that the differential of any
germ of section of satises is the conormal
bundle of a codimension one smooth foliation . When
this is the case, we say that the subbundle is integrable.
For example, the -form is not
a section of the conormal bundle of any foliation on
since vanishes nowhere. In contrast,
the -form satises
hence generates the conormal bundle of a foliation on .
Figure 1. Real picture of some
of the leaves of the foliation on
the affine -space defined by
the -form .
There is an analogue for-
mulation of the Frobenius
theorem for subbundles of
of arbitrary rank: a sub-
bundle
is the conor-
mal bundle of a smooth fo-
liation (in other words, is
integrable) if, and only if,
the differential of any sec-
tion of dened on a suf-
ciently small open subset
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1025
can be written as
where are sections of and are -forms, all dened
on .
1.1. Algebraic foliations. If is a smooth algebraic va-
riety then it makes sense to consider algebraic subbun-
dles of which are closed under Lie bracket, as well as
algebraic subbundles of
which are integrable. One
obtains equivalent denitions of what we call algebraic
foliations. Notice that we are not imposing algebraicity
of the holomorphic submersions dening nor of the
leaves of . In general, they are not algebraic. For in-
stance, if and we take as the smooth foliation
dened by the subbundle of
generated by the -form
considered above (clearly a poly-
nomial -form, hence is algebraic), then is dened by
the submersion which clearly
has nonalgebraic level sets, except for ; see Figure 1.
1.2. Singular foliations. The existence of a smooth foli-
ation on a compact complex manifold imposes strong re-
strictions on the manifold. For instance, the only smooth
hypersurfaces on projective spaces admitting smooth folia-
tions are the quadrics on the three-dimensional projective
space1(hence isomorphic to ). Moreover, there are
only two smooth foliations on . Of course, those
are the ones dened by the bers of the two natural pro-
jections to .
The sparsity of examples of smooth foliations on com-
pact/projective manifolds, invites the consideration of sin-
gular foliations. To avoid delving into the technical details
of the denition of this concept in general, we restrict our-
selves to the codimension one case. A singular codimen-
sion one foliation on a complex manifold is dened by a
collection of holomorphic -forms
, where
the open subsets form an open covering of , such that
for every the -form has zero locus of codimension
at least two and satises the Frobenius integrability condi-
tion and, whenever , the holomor-
phic -forms and differ by multiplication
by a nowhere vanishing holomorphic function. The singu-
lar set of is, by denition, the union of the zero
sets of the -forms .
When the ambient is an algebraic manifold , one can
adopt an alternative denition. If one considers the equiv-
alence relation on the set of nonzero rational -forms iden-
tifying two rational -forms when they differ by multipli-
1For the experts. When the hypersurface has dimension two, the statement is a
consequence of the classication of smooth foliations on compact complex sur-
faces by Brunella. For higher-dimensional manifolds, the statement is a conse-
quence of Bott’s vanishing principle combined with the fact that hypersurfaces
of dimension at least three have cyclic Picard group.
cation by a nonzero rational function then a codimension
one foliation on can be dened as an equivalence class
of -forms such that some (and hence any) representative
satises .
From now on, we will use the term foliation to refer to
a singular holomorphic foliation.
Figure 2. Algebraically
integrable foliation on
defined by the rational map
.
1.3. First examples.
1.3.1. Algebraically integra-
ble foliations. If 99K
is a nonconstant mero-
morphic map, then the dif-
ferential of denes a fo-
liation on . Reciprocally,
an argument of Darboux,
rened rst by Jouanolou
and then by Ghys, see
[Ghy00] and references
therein, guarantees that a
codimension one foliation
on a compact manifold
leaving invariant innitely
many distinct compact hy-
persurfaces2is dened by (the differential of) a noncon-
stant meromorphic function 99K . Codimen-
sion one foliations of this form are called algebraically in-
tegrable foliations.
1.3.2. Foliations on surfaces. When the ambient manifold
has dimension two and is algebraic, it is a trivial matter
to construct foliations, since there are many nonequivalent
rational -forms, all of which, due to dimensional reasons,
satisfying the integrability condition . A suf-
ciently general rational -form on denes a foliation
without invariant projective curves. Moreover, there are fo-
liations for which every leaf is dense in the Euclidean topol-
ogy. There are also foliations for which every leaf is Zariski
dense but not dense in the Euclidean topology. Not much
is known about the topological behavior of leaves of arbi-
trary foliations on surfaces. For instance, the minimal set
problem for foliations on [CLNS88]—is it true that the
topological closure of every leaf of every foliation of
contains a singularity of the foliation?—remains unsolved
up to date.
1.3.3. Closed rational -forms. Another class of “obvious”
examples of codimension one foliations on projective
manifolds are the foliations dened by closed ratio-
nal/meromorphic -forms. Locally, a closed meromorphic
-form is written as
2A hypersurface is invariant by a codimension one foliation on a manifold
if the restriction of is a union of leaves of .
1026 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
where are holomorphic functions, are complex num-
bers (the residues of the -form), and is a meromorphic
function, see [Per23].
On projective manifolds, or more generally compact
Kähler manifolds, basic Hodge theory allow us to write any
closed rational -form as a sum of a closed rational -form
with simple poles, known as logarithmic -forms, and a
closed rational -form without residues (the so-called -
forms of second type, which are locally exact). Recipro-
cally, as was observed by Andr´e Weil, given any formal
sum
of hypersurfaces with complex coef-
cients, there exists a closed logarithmic -form with that
sum as its residue if, and only if, the homology class of
is is equal to zero.
An important property of foliations dened by closed
rational -forms is that they admit nontrivial rational in-
nitesimal symmetries, that is rational vector elds not
everywhere tangent to the foliation and whose local ow
(wherever dened) sends leaves of to leaves of . If
is a closed rational -form dening a foliation then any
rational vector satisfying is a nontrivial ratio-
nal innitesimal symmetry. Reciprocally, if a codimension
one foliation admits a nontrivial rational innitesimal
symmetry generically transverse to it then is dened by
a closed rational -form. In general, foliations do not ad-
mit nontrivial global rational/meromorphic innitesimal
symmetries which are generically transverse to them.
1.3.4. Pull-backs under rational maps. Given a singular codi-
mension one foliation on a projective manifold and a
rational map 99K then, if the image of is not con-
tained in a leaf of , we obtain a pull-back foliation
on dened by the pull-back of a representative -form
dening with polar and zero divisor not contained in
the image of .
2. Foliations on Projective Spaces
Figure 3. Degree two foliation
on defined by the level sets
of a degree three polynomial.
The blue line is tangent to the
foliation at the marked points.
Let be a codimension one
foliation on dened by
a rational -form . Let us
denote by
the divisor (formal sum of
hypersurfaces with integer
coefcients) of poles minus
the divisor of zeros of . If
is a suf-
ciently general linear em-
bedding then is a ra-
tional -form on . Be-
sides the zeros and poles
coming from , the -form
acquires new zeros cor-
responding to the tangen-
cies of with the folia-
tion. The number of these tangencies (counted with the
appropriate multiplicities) is, by denition, the degree of
. Since the divisor of poles minus zeros of any -form on
has degree , the degree of is exactly the degree of
minus .
Starting with a rational -form on , we can lift it to
as a rational homogeneous -form. After multiplying
such -form by the dening (homogeneous) equation of
the divisor , we obtain a polynomial -form
where are homogeneous polynomials in
without a common factor. If the original -
form satised the Frobenius integrability condition, then
the same holds true for , that is . Moreover,
satises the descent condition
where
is the radial, or Euler, vector eld.
Reciprocally, starting with a polynomial -form as above,
we can set , to obtain a polynomial -form on , or
rather a rational -form on without codimension one
zeros and polar divisor equal to where is the
hyperplane .
A moment of reection reveals that the degree of the
foliation dened by equals .
2.1. Space of foliations. Let be the vector space3
formed by -uples of polynomials of de-
gree in satisfying the equation
. The subset formed by equivalence
classes of -uples for which the -form
satises the Frobenius integrability condi-
tion and has zeros of codimension at least two
is a locally closed set dened by a collection of quadratic
equations on the coefcients of the polynomials .
Problem 2.1. Describe the irreducible components of the lo-
cally closed subset dened by the Frobenius integrabil-
ity condition.
A variant of the classical Euler formula for homoge-
neous polynomials
implies that is proportional to whenever
and . It follows that for , the problem
is trivial as the integrability condition is always sat-
ised by homogeneous -forms on annihilated by the
Euler vector eld. Therefore, for every ,has
only one irreducible component.
3The vector space is nothing but
.
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1027
For , the problem is highly nontrivial, even hope-
less in its full generality, but spurred a lot of activity.
To have an idea of the complexity of the problem at
hand, a simple dimension count reveals that
and that the Frobenius integrability condition on
translates into
quadratic equations on the coef-
cients of the polynomials .
In degree zero, the variety is smooth, irre-
ducible, and isomorphic to the Grassmanian of lines on
the dual . Moreover, its points correspond to foliations
on dened the level sets of linear projections 99K .
It is unclear who were the rst mathematicians to observe
this fact.
In degree one, when , the determination of the ir-
reducible components of dates back to the 1970s
and is due to Jouanolou [Jou79]. Inspired by earlier works
of Jacobi and Darboux, Jouanolou proves that
has exactly two irreducible components. One of them pa-
rameterizes foliations which are pull-backs of foliations on
under linear projections 99K , while the other pa-
rameterizes foliations dened by pencil of hypersurfaces
generated by a quadric hypersurface and a double hyper-
plane.
The determination of the irreducible components of
, again assuming that , had to wait almost 20
years. It is considerably more involved and is the subject
of a celebrated paper by Cerveau and Lins Neto [CLN96].
Theorem 2.2. For every , the set has exactly
irreducible components.
One of the irreducible components of param-
eterizes linear pull-back foliations. The other irreducible
components parameterize foliations dened by logarith-
mic -forms (one for each partition of with at least two
summands). The two irreducible components correspond-
ing to partitions with exactly two summands (
) parameterize, respectively, pencils of quadrics and
pencils of cubics containing a hyperplane with multiplic-
ity three. The general foliation parameterized by any of
the two other components corresponding to partitions
with at least three summands is not algebraically inte-
grable. The sixth irreducible component (the so-called ex-
ceptional component) has general member corresponding
to a foliation dened by the linear pull-back under a pro-
jection 99K of the foliation on dened
by the action of afne group
on polynomials of degree at most three.
Figure 4. On the left, an isolated tangency of a leaf of foliation
with a tangent hyperplane. On the right, the induced foliation
at the tangent hyperplane. The central point corresponds to
the tangency point.
Proof of Theorem 2.2 for (sketch). Start with a codi-
mension one foliation on and consider the map from
to
which sends a point to the hyperplane
tangent to at , the so-called Gauss map of .
If the Gauss map of is not dominant, then classical
results on the geometry of (germs of) surfaces in im-
ply that every leaf of is either a cone over a curve or
the tangential surface of a curve. With this information
at hand, one veries that a foliation with a nondominant
Gauss map is either a linear pull-back of a foliation on
or its leaves form a one-parameter family of algebraic
cones with vertices moving along a line. The rst case
spans an irreducible component of for any degree
, while degree two foliations in the second case are con-
tained in the irreducible component parameterizing pen-
cils of quadrics.
When the Gauss map of is dominant, the restriction
of to a general is a foliation admitting a singular point
(corresponding to the tangency of a smooth point of
with the in question) where the foliation is locally de-
ned by a the differential of a holomorphic function of
the form ; see Figure 4. In the ter-
minology of the theory of ordinary differential equations,
such singularities are called centers. A classical result by
Dulac published in 1908, describes, rather precisely, qua-
dratic vector eld on admitting a center. In general,
degree two foliations on are not dened by quadratic
vector elds on an afne chart. This is the case if, and only
if, the line at innity is invariant. Computer-aided calcula-
tions implies that for degree two foliations on the exis-
tence of a center singularity automatically implies the ex-
istence of an invariant line not passing through it. After
moving such invariant line to innity, one is in position
to use Dulac’s classication in order to classify foliations
on with dominant Gauss map. □
Arriving at this point, it is natural to wonder why not
pursue the same strategy in order to classify degree three
foliations on . Unfortunately, it is very unlikely that
1028 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
the strategy outlined above could work in degree three.
The main obstruction is the lack of an analogue of Du-
lac’s classication in degree three or higher. The problem
of classifying foliations on with a center (the so-called
Poincar´e center problem) is a well-studied and acknowl-
edgedly difcult problem, probably harder than Problem
2.1. The equations dening the center variety are consid-
erably harder to obtain than the integrability equations
for codimension one foliations. For instance, back in
2007, Van Bothmer tried to write down, with the help of a
computer, the equations for the center problem in degree
three without success as explained in the Introduction of
[GvB07], to which we refer for further information on the
Poincar´e center problem.
3. Intrinsic Meaning of Low Degree
Instead of seeking for a generalization of Cerveau-Lins
Neto classication to higher degrees, one may try to under-
stand, in more intrinsic terms, what are the factors impos-
ing the constrained behavior of low-degree foliations on
projective spaces. The birational geometry of varieties pro-
vides a nice conceptual framework to approach this ques-
tion.
The work of Mori and others from the 1980s and its sub-
sequent developments taught us that much of the geome-
try of projective manifolds is governed by how the line-
bundle of differential forms of top degree, the so-called
canonical bundle, intersects curves. In analogy with the
case of projective manifolds, one is naturally lead to de-
ne the canonical bundle of a singular foliation as the
line-bundle of holomorphic forms along the leaves of
of degree . More formally, one denes the canoni-
cal bundle of a foliation as the dual of the determinant of
its tangent sheaf.
3.1. Negative canonical bundle. The study of algebraic
foliations having negative canonical bundle was initiated
by Miyaoka in a landmark paper [Miy87] that explores the
idea that negativity of the canonical bundle of a foliation
along a sufciently general curve should be related to the
possibility of deforming the curve along the foliation. In
rough terms, a deformation of a curve along a foliation is
a map from the product of a connected pa-
rameter space with the curve to such that for some
point ,is the inclusion of and for any given
point , the image of under is completely con-
tained in the leaf of passing through . In general,
the negativity of the canonical line-bundle of alone is
not sufcient to produce such deformations. Nevertheless,
when working with algebraic foliations one may collect the
coefcients of (nitely many) dening equations of the fo-
liations in a nitely generated -algebra , reduce every-
thing in sight modulo a maximal prime ideal of , and
obtain arithmetic shadows in positive characteristic of the
foliation and of the curve. Then one is in place to explore
particular features of foliations and morphisms dened in
positive characteristic. It turns out that after composing
the inclusion of in with a sufciently high power of
the Frobenius morphism, the sought for deformations are
shown to exist in positive characteristic and are used to
produce rational curves tangent to the leaves of the folia-
tion through a variant of Mori’s bend-and-break argument.
In conclusion, one deduces that the foliation is uniruled,
that is, through a general point of passes a rational curve
everywhere tangent to the foliation.
Miyaoka’s paper was not motivated by internal ques-
tions in foliation theory, but instead by the study of pro-
jective manifolds. His result is an important ingredient on
the proof of the so-called abundance conjecture for projec-
tive -folds. It was later extended, with different proofs, by
Bogomolov-McQuillan, Bost, and Campana-Paun. These
lead to a cone theorem for foliations by curves [BM16] and
to the uniruledness of foliations with nonpseudoeffective
canonical bundle [CP19].
In a related vein, Araujo-Druel-K´ovacs [ADK08], moti-
vated by a question of Beauville pertaining to the study of
symplectic singularities, classied foliations with canoni-
cal bundle as negative as possible: foliations of dimension
such that the canonical bundle is the -th power of the
dual of an ample line-bundle. It turns out that these are
precisely the degree zero foliations (of arbitrary codimen-
sion) on projective spaces.
3.2. Trivial canonical bundle. Codimension one folia-
tions of degree two on are examples of foliations with
trivial canonical bundle. Touzet established in [Tou08] a
classication of smooth codimension one foliations with
trivial canonical class. He builds on previous work on the
universal covering of compact Kähler manifolds with de-
composable tangent bundle [BPT06] as well as results on
the geometry of Ricci at complete Kähler manifolds. The
classication is rather precise: a smooth codimension one
foliation with trivial canonical class is, after an ´etale cover-
ing, the product of compact Kähler manifold with trivial
canonical class (seen as a foliation with only one leaf) and
a compact Kähler manifold admitting a codimension one
locally free action of an abelian Lie algebra (including the
trivial case of the action of the zero dimension Lie algebra
on an arbitrary compact Riemann surface).
Later, Touzet’s classication was extended in [LPT18] to
describe singular codimension one foliation with trivial
canonical class.
Theorem 3.1. Let be a codimension one foliation with triv-
ial canonical bundle on a projective manifold . If is not
uniruled then, after a nite ´etale covering, is the prod-
uct of a compact Kähler manifold with trivial canonical class
and a codimension one foliation induced by a codimension one
action of an abelian Lie algebra.
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1029
Although the statement is essentially the same as in the
smooth case, it is noteworthy that the algebraicity of
(and hence of ) is included as one of the assumptions. It
is conceivable that the same result holds for compact Käh-
ler manifolds, but its proof involves techniques specic to
algebraic manifolds.
Firstly, it is observed, building on complex-analytic ar-
guments due to Demailly [Dem02] and using the Bouck-
som-Demailly-Paun-Peternell uniruledness criterion
(available only to projective manifolds), that when is not
uniruled, then is automatically smooth. Next, when
is uniruled, one studies the deformation of rational curves
along to establish that the transverse dynamics of is
rather constrained. More specically, one proves that
is a transversely projective foliation (see §4.1 below for a
denition of this concept).
In parallel, one studies the reduction to positive char-
acteristic of . As observed in [CLNL07], it is relatively
easy to produce innitesimal transverse symmetries in pos-
itive characteristics. Although these symmetries cannot
be lifted back to characteristic zero in general, when the
canonical bundle is trivial this can be done in most cases.
This is used to show that is either dened by a closed
rational -form (perhaps after a ramied covering) or the
singularities of are rather constrained (admit local holo-
morphic rst integrals).
By combining the outcomes of both arguments, one
concludes that is always dened by a closed rational -
form without codimension one zeros. From there, the con-
clusion follows from a study of the Albanese morphism of
.
The study of deformations of rational curves along codi-
mension one foliations was further developed in [LPT20]
and eventually lead to a (partial) classication of the irre-
ducible components of the space of degree three foliations
on ,, see [dCLP22]. Likewise, the reduction to pos-
itive characteristic of codimension one foliations on pro-
jective manifolds is investigated in [MP23] and applied to
the detection of previously unknown irreducible compo-
nents of the space of foliations.
4. Global Structure
The structure results presented so far imply that foliations
with negative or trivial canonical bundles are either unir-
uled, hence pull-backs of foliations on lower-dimension
manifolds, or (nite quotients of) foliations dened by
closed rational -forms. Below we present a different class
of codimension one foliations which, in general, behave
differently.
4.1. Riccati and transversely projective foliations. Let
be the total space of -bundle over a complex manifold
. A Riccati foliation on is a codimension one foliations
which is completely transverse to a general ber of the -
bundle. The Riemann-Hilbert correspondence [Del70] al-
lows one to build Riccati foliations from representations
of the fundamental group of the complement of hyper-
surfaces on complex manifolds to producing a
wealth of examples with rich dynamical behavior.
Figure 5. Numerical
approximation of the real trace
of some of the leaves of a
Riccati foliation on . The
light gray vertical lines
represent open subsets of the
fibers of the -bundle. In the
picture, the foliation is
transversal to all fibers except
one. Over it one can observe
the formation of a singularity,
the point of confluence of the
leaves in the bottom half of the
picture.
A foliation on a complex
manifold is transversely
projective if there exists a
-bundle over with to-
tal space , a Riccati foli-
ation on and a ratio-
nal section 99K
of such that
. The class of trans-
versely projective foliations
includes, as particular cases,
the algebraically integrable
foliations as well as the foli-
ations dened by closed ra-
tional -forms. The study of
transversely projective folia-
tions, akin to the study of
Riccati foliations, is intrin-
sically related to the study
of representations of the
fundamental group of com-
plex manifolds to the au-
tomorphism group of .
For more about transversely
projective foliations on pro-
jective manifolds, the reader is invited to consult [LP07]
for a detailed discussion about the concept, and [LPT16]
for a description of their global structure,.
4.2. Cerveau-Lins Neto conjecture. All the known exam-
ples of codimension one foliations on projective mani-
folds are either pull-backs under rational maps of folia-
tions on surfaces or are transversely projective.
Conjecture 4.1. Let be a codimension one singular holo-
morphic foliation on a projective manifold . If is not the
pull-back under a rational map of a foliation on a projective
surface then is a transversely projective foliation.
The validity of this conjecture would impose strong
restrictions on the holonomy representation of algebraic
leaves of codimension one foliations. When is the pull-
back of a foliation on a surface, the holonomy represen-
tation of any algebraic leaf would factor (up to nite in-
dex) through a representation of the fundamental group
of an algebraic curve. If instead the foliation is transversely
projective then the holonomy representation of any alge-
braic leaf would have solvable image. The main result of
[CLPT19] provides evidence toward Conjecture 4.1 as it
implies that any representation of the fundamental group
1030 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
of a quasi-projective manifold to the group of germs of dif-
feomorphisms of satises this dichotomy. Further
evidence is provided by the main results of [CLNL07].
References
[ADK08] Carolina Araujo, St´ephane Druel, and S´andor J.
Kov´acs, Cohomological characterizations of projective spaces
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253, DOI 10.1007/s00222-008-0130-1. MR2439607
[BM16] Fedor Bogomolov and Michael McQuillan, Rational
curves on foliated varieties, Foliation theory in algebraic ge-
ometry, Simons Symp., Springer, Cham, 2016, pp. 21–51.
MR3644242
[BPT06] Marco Brunella, Jorge Vit´orio Pereira, and Fr´ed´eric
Touzet, Kähler manifolds with split tangent bundle
(English, with English and French summaries), Bull.
Soc. Math. France 134 (2006), no. 2, 241–252, DOI
10.24033/bsmf.2507. MR2233706
[CLNS88] C. Camacho, A. Lins Neto, and P. Sad, Minimal sets
of foliations on complex projective spaces, Inst. Hautes ´
Etudes
Sci. Publ. Math. 68 (1988), 187–203 (1989). MR1001454
[CP19] Fr´ed´eric Campana and Mihai P˘aun, Foliations with pos-
itive slopes and birational stability of orbifold cotangent bun-
dles, Publ. Math. Inst. Hautes ´
Etudes Sci. 129 (2019), 1–49,
DOI 10.1007/s10240-019-00105-w. MR3949026
[CLN96] D. Cerveau and A. Lins Neto, Irreducible components
of the space of holomorphic foliations of degree two in ,
, Ann. of Math. (2) 143 (1996), no. 3, 577–612, DOI
10.2307/2118537. MR1394970
[CLNL07] Dominique Cerveau, Alcides Lins-Neto, Frank
Loray, Jorge Vit ´orio Pereira, and Fr´ed´eric Touzet, Com-
plex codimension one singular foliations and Godbillon-Vey se-
quences (English, with English and Russian summaries),
Mosc. Math. J. 7(2007), no. 1, 21–54, 166, DOI
10.17323/1609-4514-2007-7-1-21-54. MR2324555
[CLPT19] Benoît Claudon, Frank Loray, Jorge Vit ´orio Pereira,
and Fr´ed´eric Touzet, Holonomy representation of quasi-
projective leaves of codimension one foliations, Publ. Mat. 63
(2019), no. 1, 295–305, DOI 10.5565/PUBLMAT6311910.
MR3908795
[dCLP22] Raphael Constant da Costa, Ruben Lizarbe, and
Jorge Vit´orio Pereira, Codimension one foliations of degree
three on projective spaces, Bull. Sci. Math. 174 (2022), Pa-
per No. 103092, 39, DOI 10.1016/j.bulsci.2021.103092.
MR4354288
[Del70] Pierre Deligne, ´
Equations diff´erentielles à points sin-
guliers r´eguliers (French), Lecture Notes in Mathemat-
ics, Vol. 163, Springer-Verlag, Berlin-New York, 1970.
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[Dem02] Jean-Pierre Demailly, On the Frobenius integrabil-
ity of certain holomorphic -forms, Complex geometry
(Göttingen, 2000), Springer, Berlin, 2002, pp. 93–98.
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[Ghy00] ´
Etienne Ghys, À propos d’un th´eorème de J.-P.
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10.1007/BF02904228. MR1753461
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ferential Equations Appl. 14 (2007), no. 5-6, 671–698,
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(French), Lecture Notes in Mathematics, vol. 708, Springer,
Berlin, 1979. MR537038
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Touzet, Representations of quasi-projective groups, at con-
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Ec. polytech. Math. 3
(2016), 263–308, DOI 10.5802/jep.34. MR3522824
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Touzet, Singular foliations with trivial canonical class,
Invent. Math. 213 (2018), no. 3, 1327–1380, DOI
10.1007/s00222-018-0806-0. MR3842065
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Sc. Norm. Super. Pisa Cl. Sci. (5) 21 (2020), 1315–1331,
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prescription of monodromy, Internat. J. Math. 18 (2007),
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[MP23] W. Mendson and J. V. Pereira, Codimension one foli-
ations in positive characteristic, Journal of the Institute of
Mathematics of Jussieu (2023), 1–46.
[Miy87] Yoichi Miyaoka, Deformations of a morphism along
a foliation and applications, Algebraic geometry, Bow-
doin, 1985 (Brunswick, Maine, 1985), Proc. Sympos.
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1987, pp. 245–268, DOI 10.1090/pspum/046.1/927960.
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(4) 41 (2008), no. 4, 655–668, DOI 10.24033/asens.2078.
MR2489636
Jorge Vit ´orio
Pereira
Credits
Figures 1–5 are courtesy of Jorge Vit´orio Pereira.
Author photo is courtesy of Dayse Haime Pastore.
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1031
EARLY CAREER
The Early Career Section offers information and suggestions for graduate students, job seekers, early career academics
of all types, and those who mentor them. Krystal Taylor and Ben Jaye serve as the editors of this section. Next month’s
theme will be Publishing and Presenting Mathematics.
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Building
Community
and Keeping
Momentum
Finding and Creating
Community in
Your Department
Alan Chang and Rachel Greenfeld
Having a network of people in your circle can benet your
career and well-being. A sense of community can make
people happier and therefore more productive. Academi-
cally, community can impact the quality of one’s research,
for example, through gaining exposure to different elds
and creating collaborations between people working in dif-
ferent areas, as well as merging perspectives of different in-
For permission to reprint this article, please contact:
reprint-permission@ams.org.
Alan Chang is an assistant professor of mathematics at the Washington Univer-
sity in St. Louis. His email address is alanchang@wustl.edu.
Rachel Greenfeld is an assistant professor of mathematics at Northwestern Uni-
versity. Her email address is greenfeld.math@gmail.com.
DOI: https://doi.org/10.1090/noti3009
dividuals. Mathematicians at every stage of their career can
contribute to the community and atmosphere in their de-
partment. While it takes some effort and initiative to build
relationships, the rewards are well worth the effort. In this
article, we share our experience and advice on nding and
creating community.
Tea time, common room, lunches. There are various
ways to create opportunities for people in the department
to interact and get to know each other. For instance, if
your department already has a weekly or daily tea time, go
to those! You might be surprised by how meaningful at-
tending them can be. Often we get to meet new people
during tea time and get exposed to other areas of research
while talking with them. The atmosphere at tea is usually
very friendly and welcoming and gives one a sense of be-
longing. If your department does not have a regular tea
time, you can ask about organizing one. There is a good
chance that your colleagues will be willing to help set one
up.
In addition to tea time, one can use lunchtime to in-
teract with colleagues. How about inviting a colleague to
have lunch together in the common room? Once, when
I was assigned to teach a class I hadn’t taught before, I
found who taught it the previous term and asked if they
wanted to have lunch together sometime. They happily
agreed, and this ended up being a very fun and productive
lunch. My colleague shared several useful tips regarding
the class I was going to teach and we talked about other
things as well. We had many more lunches together af-
terwards! Asking for teaching advice is just one of many
excuses to invite a colleague for lunch.
There are many more ways one can take advantage of
the common room in the department to interact with oth-
ers. In some places, such as Princeton University, profes-
sors and TAs even hold their ofce hours at the common
room, and often people feel comfortable participating in
the discussions that emerge although they are not neces-
sarily taking or teaching the class.
In general, don’t be afraid to speak up, introduce your-
self, and play an active role in initiating social events. You
never know when something you have to say might posi-
tively inuence or inspire another person.
Research seminars and colloquia and their dinners.
Another way to meet people in the department is to attend
seminars. I recommend regularly attending at least the re-
search seminar closest to your eld, even if some of the
topics seem far from your research. In addition to being
1032 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
Early Career
exposed to research, this gives you the chance to see your
colleagues regularly, and can help you feel like part of a
research group.
At many places, there will be a seminar lunch or din-
ner before/after the talk. These are great opportunities to
get to know people in the department. If you regularly at-
tend the seminar dinner for your research area, then you’ll
get to know people by seeing them often. At my depart-
ment, everyone is welcome to attend the dinner. Some-
times younger members of the department are shy and
need some encouragement—I do my best to encourage
people to attend.
Many departments have colloquiums, which are in-
tended for a broader audience. In contrast to seminar din-
ners, colloquium dinners are a great way to meet people
working in other elds. As a postdoc, I made the mistake
of not attending colloquium dinners until my very last se-
mester. Once I started attending, I had meaningful conver-
sations with department members, and I made some good
friends whom I only wished that I had more time to get to
know.
Working seminars / “What is. . . ?” seminars. While
research seminars often have outside speakers, it is also
good to have a way for people in the department to give
presentations to each other. Such seminars with internal
speakers are an excellent way to encourage the creation of a
community. There are many ways these kinds of seminars
can be structured, and we give some examples below.
Last semester at WashU, the analysis group had a weekly
working seminar with no particular topic—each week,
someone gave a presentation on a paper they were read-
ing. At the end of each meeting, we had everyone in the
room go around and share what they’ve been working on.
This created accountability and opened up a way for us to
learn about what others are doing. This semester, based
on the suggestion of graduate students, we are trying a dif-
ferent format: we chose a textbook to work through, and
people are taking turns giving presentations every week.
Another type of seminar that encourages the creation
of a community is a “What is... ?” seminar. At IAS, we
have such a weekly seminar. The idea of the seminar is
for a speaker to explain a basic concept in their eld to
a general audience. One of the main goals of the semi-
nar is for the people of the department—graduate students,
postdocs and faculty—to get to know each other through
their mathematics. Such a seminar has the potential to
take one’s research in new directions by making new tech-
niques and concepts available. You might end up collabo-
rating with a colleague who works in a different eld and
build new bridges in mathematics!
Math chats (Q&A) with faculty members. The WashU
math department runs an in-person “math chats” series,
which was started by graduate students. The series allows
students to get to know the professors on an academic
and personal level, and also to learn from faculty mem-
bers’ experiences and perspectives. Each meeting features
one professor, who gives a short introduction, and then
spends the rest of the time answering graduate students’
questions. When I participated as a professor, the stu-
dents asked me questions about many things—about my
research, my grad school experiences, my hobbies, my fa-
vorite Pok´emon, advice about collaboration, etc. In turn,
I asked the students many questions as well, so we all got
to know each other better.
Group chats. A way to easily reach your peers is to cre-
ate a group chat (WhatsApp, Discord, Slack, etc.). For ex-
ample, if you are a grad student, you could create one for
all grad students, or just your year. I have used these groups
to organize social events or just to share memes.
Side note: I’ve found group chats to also work very well
in small conferences, e.g., AMS sectional meetings. Mes-
sages sent in these groups have ranged from “Any plans
for dinner tonight?” and “[X] and I are hanging out in the
hotel lounge. Come join us!” to “Does anyone know how
to use an iron? I need to iron my shirt.”
Conclusion. To conclude, there are many ways to nd a
community in your department and initiate activities that
would encourage social interactions among people in your
department. Some departments already have various inter-
active events set up; if yours does not, whether you are a
student, postdoc, or faculty, don’t be afraid to take the ini-
tiative and organize one of the activities mentioned above,
or anything else you have in mind. It only takes one per-
son to make a meaningful change in the department.
Alan Chang Rachel Greenfeld
Credits
Photo of Alan Chang is courtesy of Washington University in
St. Louis/Sean Garcia.
Photo of Rachel Greenfeld is courtesy of Dan Kamoda, Insti-
tute of Advanced Studies.
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1033
Early Career
Transitioning as an
Early-Career Mathematician
Rosemarie Bongers
I am a mathematician. I majored in mathematics and
physics in college, where I fell in love with the abstrac-
tion and patterns that gave me a sense of universal truth. I
found my area of geometric measure theory while I was in
graduate school, and I continued to learn harmonic anal-
ysis as a postdoctoral lecturer. I discovered a passion for
education, curriculum development, and pedagogical re-
search around this time, leading to a teaching postdoc and
eventually a teaching professorship that I currently hold. I
am also a transgender woman who came to this personal
truth about herself at the age of 29.
For me, being transgender is a source of joy and the
deepest happiness that I’ve felt; it’s also a source of strug-
gle, challenge, discrimination, and emotional labor. This
identity and awareness permeate every aspect of my life,
my career, and my living in American society in 2024. My
goal here is to explain some of the surprising things I’ve
learned through my transition and to discuss how these
discoveries intersect with me being an early-career mathe-
matician. Although every transgender identity is unique,
we are every bit as diverse and complicated as non-trans
folks. I hope that what I write here will resonate with other
genderqueer people. I also hope to give a bit of insight into
the trans experience, and help spark conversations about
how to support the trans community in a time when the
community is the focus of political debate and frequently
subjected to extreme discrimination and legal restrictions.
0.1. Terminology. As with any mathematical paper, it’s
important to get the key denitions correct. I am a transgen-
der woman who was assigned male at birth. This means that
when I was born, someone made a choice to check the little
“M” box on my birth certicate and kicked into operation
a whole set of legal and social mechanisms—all without
asking me for my thoughts. For most people, this is okay
and these folks are called cisgender, or just cis; all it means is
that your internal sense of self matches how society treats
you. If you’ve never really thought about these things or
had a complaint with your assignation, then that’s pretty
typical. On the other hand, my internal sense of identity is
female, regardless of what decisions were made more than
30 years ago. This means that I am more at peace, more
fullled, and just plain happier living as a woman.
Rosemarie Bongers (she/they) is an assistant teaching professor at UC Merced.
Her email address is rosemariebongers@ucmerced.edu.
DOI: https://doi.org/10.1090/noti3006
Some important notes need to be mentioned here.
First of all, gender is different from sexuality. Gender
is about our internal sense of self, whereas sexuality is
about who we want to have relationships with. Many
transgender people are straight, gay or lesbian, asexual,
or on any variety of spectra of desire for relationships.
Secondly, gender itself is a broad spectrum that incorpo-
rates all sorts of expression (both along the traditional
male/female binary as well as directions of being agen-
der). Thirdly, there is a huge amount of diversity even
in terms of transgender identities: after nding out that
one identies as being trans, some people dress differently;
some change their pronouns (e.g., from he/him/his to
she/her/hers or they/them/theirs); some use medical inter-
ventions such as hormone replacement therapy or gender-
afrming surgery; some do nothing. In any case, the spec-
trum of trans identities is every bit as rich, complicated,
and messy as cis identities—so it’s always dangerous to as-
sume that you know a person’s thoughts or feelings just
from knowing their gender identity or which pronouns
they use.
Likewise, it’s important to discuss pronouns and lived
names. Pronouns are not the same as gender; many peo-
ple use multiple sets of pronouns (such as he/they) and
this may differ from a stereotypically masculine or fem-
inine presentation. Likewise, many trans people change
their name, and it’s common to use it as a lived name for
a long time before it is legally changed (if ever! Remem-
ber that not everyone does this). I do not refer to these as
preferred pronouns or names because they truly are lived;
if a trans person tells you their name or pronouns, listen
to them and use them. Being misgendered or deadnamed
(i.e., unwillingly referred to by your previous name) can
be profoundly insulting and painful experiences that erase
our identities and tell us that our truths do not matter.
When we make mistakes, it’s frequently best to address
them, apologize for them, and move past them.
Finally, to emphasize: every transgender life and experi-
ence is different. I speak for myself and myself alone; there
are many ways in which I experience privilege (e.g., as a
white, able-bodied, native English speaker in academia)
and many ways in which I do not. I do not speak
for all transgender people, and there are so many trans
authors—especially those from indigenous, BIPOC, and
two-spirit communities—whose work should be read, dis-
cussed, and elevated.
0.2. My story: the joy of being trans. Although I didn’t
have the terminology to describe it then, many of my old-
est memories revolve around wanting to be a girl—as if it
was something that I could aspire to. Due to the social
expectations and religious pressures in my childhood, I
chose to suppress this. To act “normal” and to t in as best
as I could. But suppressing the complicated parts of your
1034 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
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life also mean that you don’t get to fully experience the
happy, exciting, or awe-inspiring parts either; it just emo-
tionally deadens you. I also didn’t know that being gay,
or queer, or transgender was something that was okay; as
with many others who grow up in a religious or socially
isolated home, I didn’t know any out queer people until I
was late into graduate school.
My feelings about being out of place and of trying to
play-act as a man pushed me into deep depression and anx-
iety, and I didn’t even know what was wrong. I compare
this to the feeling of a left-handed person who was forced
to write with their right hand; something feels wrong, and
you don’t quite get how everything comes so easily to other
people, and you’re never quite as good at it as the others. It
took the isolation of the pandemic (between online teach-
ing, not seeing family or friends for months, and the con-
stant background of fear) to nally break me and force me
to face the deepest source of my unease in the world. I
came out rst to myself, then to my partner, then to some
of my closest friends. Work and family followed shortly
after that and I began to transition.
I am incredibly grateful that my partner is one of the
kindest, most caring people that I know; her support kept
me alive through some incredibly dark times. I am also
proud of the community of friends that I was a part of;
without them, I could never have started to live my au-
thentic truth. As I initially came out, I was also deeply
fortunate to work in a department where many of my col-
leagues were queer and the culture was open, validating,
and supportive. There was never any doubt that I would
be anything less than an equal, valued, and trusted col-
league and friend. At the time, I didn’t really know any
other transgender mathematicians; but believing that at
least within my department I would be validated and loved
made all the difference.
A few months later, I began the process of medical
transitioning. As I grow through this experience, I am so
deeply grateful that I could start the process. I have no
regrets about my transition, save that I did not come out
sooner. As people have started to naturally see me the way
I am on the inside, I feel calmer and increasingly at peace.
There are so many moments of joy that derive from this
and from the simple feeling of relaxation in my body.
In many ways, I am now “fully” socially transitioned.
I dress as I wish, I act as I wish, I speak as I wish. My
name has been legally changed. For the rst time in my
life, I am unapologetically me. There is still a moment of
joy every time someone calls me “Rose” or “Rosemarie.”
There is still a moment of happiness in seeing myself in
the mirror, being what I dreamed of. Many cis commen-
tators reduce being transgender to dysphoria, that feeling
of the distress from a mismatch between body and mind.
But the counterpart to that is euphoria; the deep fulllment
and happiness and love that I feel when I just get to be me
in the deepest ways possible. Moving through society as
my true self helps me to realize just how much joy there is
in life; to paraphrase the video essayist Abigail Thorn, liv-
ing in my body is nally a source of relaxation and peace.
This is not to say that my life is problem-free; there are
many challenges, difculties, and emotional labors that I
face due to my femininity and identity. But I believe that it
is important to center joy as a core part of the transgender
experience—because it is.
0.3. On coming out. Many portrayals of coming out focus
on it as a singular moment in a queer person’s life—the
moment where they share their identity with the world;
but this is far less complicated than the actual experience.
For many queer people, myself included, it is a years-long
process involving dozens of emotional conversations and
moments of uncertainty. This is doubly true for trans
folks, especially when there are changes in lived names
or pronouns—as there were for me. The inconsistency be-
tween lived name and legal name can forcibly out a per-
son; simply having to show ID or ll out an application
on MathJobs can lead to this; and we have no control over
it.
My coming out process started small. I consider myself
to be the rst person I came out to—because it was a strug-
gle even to admit to myself that I was different. I then came
out to some of my closest friends, then to my coworkers,
then to my family, and then more broadly. Each conversa-
tion is exhilarating and terrifying at the same time because
you don’t truly know how the person you’re telling will
handle you sharing your deepest truth. These are some
of the times in my life when I have felt the most vulnera-
ble: in that moment between saying the words and hear-
ing the response. On the other hand, when people have
responded with joy (one friend in particular screamed in
happiness!), I had some of the highest moments of my life
until now.
But coming out is a huge challenge. Most people are not
transgender, and many people still hold strong prejudices.
There is genuine fear in coming out in the wrong con-
text; transgender people are routinely red, discriminated
against, shunned or disowned, or physically attacked due
to their identities. In the current political environment
where transgender people are the “demon” a major polit-
ical party ghts against, this moment is especially fraught.
In many states, it remains legal to discriminate against
trans people in housing and employment and transgen-
der people face high rates of poverty and marginalization
throughout the United States. This is also a time when
there are dozens or hundreds of new laws that focus on
restricting trans rights and which work to marginalize or
erase the community.
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1035
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0.4. Transitioning on the job market. I was applying for
jobs during the height of my social transition. In the space
between when I submitted my applications for faculty po-
sitions and the time that I interviewed for them, I changed
the name and pronouns that I use. I was also experiment-
ing with new ways of dressing, starting speech therapy, and
learning so much about being myself. It came at a time
of incredible emotional labor but also happiness at being
able to follow my dreams.
I chose to center my transgender identity within my ap-
plication materials. I talked about how being trans had
shaped my perspective in my teaching statement and my
diversity/equity/inclusion statement. I included the pro-
nouns I was using at the time on my CV. It would have
been impossible to read my documents without knowing
that this was a core part of me that shaped my experiences.
I was incredibly nervous about doing this; in a sense, this
was the rst time I was coming out to a broader mathemati-
cal community outside my home department, and I didn’t
know how people would take it. I thought that if I centered
it, I could at least screen out a transphobic department or
university without putting much labor into the process.
Happily, most of my experiences were very positive. In
the job I currently hold, I felt so incredibly supported all
the way through the interview process. The committee
never called me the wrong name or made me feel like
an other. The department chair offered to connect me to
queer faculty on campus who had experience living in the
area. They made efforts to make me feel welcome and safe
and that being transgender was an asset and not a mark
against my application. I feel beyond fortunate to have
the job that I currently hold, and my department’s com-
mitment to me has been apparent throughout my rst year
here. My department and university clearly work to sup-
port people with marginalized identities, and I have found
a loving and accepting queer community that help me to
work in the spaces that I truly care about.
Unfortunately, this is not the case in every department.
As I met search committees, it became painfully clear to
me that for many departments a commitment to diversity
and equity are merely lip service or words on a departmen-
tal webpage. In one on-campus interview, I was repeat-
edly misgendered by the search committee to their faculty.
While this may seem like a small word choice or slip-up,
they are reections of how important these issues are to a
department. The indications that a department and a uni-
versity give can really show the difference between paying
lip service to DEI issues and being honestly committed to
doing the hard work of creating an afrming and safe en-
vironment.
0.5. Transitioning in the classroom. Before I transi-
tioned, I had already been teaching for nearly a decade as
a graduate TA and as a postdoc. I t exactly the stereotype
of a slightly absent-minded mathematician—down to the
plaid collared shirt and cargo pants. This also meant that
students immediately viewed me as competent and intel-
ligent, even when I was only a graduate student; it also
meant that they were unlikely to talk to me as a human
being, to come to me for support, or to show me who they
truly were.
Now I am visibly queer in the classroom; I make a point
of briey coming out on the rst day of class and to build
community around the idea that everyone’s whole self is
welcome. It is important that students can see people of
all gender identities at the front of a math classroom, be-
cause there are students of all gender identities in a math
classroom. One of my proudest teaching moments was at
the end of a semester where a nonbinary student told me
how happy they were to be in my class, because they had
never seen someone like them as a teacher. Visibility and
representation matter.
Teaching as a woman is a vastly different experience for
me than teaching as a man was, and the labor of teach-
ing has become so different and in many ways has in-
creased. Students connect with me much more now, and
are more willing to talk to me about the challenges that
they face. This leads to deeper connections and more im-
pactful teaching, but the unobserved labor of supporting
students through crises or anxiety can have terrible impacts
on faculty mental health and burnout. On a negative note,
students are far more willing to openly doubt my compe-
tence in a eld I’ve taught for years. I face sexist or de-
meaning comments routinely; in a surprise to me, very few
of these have to do with transitioning. The vast majority
come simply from being a woman in a mathematics class-
room.
0.6. The silent labor of being transgender: thoughts
for our allies. While transitioning has been the greatest
source of joy in my life, being trans involves a lot of labor
that does not fall on cisgender colleagues. Depending on
the steps that a person takes in their transition, we may
face dozens or hundreds of hours of legal and administra-
tive work for a name change. Some of us have frequent
medical appointments to manage hormone levels. Some
go through hundreds of hours of painful hair removal,
or months-long recoveries from procedures. While being
deadnamed or misgendered might take a single word on
the part of a colleague, it can have a strong emotional im-
pact of reminding you how you are other and how your
true self is not seen, tolerated, or loved. A political group
demonstrating on campus might be a small distraction for
a cisgender professor, but when such a group is actively
working to criminalize or erase transgender identities, it
is impossible to ignore. Even routine conference travel
can become challenging due to harassment during an ID
check and extra searches from the TSA. Many of us who are
1036 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
Early Career
transgender women also face the burden of simply being
a woman in academia, and have to deal with our opinions
being automatically less valuable and learn what it’s like
to be constantly spoken over. In any case, we face a natu-
rally discriminatory environment where our concerns are
not viewed as important, by default.
Because transgender people are a minority in most aca-
demic spaces (and most spaces within society, period), it is
incumbent upon our cisgender allies to learn how to sup-
port our community. Building an inclusive department or
inclusive pedagogical practice isn’t just a political goal; it
has genuine impacts on people’s lives. It creates a space
where we can all contribute, all bring our knowledge and
creativity, and all explore the beauty of mathematics. As
such, I wanted to share some thoughts for mathematicians
about how you can help:
•Learn about transgender identities by reading and
hearing their stories. Trans identities are beautiful and
diverse and have been a part of humanity since the
beginning. And do remember that every transgender
person is different and none of us can speak for all of
us.
•Remember that we are more than our gender identi-
ties; we are just as complex, fascinating, and messy as
anyone else. Don’t reduce us to just our trans-ness,
and don’t expect every trans person to educate you on
trans issues.
•Calling someone the wrong name or wrong pronoun
is something that unfortunately happens regardless of
best intentions; if you do it, make a quick, sincere, and
direct apology and move on. Don’t use it as an oppor-
tunity to bring up every moment that you’ve misgen-
dered a trans person. Better yet, ask the person how
they want it to be handled.
•If you see someone being called the wrong name or
pronoun, address it. It can be exhausting for that la-
bor to always fall on the queer folks in the community.
This goes for pretty much any kind of discriminatory
behavior.
•Remember that while intent matters, so does impact.
Something might feel minor to you but be emotion-
ally devastating to another person. Be willing to do
the work to address this.
•Intentionally create welcoming spaces where trans
people don’t have to worry if they’ll be safe or com-
fortable. Work with your campus’s LGBTQ+ center or
its analogue to gure out how your department can be
better.
•Think about intersectionality. Although our strug-
gles are different, many of the things that the com-
munity can do to support women, racially marginal-
ized groups, or disabled colleagues, are the exact same
things that the community can do to support transgen-
der people. Inclusive practices are often exactly the
same as good practices.
•The responsibility to address transphobia belongs to
all of us. If you aren’t sure how you can contribute,
then read, learn, and discuss these issues.
0.7. Conclusion. I hope that there are two takeaways that
you’ll keep: that being transgender can be a deeply joyful,
satisfying, and fullling experience in a beautiful life. And
that there is an incredible emotional labor that comes with
being trans or actually transitioning. Our transgender col-
leagues deserve the support of their friends, their depart-
ments, and the mathematical community at large. I know
that I am incredibly fortunate; I have had the privilege to
work within departments that are afrming and am sur-
rounded by a community that loves me as I am. Not every
transgender person is so fortunate, and we, as the math-
ematical community, must do the hard work to bring in
everyone.
Rosemarie Bongers
Credits
Photo of Rosemarie Bongers is courtesy of Rosemarie Bongers.
Do Mathematics Every Day
Daniel J. Thompson
A fact of life in our profession is that we often spend
extended periods of time facing two opposite challenges.
The rst challenge many of us face is multimonth periods
of time spent with extremely full schedules with teaching
commitments, committees, seminars, and the whirlwind
of activity of the academic year. This poses obvious chal-
lenges to our research productivity and schedule. The sec-
ond challenge, which may receive less attention, is long
stretches of time when our schedules are completely open,
and here the challenge is for us to nd ways to organize our
time efciently and be productive. While the second situa-
tion may be considered “optimal” for research and getting
Daniel J. Thompson is a professor of mathematics at The Ohio State University.
His email address is thompson.2455@osu.edu.
DOI: https://doi.org/10.1090/noti3008
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1037
Early Career
a lot done, it certainly comes with its own set of potential
pitfalls. In this article, I want to share some thoughts on
how to keep our mathematical res burning consistently,
and also how to balance an ambitious research schedule
against the need to recharge and to be mindful of our well-
being.
Semester-pocalyse. We often face stretches of time
when we have far more on our plates than we can reason-
ably do. This might be for a stretch of weeks, a couple of
months, or even a whole semester. If you are reading this
as an early career person, I’m sure you can relate, but un-
fortunately this only gets worse with more seniority. So
what can you do to keep your mathematics moving for-
ward when overloaded with myriad other duties? My ad-
vice is deceptively simple.
Do mathematics every day. By this, I mean to con-
sciously set aside time for mathematics every day, and to
do this no matter what else is going on (barring a major
crisis or something really exceptional going on). I used
to do this intuitively—I’ve always enjoyed working in cof-
fee shops, and even when I’ve been really busy I would
often relocate to a coffee shop for an hour and work on
mathematics—the change of scenery would free me from
feeling the need to reply to emails and remove the chance
of interruptions. My caffeine dependency thus led to me
doing consistent daily research even during the busiest of
times!
In recent years, I’ve carved out research time more in-
tentionally and systematically. This came from a break-
through in my bass-playing. I’ve been a dedicated bassist
since I was a teenager, but my playing had been languish-
ing due to a lack of practice for several years (correlat-
ing with starting a tenure-track position and having kids).
What used to be possible for me in terms of nding prac-
tice time was no longer possible. I rebooted my playing
during the pandemic, taking an online course on the Dis-
cover Double Bass platform with jazz bass maestro John
Goldsby, who had this to say about practice: “Practicing 15
minutes a day, every day, is better for your development than
two hours of hectic bass playing every weekend.” This was a
lightbulb moment for me. For years, I’d felt that I was too
busy to practice. The idea that I could make progress by
doing 15 minutes of serious practice each day was a rev-
elation. What kind of crisis would have to be going on
that I couldn’t nd 15 minutes in my day? It would have
to be quite a crisis if I couldn’t nd just 15 minutes! An-
other aspect of this is that 15 minutes is the oor, not the
ceiling—if the going is good, you just keep on going. Over
the course of a year or two, this approach worked wonders
for my playing, and now I’m swinging my way through
jazz standards and burning my way through the bluegrass
songbook. I got to thinking about how to apply this to
mathematics research.
A common mistake that we make in research is the same
one I was making with my music practice. It’s a common
trap to feel like we need a substantial block of uninter-
rupted time to get in the zone and make progress. For
research, maybe we are waiting for a free day or even a
free week so we can really get in the zone. Those are cer-
tainly optimal conditions, and those big blocks of time are
great to have when they’re available. However, it’s impor-
tant to let go of the idea that you need to wait for optimal
conditions to do research. Being able to make progress
in nonoptimal conditions is the way to produce research
consistently over the long-term.
My advice to you during busy periods is to decide how
much research time each day can be a realistic oor for
your research time. Is it one hour? Is it half an hour?
Maybe even carving out two hours each day would be re-
alistic depending on your schedule. The point is to gure
out what can realistically be done EVERY workday based
on your schedule for the semester, and to stick to it as re-
ligiously as possible. This will keep your research moving
along, and when a block of uninterrupted time does open
up, you are ready to make the most of it. Otherwise, when
that uninterrupted block of time comes up, you risk need-
ing most of the time just to get back up to speed on your
projects, and there may be little time remaining to make
further progress.
Let your subconscious do some of the work. A com-
mon misconception is to think that we’re only working
when we’re sitting in front of the computer, or reading a pa-
per, and actively concentrating. In fact, the subconscious
often does a lot of the heavy-lifting. The “active work” of
concentrating on reading a paper, or working through a
calculation, is sometimes most useful as a process of get-
ting the ideas into our head. We may not know how to
solve a problem when we’re consciously trying to do it.
When we step away from a problem we’ve concentrated
hard on, and come back the next day, often our subcon-
scious has been working away in the meantime, and sud-
denly we know what to do next. This is another benet of
the ‘Do math every day’ approach. Each day that you learn
some mathematics, even for a relatively modest amount of
time, your subconscious will continue to work away on it
while you are teaching your classes or busy with other du-
ties.
When not to do mathematics every day. Burnout hap-
pens in our profession. We’re running a marathon, not
a sprint, and it’s important to have the tools to keep on
going indenitely. My previous advice to do math every
day aims to stimulate your subconscious to keep on work-
ing on math even while you’re doing other tasks. Some-
times you just need to rest and to let your mind be quiet.
Without rest, stress becomes cumulative and can lead to
chronic anxiety issues. The classic recommendation by
1038 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
Early Career
psychologists is to give yourself the following periods of
downtime [1]: One hour per day, One day per week, One
week out of every twelve to sixteen weeks.
Taking a full day away from research each week is cer-
tainly important. It’s probably difcult for most people
to take off a full week every quarter; I haven’t been doing
that. However, that is the classic evidence-based advice
from psychologists, and maybe we would all benet from
it when possible. Taking at least a couple of week-long
breaks in the year seems realistic and in the spirit of this
advice. During these times, you want to rest your brain
as completely as possible so you can come back fresh and
strong when you return to work.
Balancing semester duties. When things are really
busy, it is an unfortunate fact of university life that other
tasks squeeze our research time. Our teaching and service
work have hard deadlines that push these duties to the top
of our priority list. To counteract this, it’s good to create
some rm deadlines in our research too. Regular meetings
with collaborators are a great way to provide this, since it’s
natural for us to want to have some progress to report at
the meeting. Giving a talk on your research, or an exposi-
tory talk on an adjacent subject can be another great way
to generate momentum. Deadlines like these can guide
how we use our research time and help us use it efciently.
While adding research deadlines is useful for direction and
motivation, it can be freeing to balance this by occasion-
ally going “rogue” and thinking about some interesting
mathematics even if it is not directly useful for a current
project. For example, learning what a colleague is work-
ing on or spending a little time with an interesting paper
that jumped out at you from the daily arXiv listing is a
great way to stay productive when you need a break from
your main research projects and deadline-oriented work.
If you’re lucky, you might even nd some inspiration for a
great new project.
If service has taken over your schedule to the point
where research feels almost impossible, then I would ad-
vise looking at what you can let go of in the future. Per-
sonally, I always try to complete the service assignments
I committed to as best I can, even if I nd out the hard
way that I took on more than I should. However, if I have
taken on too much, I try to make a change for the following
year. Most things we sign up for, committee assignments
for example, are for a xed amount of time. If you nd
yourself doing an unreasonable amount of service during
the semester, then in most cases it is absolutely reasonable
to let some things go the following year to preserve more
of your time for research. Alternatively, if you anticipate
having an unusually demanding service load, and you’re
unwilling or unable to let anything go, you can request to
teach courses you’ve taught before to reduce time spent on
teaching preparation.
Wide open spaces—summer and sabbatticals. Being
productive on research when we are NOT busy with teach-
ing, committees, etc., is an underrated challenge. Many
of us oscillate between the extremes of our schedules be-
ing completely full and mostly empty as semesters begin
and end, and it can be a challenge to adapt. We often
begin our summers, or a semester without teaching du-
ties, with high hopes for extraordinary research produc-
tivity. Indeed, these are the times when one can really
make a lot of research progress, but these periods can also
be overwhelming, particularly when paired with lofty self-
expectations. The “do math every day” mantra still applies,
but now the challenge is to use your research time well and
also to be mindful about staying healthy.
I recommend creating a research “master plan” for your
summer or semester. Not only should you create a list of
the broad projects that you would like to work on, but
you should write down a couple of pages worth of spe-
cic research tasks, e.g., “understand paper X; polish the
proof of Lemma 3.7; work on the introduction of paper
Y; work on the main technical result needed for project Z,”
and research-related tasks, e.g., “schedule regular meetings
with collaborator A; email expert B asking them a ques-
tion.”
If your to-do list solely consists of “do great math re-
search today,” that will likely be overwhelming, and some
days it will be unclear where to start. We often lose some
perspective and mental clarity if we’ve just spent a couple
of days working on a technical lemma, or doing a Math-
SciNet deep dive into the literature on a topic we’re learn-
ing. When we hit a wall on writing up a technical lemma,
or trying to absorb all the information in a literature search,
it can be valuable to change gears. That is where your to-do
list of specic tasks becomes useful to fall back on. If you
hit a wall with one task, or you’re unsure how to proceed,
just consult your list and pick whatever task looks good
to you to work on that day. Some people go further and
schedule their days into blocks of time for specic tasks. If
that style works for you, then great. Personally, I prefer to
leave my time unstructured, but to use my list of specic
tasks to guide me and keep me on track and productive.
It’s also important to keep up with non-math activities
to help you stay mentally sharp and healthy. This could
be exercise, music, social activities, or just getting outside.
This is a good idea for overall well-being, but it’s also good
for the mathematics itself. Judiciously chosen time spent
on non-math activities helps our subconscious organize
and process all the hard work we do.
Let me nish with a quote from legendary record pro-
ducer Rick Rubin [2]:
Take the example of an album. If you’re a musician
struggling with ten songs, narrow your focus to two.
When we make the task more manageable and focused,
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1039
Early Career
a change occurs. . .. Going from two to three is easier
than going from zero to two. And if you happen to
get stuck on three, then skip it and get four and ve
done. Complete as many elements of the project as you
can without getting hung up... Often the knowledge
we gain from nishing the other pieces becomes a key
to overcoming earlier obstacles.
References
[1] Edmund J. Bourne. The Anxiety and Phobia Workbook, Sev-
enth Edition. New Harbinger Publications, 2020.
[2] Rick Rubin. The Creative Act: A Way of Being. Penguin,
2023.
Daniel J. Thompson
Credits
Photo of Daniel J. Thompson is courtesy of Neal Havener.
Productivity and Time
Management in Research
Steven Senger
I’m relaxing in a coffee shop overlooking Ngo
.c Kh´anh Lake
in Hanoi, sipping the last of my ba
.c xiu, an absurdly de-
licious coconut cream coffee drink, as a summer-long re-
search program at the Vietnam Institute for the Advanced
Study of Mathematics draws to a close. I’m reecting a bit
on how I got here, and how I can keep producing enough
mathematics to hopefully get asked back someday. I came
into our trade by accident, as many of us do. One of my
college buddies happened to be a mathematician, and be-
cause I was an electrical engineer by training, I gured I
should be able to at least understand what some of the big
ideas were. Even though math had never been my focus be-
fore, I registered for a graduate program in engineering and
applied mathematics at the same school where I had n-
ished my undergrad without really thinking much about
Steven Senger is an associate professor at Missouri State University. His email
address is stevensenger@missouristate.edu.
DOI: https://doi.org/10.1090/noti3011
what that would mean for my future; I really just enjoyed
school and wanted to keep learning.
Despite feeling like I was a few steps behind in my math-
ematical foundation, I always thought I had some useful
physical intuition because of my engineering background,
particularly when it came to Fourier analytic techniques.
Partially due to an all too common blend of pride and self-
doubt, when I started graduate school, I spent more energy
on traveling around to play music and go rock climbing
than on studying. While I did ne in my coursework, I
proceeded to get soundly outclassed by my fellow students,
who seemed to cut through research roadblocks by apply-
ing ideas from this or that paper that I’d never heard of.
“So the support isn’t compact? That’s ne, we’ll just con-
sider this Radon-Nikodym derivative. Oh, you’re worried
about some pesky symmetries? Just get a bound using the
orbit-stabilizer theorem.” In hindsight, this is all standard,
but when I was just getting started, I was completely over-
whelmed by the relative ease with which my compatriots
navigated the same challenges that kept me up all hours of
the night.
I am extremely fortunate to have a world-class advisor,
Alex Iosevich, who was careful to meet me where I was
in my research journey, and instead of chastising me for
not studying hard enough, he merely gave me objectively
reasonable research problems, and when I couldn’t solve
them, I worked out for myself that I needed to decide what
was important and prioritize accordingly. I played fewer
shows and spent less time climbing but I still kept a bal-
ance. Slowly, the tricky calculations began to seem rou-
tine, and I got a better sense for which barriers were just
better left for another day. So instead of a series of brilliant
epiphanies, I got a feel for how to make slow and steady
progress on a number of fronts and increase my personal
chances of producing a steady stream of research ideas.
However, during my postdoc, I learned that this focus-
ing has limits. I pretty much ditched my hobbies entirely,
and put everything into mathematics. Somehow, even
though I was dedicating more time and energy than ever
to research, all of my projects seemed to grind to a halt.
So I worked even harder, applied for grants, designed and
taught a well-received graduate topics course, and ended
up developing a severe anxiety disorder. After a partic-
ularly nasty attack I passed out in my ofce and took a
ride in an ambulance. I think it’s funny that after working
jobs in skydiving, rock climbing, music, and even being a
bouncer for a while, my most dangerous professional haz-
ard was in mathematics. I am so grateful to have a solid
network of people to support me, and while I still have
issues, I have learned to not work myself so hard all the
time.
With these adventures, I found good approximate up-
per and lower bounds for how hard I can work and still
1040 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
Early Career
produce. However, as soon as I got a handle on life after
graduate school, the game changed again with my tenure-
track appointment in 2014. I am very lucky that I got my
top choice, Missouri State University. It’s a state school,
close to my family, with research programs similar to my
own. The thing is, once you’re a faculty member, the com-
mittee meetings and teaching preparation balloon up with-
out warning. Okay, I heard plenty of warnings to guard my
time, keep my pedagogical expectations reasonable, and
not to let too many committees hop on my plate, but I
was an alacritous new member of the team, and wanted
to prove my worth! Luckily, my mentors both within the
department and beyond were careful not to let me bite off
more than I could chew. After a few semesters, I began to
appreciate their wisdom more personally.
Of course, teaching and service tasks took precious time
and energy. One of the pitfalls for the budding researcher
is working hard for a day or a week, or whatever, and feel-
ing so accomplished with their hard work on teaching, ser-
vice, and just surviving, that research can seem to be a lux-
ury task for another day. The thing I’ve noticed is that the
longer breaks I take between serious research pushes, the
harder it is for me to get moving forward again. It’s al-
most like exercising. If you skip your exercise routine too
many times, it gets much harder to get it moving again. So
my two-pronged approach to research praxis is consistency
and variety.
For me personally, I really need to think about research
for a good chunk of time at least once a week, or I am
likely to lose whatever framework for a problem I had in
my head, as well as atrophy my creative reasoning abilities.
During busier parts of the semester, this may be as sim-
ple as rereading notes from previous weeks, or going back
through the deltas and epsilons of a current manuscript.
A while back, my friend Paul Baginski suggested that we
meet up once a week over Zoom to discuss our (very dif-
ferent!) mathematics research programs. Even though he
and I don’t collaborate on manuscripts, I feel a responsibil-
ity to have something new to tell him each week. This gives
me a deadline for my research, which keeps me motivated
during the week. Teaching, service, and other activities
tend to have deliverables with regular deadlines: grading
student quizzes within one week, nishing the report for
the ad hoc subcommittee, or even submitting an abstract
for an upcoming talk. For better or worse, the deadlines
in research are often exible, and the expectations can get
nebulous at best. As a result, they can sometimes get lost
in the shufe. How many times have you heard, “I’m still
thinking about that,” or, “This week I’ve really been trying
to read,” from a collaborator in a research meeting? Meet-
ing regularly with Paul forces me to have concrete things
to say each week, even if the concrete thing I say is, “I failed
to get research done this week.” When I have to acknowl-
edge it out loud, it reminds me to prioritize it in the week
to come.
While the idea of regularly attacking a problem makes
sense in an abstract, maybe even romantic way, the real-
ity of slamming your head into the same expression every
time you sit down to do research can be punishing at best.
So I try to make sure that I always have a curated selection
of active projects, with difculty/feasibility ranging from,
“simply turn the crank and the student gets a good start on
a paper,” through, “this is fun, but I don’t know how soon
we’ll nish,” all the way to seriously difcult problems that
I mostly study for fun. My goal is to make sure I have a
moving pool of ideas that I can use as vehicles to shep-
herd students through their rst projects, but that I can
occasionally spin off into serious research papers. Then I
separately have hard conjectures that I can try new ideas on
after working through some of the more standard projects.
The hope is that these feed and support one another, so
that simpler cases come out of hard problems to provide
projects for students, and the surprising pieces that come
out of student projects can give me new angles of attack
on harder problems. This way, I can have a fairly steady
stream of papers, but I never feel like I’m just beating the
same dead horse.
To put this in perspective, here is a sketch of some of
my current research program. At the top, I have a few in-
sights on the celebrated Erd˝os unit distance problem (How
often can the most common distance occur in a large -
nite point set in the plane?) that I haven’t yet seen written
down anywhere, so I can always entertain myself by push-
ing on those ideas. I certainly think about it every day,
but I maybe push on it in earnest once a week. Now, this
particular problem hasn’t seen concrete progress since the
eighties, so I don’t put all my eggs in that basket. There are
a number of related problems that have seen some trac-
tion recently, and I see these as a safer investment of my
time and energy. For example, instead of a single distance
determined by a pair of points, we could consider mul-
tiple distances determined by 𝑘-tuples of points. This is
still challenging, but not so hopeless, so I can publish pa-
pers on problems like these with my colleagues. If we get
started proving things, and it seems like we’ll mostly be
applying standard techniques to get results, then I’ll invite
some students along for the ride, and I can put more en-
ergy into setting up an environment where they can read
some papers, learn some things, and prove some of the
theorems.
Of course, when I do have time to push hard, I like to go
to a coffee shop, put on headphones, and melt into a prob-
lem. The background bustle of the shop keeps me from get-
ting distracted by silence, and any music that I can’t sing
along to will prevent my ears from latching onto any pass-
ing conversations. Everyone has different preferences, but
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1041
Early Career
I think it is important to experiment and get a sense for
where you can be the most effective for each task. For ex-
ample, I can proofread a draft of a manuscript just about
anywhere, like while resting between sets at the climbing
gym, but I have a hard time coming up with new ideas un-
less I am really insulated from distractions. When I don’t
have the best conditions for hard thinking, but want to
be creative, I like to type up notes. Even if a particular idea
didn’t work with one problem, I can sometimes recast that
idea successfully in a different context, and that’s much eas-
ier to see when you’ve clearly elucidated it in a beautifully
typeset document.
I also think that many creative pursuits follow natural
inhale/exhale cycles. Sometimes I’m taking more ideas in,
either by reading papers that seem interesting on arXiv, try-
ing to brush up on this or that topic that I haven’t seen
in a while, or even just refereeing papers. I nd that I
don’t write as much or have as many new ideas during
these times, but I perceive that they give me the fuel for
the opposite end of the cycle, which is when I’m grinding
out the last few lemmas on a manuscript that should have
been submitted weeks ago, trying to pull together a new
big project with some collaborators, or just playing around
with undergrads and nding connections that I hadn’t sus-
pected. I nd that these times last longer and have more
output that I am proud of directly after a longer, harder in-
put phase. So I don’t get particularly disturbed by a week
or two without a concrete breakthrough, assuming that I
have managed to digest or resurrect some signicant ideas.
I guess what I’ve come to believe after almost twenty
years of working on research is that I can’t force success,
but I can get a sense for what conditions make me more
likely to make new connections and communicate them
effectively. It is really like art in that I won’t necessarily
know when something wonderful is about to come out,
but I know nothing will come out if I don’t at least show
up and consistently push on a variety of ideas.
Steven Senger
Credits
Photo of Steven Senger is courtesy of Riley McCullough.
How Does Your Daily Life
Change When You Become
the Graduate Coordinator?
Chun-Kit Lai
When I got tenured, I was recommended to become a
graduate coordinator in my University (San Francisco State
University, SFSU). This has been by far the most satisfying
departmental service I have ever done. My life has become
much busier after taking on this role in addition to doing
research and teaching, but the experience and the outcome
have been rewarding and unforgettable.
The ultimate degree that SFSU provides is a master’s in
mathematics. Every year, we attract a diverse pool of ap-
plications from different ethnicities, geographic locations,
and ages (even retired people apply to our graduate pro-
gram!). The rst job of the graduate coordinator is an-
swering all the questions raised by these applicants. To
save time in replying to these emails, it is important to
make sure the department website is informative and at-
tractive. The next major task is reviewing these applica-
tions after our rst priority deadline (the deadline that
we will start reviewing our rst pool of applications, but
we will still accept applications after this deadline), which
could take a couple of days. All these applicants have differ-
ent expectations and qualications, but a common theme
is that they have a passionate heart toward mathematics
and mathematics teaching. They expect our program to
ll their hearts. When I see them accept the offer and later
come to my ofce in person, I regard them as joining our
family. We will teach them advanced mathematics, but
indeed their presence also changes how we deliver our lec-
tures and shape our curriculum.
Another major task as graduate coordinator is to help
them choose the right courses and nd an advisor for their
thesis. There are set courses our students can choose from
and there are also exible electives for our students to take.
Whenever a student comes for advice, it is important to
listen carefully to what they need. In particular, I will try to
gain some insight into what their mathematical interest is.
I then direct them to my colleagues whose interests overlap
with theirs. It is also my job to make sure that they are on
track to nish all required courses.
Graduate students in SFSU normally stay for only two
years, some three years. They are like a close family doing
homework together, talking about their research projects
together, having fun together, and nally graduating to-
gether. After their graduation, some continue on to PhD
Chun-Kit Lai is an associate professor at San Francisco State University. His
email address is cklai@sfsu.edu.
DOI: https://doi.org/10.1090/noti3010
1042 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
Early Career
programs, some become lecturers in community colleges,
some work in industry and some continue to enjoy their
retirement. We may or may not have time to get together
again, but our students all leave with a satisfying memory
of their life at SFSU. I have signed more than 50 gradu-
ation forms in my ve years of service. Most of our stu-
dents come in with minimal undergraduate mathematics
background, but they graduate with a thesis on a cutting-
edge research topic. Seeing their growth and their success
is the most fullling experience in this position of graduate
coordinator.
Chun-Kit Lai
Credits
Photo of Chun-Kit Lai is courtesy of Sin Yee Chau.
Dear Early Career
I had been working on a paper for some time, and a
colleague gave me a few ideas and suggestions which
have improved the paper quite a bit. Should I offer
coauthorship?
—Confused author
Dear Confused author,
Firstly, congrats on the interesting developments
in your research! Your question is delicate, and there
is no one-size-ts-all answer, but I can say that in my
experience being generous with offering coauthor-
ship has been a good decision in the longer term.
However, one should only do this when appropri-
ate.aEssentially my guideline is that if the results
and their proofs would be present without the discus-
sions with your colleague, and the discussions have
streamlined the presentation, then including an ac-
knowledgement might be more appropriate.bHow-
ever, if the discussions have led to signicant changes
to the main results or the structure of the mathemat-
ics, then it would be benecial for you to have a
conversation with the colleague as to whether they
would like to be included in the paper as a coauthor.
If you have been working on the paper for a while,
this can be a difcult decision to arrive at, as you
may feel like the effort level has not been equitable.
It is benecial to have colleagues who are generous
with their ideas, and you do not know how much ef-
fort they have made in arriving at their suggestions,
and so thinking in this manner can be not an accu-
rate reection of “work done.” In my experience, ob-
servations of colleagues which at the time felt like
they came out of nothing, were, in fact, modica-
tions of ideas they had attempted to execute in dif-
ferent contexts several times previously. Their obser-
vations were really the result of thousands of hours
of work.
Assuming the observations can be separated from
the text without disturbing the ow of the exposition,
a common alternative is to keep the authorship of
the paper as it is, but to include the observations in
a coauthored appendix. This is particularly suitable
if
1. the conversations have led to applications of
your main results that you did not envisage,
but have not modied the primary results them-
selves, or
2. if you need a modication of a “standard” tech-
nical tool from another area, and the colleague
has helped you with this due to their expertise
in that area.
Finally, I can also attest to the fact that if you offer
an important tool for people to use in a paper, only
to not be invited to be a coauthor, then you can be
a bit weary of communicating with them going for-
ward. Consequently, if you value the potential col-
laborative future with this colleague, then being gen-
erous with offering coauthorship at this stage can be
hugely benecial for you.
—Early Career editors
Have a question that you think would t into our
Dear Early Career column? Submit it to Taylor
.2952@osu.edu or bjaye3@gatech.edu with the
subject Early Career.
DOI: https://doi.org/10.1090/noti3007
aThe AMS Ethics Guidelines, which broach coauthorship briey,
may be found here: https://www.ams.org/about-us
/governance/policy-statements/sec-ethics.
bBeing generous with acknowledgments is a good policy. If some-
one has helped you with any technical aspect of the paper (be
it mathematics, references, or typesetting), then acknowledging
their assistance is important to show that you valued their con-
tribution. Additionally, if you are giving a seminar talk and you
arrive at a particular moment when someone’s guidance helped,
you should feel free to give them a shout-out.
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1043
Eli Goodman (1933–2021)
and Ricky Pollack (1935–2018)
J´anos Pach, Micha Sharir, Noga Alon,
and Andreas Holmsen
Twin Primes: Eli Goodman
and Ricky Pollack
J´anos Pach
In my eyes, Eli (Jacob) Goodman and Ricky Pollack were
inseparable. Exactly when and where they rst met was a
matter of discussion between them. Was it in the alcoves
of City College, playing chess or in the NYU library, lis-
tening to recordings of classical music? Eli’s father was a
well-known secular Jewish scholar, who published exten-
sively in Yiddish and English. Ricky was one of the “red
diaper babies,” his parents were Communists, constantly
harassed by the authorities. Both of them were passion-
ately interested in mathematics, in music, and in literature.
Both of them played the piano, Eli at a semiprofessional
level. After attending the Bat Mitzvah of one of Eli’s daugh-
ters and listening to the band Klezmatics, Ricky started tak-
ing clarinet lessons from David Krakauer. He would not
travel anywhere without his clarinet. Eli went even further:
he got a degree in composition and he cofounded the New
York Composers Circle. His pieces were performed by lead-
ing musicians and were recorded.
Both of them had brilliant supervisors, but they did
not have an easy start in mathematics. Eli’s supervisor
Communicated by Notices Associate Editor Emilie Purvine.
For permission to reprint this article, please contact:
reprint-permission@ams.org.
DOI: https://doi.org/10.1090/noti2997
J´anos Pach is a professor of mathematics at HUN-REN R´enyi Institute of Math-
ematics, Budapest, Hungary and ´
Ecole Polytechnique F´ed´erale de Lausanne,
Lausanne, Switzerland. His email address is pach@cims.nyu.edu.
Figure 1. Eli and Ricky, the main organizers of the 2008
Discrete Geometry Conference in Oberwolfach, Germany.
was Heisuke Hironaka, who later got the Fields Medal
for groundbreaking discoveries in algebraic geometry. For
many years, Eli worked relentlessly on a conjecture that
turned out to be false. His favorite teacher at NYU was
Harold N. Shapiro, a number theorist, who collaborated
with Paul Erd˝os and Richard Bellman, and loomed large
in mathematical circles. Ricky became his student. They
spent a lot of time together. It was easy to learn from
Shapiro, but difcult to shine next to him. By the mid-
seventies, both Eli and Ricky were ready to venture into a
new eld that they felt was their own.
They were lucky: during their sabbaticals they hit on
roughly the same subject and shortly afterward they found
out about their common interest. At McGill University,
Montreal, Willy Moser told Ricky about the happy ending
problem of Erd˝os, Esther Klein, and George Szekeres, and
he almost instantly got obsessed with it. Is it true that any
set of points in general position in the plane has
1044 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
elements that form the vertex set of a convex -gon? If yes,
this bound would be best possible. We still do not know
the answer to this question, but a few years ago Andrew
Suk showed that points always sufce. Eli had
been searching for a simple geometric question that can
be approached by encoding the underlying congurations
and translating the problem into a purely combinatorial
one. The happy ending problem appeared to be a perfect
candidate. They started to explore these ideas by introduc-
ing (rediscovering) the notions of order types and allow-
able sequences. They did not get any closer to the proof of
the Erd˝os-Szekeres conjecture, but the approach quickly
yielded fruit. They proved Grünbaum’s conjecture that ev-
ery set of eight pseudolines (two-way innite curves, any pair of
which cross precisely once) is stretchable [7]. They continued
to make progress related to a number of other questions
raised in Branko Grünbaum’s classic 1972 treatise, Arrange-
ments and Spreads [14]. However, the most elegant early ap-
plication of the method of allowable sequences was found
by Peter Ungar [18], a legendary problem solver of Hungar-
ian origin. He proved that any set of points in the plane,
not all of which are collinear, determine at least distinct direc-
tions, provided that is even. His argument was reproduced
by Aigner and Ziegler in their popular volume Proofs from
THE BOOK. Erd˝os often said as a joke that God kept a book
with only the most elegant mathematical arguments, and
he rarely allows anyone to have a glance into it.
In 1980, Ricky and Eli started a geometry seminar at
Courant Institute (NYU) which was attended by faculty
and students of many universities from the Greater New
York area, including Rutgers, Princeton, Columbia, CUNY,
Stony Brook, and Pace University, and by researchers from
Bell Labs, AT&T, and IBM. Over the years, when passing
through the Big Apple, almost all important gures work-
ing in combinatorics, discrete geometry, computational
geometry, or convexity gave a talk in this seminar. The
abstracts of these talks were widely circulated. In the pre-
internet era, anyone following these announcements had
a pretty good overview of the most exciting new develop-
ments in our eld. It was in this seminar that Peter Ungar
learned about allowable sequences, which enabled him to
prove his above-mentioned theorem on directions, origi-
nally conjectured by Scott. Twenty-ve years later, Rom
Pinchasi, Micha Sharir, and I managed to settle Scott’s
problem in three dimensions [15]. All three of us attended
the meetings of this seminar for years. I also had the priv-
ilege of co-organizing it in the rst decade of the 21st cen-
tury.
In the beginning, it was not clear whether such a sem-
inar would ever y. As Joe Malkevitch recalls, Ricky
doubted if anyone would show up if they “put out a shin-
gle.” Yet people did come, and they came in ever grow-
Figure 2. J ´anos Pach, Ricky, and Eli at the International
Conference on Intuitive Geometry, Si ´ofok, Hungary, 1985.
ing numbers. Why? It is hard to deny that the charis-
matic personalities of Ricky and Eli played a big role in this.
They picked the right speakers and fascinating topics, and
they had a good nose for signicant new developments in
the subject. Sometimes they were wrong, especially Ricky,
who easily fell in love with a new problem. But this only
added to the thrill of novelty and discovery. The topics
covered in the seminar have opened up new avenues of re-
search for most participants, including many established
senior mathematicians and computer scientists. A touch
of luck has also contributed to the remarkable success of
the seminar. The early 1980s witnessed an explosive surge
in computing power, which resulted in a sustained appreci-
ation for algorithmic techniques, an appreciation that has
only grown stronger over time. NYU had a Robotics Lab,
codirected by Jack Schwartz and Micha Sharir, who laid
the mathematical foundations for motion planning. As
recognition of their work, in 1986, they were invited to
speak at the International Congress of Mathematicians in
Berkeley. Many practical questions such as the so-called
piano movers’ problem, visibility and ray-shooting prob-
lems raised deep questions about arrangements of points,
lines, curves, convex sets, and other geometric objects in
Euclidean spaces. It turned out that some closely related
questions, with deep ties to number theory, functional
analysis, discrete geometry, and information theory, had
been investigated before by Gauss, Hilbert, Minkowski,
Fejes T ´oth, Rogers, Conway, Erd ˝os, Lov´asz, Spencer, Sze-
mer´edi, Trotter, and others. Many of their results proved
to be applicable in the design of efcient geometric algo-
rithms. A new eld of computational geometry was born.
It was also popularized by Ron Graham and Frances Yao’s
concise and elegantly written survey, titled “A whirlwind
tour of computational geometry,” published in the Ameri-
can Mathematical Monthly.
The number of graduates in computer science far sur-
passed the number of mathematics graduates. The new
generation of computer scientists were equally well-versed
in discrete mathematics and geometry as their counter-
parts in mathematics. They were familiar with the happy
ending problem and the probabilistic method (or the
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1045
random sampling technique, as it was called in computer
science), they learned about the Szemer´edi-Trotter theo-
rem on the number of incidences between points and lines
and Lov´asz’s theorem on halving lines. They not only
knew about these results, but soon improved on them!
In particular, in a seminal paper presented at the 2nd An-
nual Symposium on Computational Geometry in 1986,
David Haussler and Emo Welzl borrowed a technique for
set-systems of bounded Vapnik-Chervonenkis dimension,
and applied it to a wide range of questions in geometry.
This paved the way for a series of new discoveries, includ-
ing far-reaching extensions of the Szemer´edi-Trotter the-
orem and a substantial improvement to Lov´asz’s upper
bound on the number of halving lines (by Clarkson, Edels-
brunner, Guibas, Sharir, and Welzl, and by Dey, all of
whom are computer scientists!) The relationship between
discrete geometry and computational geometry proved to
be mutually benecial and resulted in remarkable break-
throughs on both sides.
It was perhaps Ron Graham, who rst suggested that
the time was ripe to launch a journal devoted entirely
to discrete geometry. He must have talked to some pub-
lishers, as in 1984, Cambridge University Press, Wiley,
and Springer-Verlag all expressed interest in such a project.
Ricky and Eli negotiated with all of them. Forty years ago,
scientic publishing was a completely different business
than what it is today. Almost all mathematics editors held
PhDs. They were deeply embedded in the mathematics
community, they had a good sense of scholarly quality and
commercial value. Striking a balance between the two was,
of course, crucial, but their primary goal was the advance-
ment of science. Finally, the project was embraced by the
late Walter Kaufmann-Bühler from Springer, about whom
Ricky and Eli always spoke with the greatest admiration.
Condent that the marriage of the classical subject of dis-
crete geometry and the newly emerging eld of computa-
tional geometry would be fruitful and long-lasting, they
named the new journal Discrete & Computational Geometry
(DCG). Through their seminar, the organization of numer-
ous conferences, and their groundbreaking mathematical
and editorial work, Ricky and Eli nurtured a thriving com-
munity that kept the subject and the journal as fresh and
vibrant as ever. After all these years, it still lls me with
pride to have contributed two papers to the inaugural issue
of Discrete & Computational Geometry in 1986 and to have
had the honor of serving for several years, alongside Ricky
and Eli, as coeditor-in-chief of the journal they founded.
Ricky and Eli made many important discoveries in dis-
crete and computational geometry, convexity, geometric
transversal theory, and real algebraic geometry. (Some of
their achievements will be mentioned below, by Micha
Sharir, Noga Alon, and Andreas Holmsen.) However, they
Figure 3. Participants of the first Computational Geometry
Conference at Bellairs Institute of McGill University,
Holetown, St. James, Barbados, in 1986. Micha Sharir in the
middle, Ricky above him, in red shirt, at the top.
always considered their most important legacy to be the
creation of the journal and a large, friendly community of
researchers around it. In their own ways, both of them
were fundamentally social creatures, for whom mathemat-
ics, as Ricky’s son, Danny, once put it, was an “intensely
social enterprise.” They were born in New York, and un-
til the very last years of their lives, they both lived in New
York. They could not imagine moving anywhere else. New
York was their natural, cultural, and mathematical habitat.
Shortly before his death, Eli completed his excellent novel,
which has just appeared in print [5]. The protagonist is a
professor of mathematics from New York City, who myste-
riously disappears from his Manhattan apartment. From a
short article published in the Times, one can learn that
He was last seen there three weeks ago at a party in his
honor, but failed to show up for his classes the following
Monday. A police spokesman indicated that no signs of
disturbance were found in his apartment and that no
correspondence has turned up that might indicate his
whereabouts.
Eli and Ricky have disappeared from the New York scene,
but their huge social and professional footprints are des-
tined to linger for generations to come.
1046 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
Figure 4. Participants of the Monte Verità Conference on
Discrete and Computational Geometry, in Ascona,
Switzerland, 1999. In the middle, Ricky, wearing a red striped
shirt. Eli is the 4th from the right.
A Tribute to Ricky Pollack
and Eli Goodman
Micha Sharir
Ricky Pollack and Eli Goodman were, in many aspects, the
founding fathers of discrete and computational geometry,
as a thriving, active, and mainly interactive research area.
Before turning to their scientic achievements, I would like
to highlight two aspects of their leadership and inuence
on the eld, which were not emphasized in the previous
contribution.
A. The Discrete and Computational Geometry Confer-
ence series. Computational Geometry was a young eld
in 1986 when Eli and Ricky launched the journal DCG.
Discrete Geometry had been around for several decades,
but bringing the two elds together was to a large extent
the work of Eli and Ricky. Alongside the journal, they si-
multaneously launched a conference called Discrete and
Computational Geometry, in Santa Cruz, CA, which had a
huge impact in dening and directing the eld, as a com-
mon discipline. It was so successful that they continued
the tradition, by organizing two follow-up conferences,
one at Mt. Holyoke, MA (Discrete and Computational Ge-
ometry Ten Years Later, 1996), and one at Snowbird, UT
(Discrete and Computational Geometry Twenty Years Later,
2006). A fourth conference, Discrete and Computational
Geometry Thirty Years Later, was organized (by others) at
Monte Verita, Switzerland, in 2016.
B. The books. Each of them contributed a major book
in this area. Eli (with Joe O’Rourke) edited the Handbook
of Discrete and Computational Geometry [6], a monumental
1500-page collection of papers surveying all aspects of the
eld. It later mushroomed into an even bigger collection
(1937 pages), with Csaba T´oth as a third editor.
Micha Sharir is a professor emeritus of computer science at Tel Aviv University,
Tel Aviv, Israel. His email address is michas@tauex.tau.ac.il.
Ricky was a coauthor, with Saugata Basu and Marie-
Fran¸coise Roy, of another extremely inuential book, Al-
gorithms in Real Algebraic Geometry [2], which I will discuss
later.
The major networking ventures undertaken by Eli and
Ricky had an enormous effect on the community, as
did their technical contributions to discrete and compu-
tational geometry. They spanned many topics, including
order types and allowable sequences, Helly-type results, al-
gorithms in real algebraic geometry, and a variety of appli-
cations in computational geometry.
A common thread in many of their works is the use of
topological considerations in the analysis of structures in
discrete geometry. Most notably, they looked for topolog-
ical generalizations of standard concepts, such as pseudo-
lines instead of lines, topological planes, and more. Per-
haps the rst example of such studies is their groundbreak-
ing work on allowable sequences, which I will mention
shortly.
In the remainder of this note I would like to combine a
brief review of some of the major achievements of Eli and
Ricky with illustrations of how these have inuenced my
own research. I divide the discussion into four themes.
1. Allowable sequences and order types. Take a set of
points in the plane, and project it onto a line . In gen-
eral, we get a sequence of distinct points on . As
we rotate , the sequence does not change combinatori-
ally, except at certain critical orientations of , at which a
block of consecutive elements of , or several blocks si-
multaneously, collapse into points and then are reversed.
This evolution of as rotates is called an allowable se-
quence; see [11]. Many interesting properties of can be
deciphered out of its allowable sequence, but the most in-
triguing question is whether a given allowable sequence
(a sequence that obeys the evolution rule given above) is
realizable, that is, whether it comes out of an actual set
of points. The answer is that most sequences are not
realizable, and that deciding whether a given sequence is
realizable is PSPACE-complete (a computational complex-
ity term, meaning, roughly, “intractable”). This however
was not known at the time when they were working on
the problem, and they were very excited about nding an
effective solution to the decidability problem. As a matter
of fact, when I rst met Ricky in 1982, he enthusiastically
gave me a “research announcement” (as these things were
called those days) where he and Eli obtained an effective
(albeit, sadly, wrong) solution.
Let me switch to the related topic of order types, which
generalizes these concepts to higher dimensions. For
example, in the plane, the order type of a set of
points species the orientation (left turn, right turn, or
straight) of every ordered triple of points of . Order types,
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1047
introduced by Eli and Ricky, can be regarded as a con-
cise purely discrete way of representing the “essence” of
point congurations or, dually, of arrangements of lines
(in the plane) or of hyperplanes (in higher dimensions).
In fact, in this dual setting, order types can naturally be de-
ned for arrangements of more general curves and surfaces,
most notably for arrangements of pseudolines and pseudo-
hyperplanes. Again, the question of the realizability of or-
der types was a major topic of study. It was another mani-
festation of their interest in how basic discrete and combi-
natorial properties can be studied in a purely topological
context, that had motivated many of their joint works.
Let me mention a fairly fresh result (Comput. Geom.,
2023), which is based on Eli and Ricky’s pioneering work
on order types. We studied subquadratic algorithms for
some 3Sum-hard geometric problems in the algebraic de-
cision tree model. These are problems that are at least as
hard as the 3Sum problem: determine whether a set of
real numbers has a triple that sums to zero. These prob-
lems include collinearity testing, i.e., determining whether
a set of points in the plane has a collinear triple. This
problem is not known to have a subquadratic solution in
the standard real-RAM model, and our work gave such a
solution, for a restricted kind of collinearity testing (also
known to be 3Sum-hard), in the algebraic decision tree
model, where we only count algebraic sign tests involving
the input data.
We worked in the dual plane, where we have a set of
lines (or other curves), and we want to preprocess the
arrangement for fast point location. This is of course
a well known problem, which can be solved with
storage and query time, using line sweep and per-
sistent search trees. However, to achieve this performance,
one needs, among other things, to sort the vertices of
by their -order, and each comparison in this sorting in-
volves four input lines, two for each of the two vertices
that are being compared. For our application, we wanted
to obtain algebraic comparisons involving the input data
that depend on a smaller number of elements, and for that
it was crucial to reduce the number of lines involved in a
comparison. The theory of order types was the tool that
we needed. The order type information gives us the or-
der of the vertices of along each line of , and each
comparison that these sortings perform involves only three
lines. This seemingly unimportant difference was crucial
in improving the running time of our algorithm. This is
just one, personal application, among many others, of the
beautiful theory of order types; see [11].
2. -sets. Eli and Ricky’s papers on this topic have opened
up a rich area of research on -sets in congurations of
points, and of levels in arrangements of curves and sur-
faces, in the plane and in higher dimensions. A -set of a
Figure 5. Ricky, Eli, and Peter McMullen at the
AMS-IMS-SIAM Summer Research Conference on Discrete
and Computational Geometry, Santa Cruz, CA, July 1986.
set of points in the plane, say, is a subset of size that
can be cut off its complement by a half-plane. In a dual
setting, the -level in an arrangement of a set of lines
(or other curves) in the plane, say, is the set of all vertices
and edges of the arrangement of that have exactly
lines below them.
What Eli and Ricky showed was that the number of
at-most--sets, namely the overall number of -sets, for
, is , which is an asymptotic worst-case
tight bound; another proof with a tighter bound was given
later by Alon and Gy˝ori. As it turned out, this notion plays
a crucial role in the analysis of randomized algorithms
in computational geometry, and in many other computa-
tional and combinatorial problems in geometry; such as
the celebrated probabilistic analysis technique of Clarkson
and Shor.
3. Hadwiger-type theorems and geometric permuta-
tions. Hadwiger’s theorem gives a necessary condition for
a nite collection of pairwise disjoint convex sets in the
plane to have a line transversal (i.e., a line that crosses of
all of them): If there is a linear ordering of the sets such
that every triple of sets is met by a directed line in the
corresponding order, then the entire collection has a line
transversal. In a remarkable work, Eli and Ricky extended
this result to arbitrary dimensions, giving a condition for
the existence of a hyperplane transversal in terms of the
multidimensional order type of the input sets, replacing
the one-dimensional sorted order.
Eli and Ricky were also interested in line transversals
in higher dimensions; see [13]. A line transversal to any
collection of disjoint convex sets meets all the sets in a
given order or its reverse, depending on the direction of
the transversal. This pair of orders (permutations) is called
ageometric permutation. The study of geometric permuta-
tions, mainly to derive upper and lower bounds on the
number of such permutations, took off from their pio-
neering work, and I have been involved in some of these
1048 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
studies. There are still many open challenges to better un-
derstand the structure of geometric permutations.
4. Algorithms in real algebraic geometry. A real semi-
algebraic set is a region in that is dened by a Boolean
combination of a nite number of polynomial equalities
and inequalities. Given such a set , how can we process
it algorithmically? How do we determine whether is
empty? Compute its connected components? Find a point
in each connected component? These and many other ba-
sic problems in Computational Real Algebraic Geometry have
been studied by Ricky, together with his colleague Marie-
Fran¸coise Roy and his student Saugata Basu. This work cul-
minated in their monumental book Algorithms in Real Alge-
braic Geometry [2], which has become, in a sense, the bible
of this area, containing all the basic tools, techniques, and
algorithms in computational real algebra and algebraic ge-
ometry. It is remarkable that the book is publicly available
via open access, as demanded by the authors.
I would like to nish by mentioning some of my joint
work with Eli and Ricky, most of which are only with Ricky.
An exception is a work with both of them on the space
of hyperplane transversals to a family of separated and
strictly convex sets in , where we show that the maxi-
mum combinatorial complexity of this space is .
A main feature of the analysis, mostly contributed by Eli
and Ricky, is the analysis of the topology of the space of
common tangents, of a special kind, of a collection of such
sets. It is yet another manifestation of their interest in
studying topological aspects of discrete geometry.
Of my work with Ricky, I would like to mention one
on counting and cutting cycles of lines in space. This was
a notoriously difcult problem, which was solved much
later, where the goal was to break the lines in a set of
lines in into the smallest (or, at least, a small) num-
ber of pieces, so as to eliminate all the depth cycles be-
tween them. The newly derived upper bound is close to
, which is nearly tight in the worst case. However, back
when the paper with Ricky appeared, only very partial re-
sults were known. The paper was accompanied by a paper
of Pach, Pollack and Welzl, in which they showed that a
pattern of lines in space cannot be completely weav-
ing, namely that it is impossible for each line to alternate
between passing above and below the lines in the other
set in order. To experiment with this nding, they went
out to buy some toy sticks to physically test how they can
weave. Without thinking too much, they naturally bought
sticks... .
Several other works with Ricky are on quasi-planar
graphs, on arrangements of Jordan arcs with three intersec-
tions per pair, and various problems on simple polygons.
All this goes to show that Ricky and Eli were very curious
and open-minded, and were interested in basically every-
thing. Working with them was fun and very inspiring.
The community at large, and I personally, are still reel-
ing from the loss of both of them, and sorely miss their
leadership and great science, not to mention friendship.
Eli Goodman, Ricky Pollack,
and Semivarieties
Noga Alon
The friendship and collaboration between Eli Goodman
and Ricky Pollack has been rare and productive, spanning
decades of joint work and including the foundation of a
leading journal and the organization of meetings and an
active research seminar. Their joint papers stimulated a
considerable amount of follow-up work. In this section,
we focus on one of their beautiful contributions and de-
scribe some of its many subsequent developments.
The basic idea appears in a remarkable short note [9],
where Goodman and Pollack observed that a theorem of
Milnor in real algebraic geometry can be used to provide
an elegant nearly tight asymptotic estimate for the number
of (simplicial) polytopes with vertices in . Their ap-
proach paved the way to a signicant amount of additional
results in combinatorics, discrete and computational ge-
ometry, and related areas, obtained by applying powerful
tools from real algebraic geometry. It is natural to specu-
late that the background of Goodman in algebraic geome-
try demonstrated by his inuential early work in the sub-
ject [3] helped in the early development of this approach.
Below I describe the background to Goodman and Pol-
lack’s work on convex polytopes, followed by their results,
and two research directions inspired by their work.
1. Connected components and sign patterns. There are
several results that provide upper bounds for the num-
ber of connected components of real varieties or semi-
varieties. Following such estimates by Ole˘ınik–Petrovski
(1949), Milnor (1964), and Thom (1965), Warren [19]
proved that if , and ,
is a set of polynomials of degree at most in real vari-
ables, then the number of connected components of the
semivariety
for all
is at most .
Noga Alon is a professor of mathematics at Princeton University, Princeton,
New Jersey, and Tel Aviv University, Tel Aviv, Israel. His email address is
nalon@math.princeton.edu.
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1049
For each point , the sign pat-
tern of the polynomials at the point is the vector
signsign . Let
denote the total number of distinct sign patterns of the
polynomials , as ranges over all points of . Since
the sign of each polynomial cannot change in any con-
nected component of , it follows that for ,, and
as above, . In applications it is
sometimes desirable to bound the number of sign pat-
terns signsign, where here
ranges over all points of (including those in which
some of the polynomials vanish). It is not difcult to
show (see [1]) that the result of Warren implies that this
number does not exceed .
2. Counting polytopes and congurations. Let
denote the number of (combinatorial types of) -
polytopes on labeled vertices and let denote the
number of simplicial -polytopes on labeled vertices
(that is, polytopes in which all facets are simplices). The
problem of determining or estimating these two functions
(especially for -polytopes) has been the subject of much
effort and frustration of nineteenth-century geometers as
described, for example, in the book Convex Polytopes by
Grünbaum. Despite these efforts, for any (and
large ) the best known upper bound for both and
has been exponential in . This estimate
follows from the upper bound theorem for convex poly-
topes. The remarkable result of Goodman and Pollack
[8, 9] improved it dramatically to a bound exponential in
. They started by bounding the number of order
types of congurations of (labeled) points in , dened
in what follows.
If is a sequence of points in , with
for each , we say they have a positive ori-
entation if the determinant of the matrix where
for each , is positive. If the determinant is nega-
tive they have a negative orientation, and if the determi-
nant is they lie on a common hyperplane. The order
type of a conguration of labeled points
in is a function from the set of all -subsets of
to , where for with
is or according to
the orientation of the points .
Let denote the number of distinct order types of
congurations of labeled points in . Note that
is the number of sign patterns of
polynomials of
degree in the real variables ,,
which are the coordinates of the points. The polynomials
are just the determinants det , where
for all and .
Therefore, the estimate of Warren (and its slight exten-
sion for the total number of sign patterns) shows that
.
This immediately supplies a similar bound for the num-
ber of -polytopes on points. Indeed, the order
type of a conguration that spans determines which
sets of its points lie on supporting hyperplanes of its con-
vex hull. Hence, the order type of a conguration on a set
of points in which is the set of vertices of a convex
polytope determines its facets and its complete combi-
natorial type.
3. Signrank. The sign-pattern of an by real matrix
with nonzero entries is an by matrix
of entries where sign . For an
by matrix of entries, let denote the minimum
possible rank of a matrix such that . Dene
is an by matrix over .
The problem of determining or estimating , and
in particular , was raised by Paturi and Simon in
the early 80s, motivated by the study of the so-called
unbounded-error probabilistic communication complex-
ity of a Boolean function of bits. Alon, Frankl, and
Rödl (cf. [1]) proved in 1985 that
and that if and then
The lower bounds in both estimates are derived from
the estimate of Warren by a simple counting argument.
4. Semialgebraic properties. A graph property is any fam-
ily of graphs closed under isomorphism. Such a family
is called semialgebraic if every vertex is a point in a real
space of bounded dimension, and the adjacency of two ver-
tices is determined by the signs of a nite set of bounded
degree polynomials in the coordinates of the correspond-
ing points. This can be extended to hypergraphs, but for
simplicity we focus here on the case of graphs. Natural spe-
cial cases of such properties are intersection graphs of sim-
ple geometric objects, like segments or disks in the plane,
boxes in and more. The speed of a family is the func-
tion , where is the set of all graphs with
vertices in the family. The results about the number of
sign patterns of polynomials described here imply that the
speed of any semialgebraic family of graphs satises
, where is a constant that depends
on the dimension and the degrees of the polynomials in
the denition of the property. Examples can be found in
[1] and the references therein. A result of Sauermann [17]
shows that under mild conditions the estimate obtained
for the constant by applying Warren’s theorem
is tight. Besides their modest speed functions, it turns out
1050 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
that semialgebraic graph properties are simpler than gen-
eral families of graphs in many respects. The study of their
Ramsey properties and the investigation of additional ex-
tremal questions for such families received a considerable
amount of attention in the last decade. It will surely keep
being the subject of future research, like other topics initi-
ated by the work of Eli Goodman and Ricky Pollack.
Eli Goodman, Ricky Pollack,
and Geometric Transversals
Andreas Holmsen
One branch of combinatorial geometry in which the work
of Eli and Ricky had a tremendous impact is what we call
geometric transversal theory. This line of research, an off-
shoot of Helly’s theorem, was initiated in the 1930s by Vin-
censini and Santal´o, and explored further in the 50s and
60s by a number of prominent geometers such as Grün-
baum, Hadwiger, Klee, and Danzer. The famous survey,
“Helly’s theorem and its relatives,” gives a detailed account
of the state of affairs in 1963, and motivated further study
throughout the 70’s and 80’s.
Eli and Ricky’s 1988 paper “Hadwiger’s transversal theo-
rem in higher dimensions” stands as one of the milestones of
geometric transversal theory. Their beautiful result related
the (at the time) novel notion of order types to the classical
study of geometric transversals, and paved the way for re-
search directions in discrete and computational geometry
which bear fruit to this day.
The study of geometric transversals originated with
Helly’s theorem, which asserts that for a family of at least
compact convex sets in , if every mem-
bers can be intersected by a point, then the entire family
can be intersected by a point. Can a similar theorem be
true if the property “intersected by a point” is replaced by
“intersected by a line,” or a plane, or more generally a -
dimensional afne at?
This was the problem, posed by Vincensini in 1935, that
initiated the study of geometric transversals, but it did not
take long before Santal´o realized that no such “Helly-type”
theorem can exist for -ats when . While this situa-
tion may seem somewhat discouraging, it did not prevent
further study of geometric transversals. Indeed, Santal ´o
showed that if we restrict ourselves to families of axis par-
Andreas Holmsen is a professor of mathematics in the Department of Mathemat-
ical Sciences, KAIST, Daejeon, South Korea, and is a member of the Discrete
Mathematics Group, Institute for Basic Science (IBS), Daejeon, South Korea.
His email address is andreash@kaist.edu.
1
2
3
4
Figure 6. The line transversal induces the ordering
.
allel boxes in , then the “Helly number” for line transver-
sals becomes , and for hyperplane transversals
it becomes . In fact, there has been extensive
work on geometric transversals which investigates Helly-
type theorems under various restrictions on the geometric
shapes of the sets in the family.
Rather than focusing on the geometric shape of the
sets, Hadwiger’s approach has a more combinatorial a-
vor. Suppose a nite family of pairwise disjoint convex sets
admits a line transversal. By orienting this line, it induces
an ordering on the family, namely the order in which it
meets the sets. In particular, any three members of the
family are met by a line which is consistent with the given
ordering. (See Figure 6.) What Hadwiger showed is that,
in the plane, this obvious necessary condition is also suf-
cient:
Theorem. A nite family of pairwise disjoint convex sets in
the plane admits a line transversal if and only if there exists a
linear ordering of such that every three members of are met
by a line consistent with the ordering.
By the mid 1980’s, Eli and Ricky had been investigating
order types of point congurations in for several years,
when it dawned on them that the linear ordering in Had-
wiger’s transversal theorem was simply a one-dimensional
order type. They noticed that by making a bijection be-
tween a nite set and a point conguration in , the order
type of the point conguration induces what they called a
-ordering of the set, which for is precisely a linear or-
dering. This meant that they had just the right tool to gen-
eralize Hadwiger’s transversal theorem to higher dimen-
sions! All that was missing was the right analog of pair-
wise disjointness, and the natural condition they found
was to dene a family of at least convex sets in to
be -separated if no of them admit a -
transversal. In particular, being -separated means that
no two members have a 0-transversal, i.e., the members
are pairwise disjoint. A consequence of the denition is
the following: If a -separated family of convex sets
in admits a -transversal, then by choosing one point
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1051
from each set within the given -transversal, we obtain a
point conguration in , and the order type of this con-
guration is independent of the choice of points. In this
way, a -transversal naturally induces a -ordering of the
family. These observations led Eli and Ricky to their cele-
brated generalization of Hadwiger’s theorem:
Theorem ([10]).A nite -separated family of convex
sets in admits a hyperplane transversal if and only if there
is a -ordering of such that every members of
are met by a hyperplane consistent with the -ordering.
Another important development was made a couple
years later by Wenger, a PhD student of Ricky’s, who was
able to remove the disjointness assumption from Had-
wiger’s transversal theorem. This requires a clarication
of what it means for a line to intersect a family of convex
sets consistently with an ordering, since such an ordering
may no longer be uniquely determined. Wenger showed
that it sufces for some choice of points to be consistent
with the ordering, and shortly after, Pollack and Wenger
extended this to higher dimensions as well. By an elegant
proof from “the book,” which combines order types, com-
binatorial convexity, and the Borsuk–Ulam theorem, they
proved what we now call the Goodman–Pollack–Wenger
theorem:
Theorem ([16]).A nite family of convex sets in admits
a hyperplane transversal if and only if for some ,,
there is a -ordering of such that every of the sets are
met by some -at consistent with the -ordering.
Over the years there have been a number of further
generalizations and extensions of Eli and Ricky’s break-
through result. Some of the highlights include:
Anderson and Wenger (1996): Replaces the -
ordering by the more general concept of an acyclic ori-
ented matroid.
Arocha et al. (2003): Shows that not only do we get
a single hyperplane transversal, but in fact “many” of
them, captured by what they call a virtual -transversal.
Arocha et al. (2008): Gives a colorful version of Had-
wiger’s transversal theorem in the spirit of the B´ar´any–
Lov´asz “colorful Helly theorem.”
Cheong et al. (2023): Proves the colorful version of
the Goodman–Pollack–Wenger theorem conjectured
by Arocha et al.
McGinnis (2023): Establishes an analog of the
Goodman–Pollack–Wenger theorem for hyperplane
transversal in .
For nearly two decades, Eli and Ricky (with various col-
laborators) continued working on geometric transversals,
exploring Helly-type theorems, the topological structure
and combinatorial complexity of the space of transver-
sals, and convexity on the afne Grassmannian. Their sur-
vey [13] joint with Wenger, documents the explosion of
work in geometric transversal theory that had taken place
in the years following their breakthrough papers on the
generalizations of Hadwiger’s transversal theorem.
In the paper “Foundations of a theory of convexity on afne
Grassmann manifolds,” Eli and Ricky asked whether there
is a convex hull operator, conv, on the space of -
dimensional afne ats in , which naturally extends the
standard convex hull operator for points, and satises gen-
eral properties such as: monotonicity,idempotence,antiex-
change, and afne invariance. (For precise denitions see
[12] or the expository article [4].)
Their solution was as natural as the question: Fix an
integer . For a set of -ats in , dene its
dual,, to be the family of all convex (point) sets which
meets every at in . For a family of convex (point) sets,
dene its dual, , to be the set of all -transversals to the
family . Now dene the convex hull of a set of -ats
in to be its double dual, that is, conv.
It turns out that this notion of convexity satises the
four properties stated above, and indeed, when restricted
to the case , it reduces to the standard convexity. For
a rich theory emerges which is closely tied to cen-
tral questions in geometric transversal theory, and their pa-
per explores some interesting examples ranging from rul-
ings on a hyperboloid to certain Schubert varieties. In fact,
many sophisticated constructions and counterexamples in
geometric transversal theory can be traced back to this con-
vexity structure.
Today geometric transversal theory is an active area of
research. In the last decade, we have witnessed an emer-
gence of new and exciting directions motivated by recent
trends in discrete and computational geometry, as well as
developments on research problems dating back to Eli and
Ricky’s seminal work in the area.
References
[1] Noga Alon, Tools from higher algebra, Handbook of com-
binatorics, Vol. 1, 2, Elsevier Sci. B. V., Amsterdam, 1995,
pp. 1749–1783. MR1373688
[2] Saugata Basu, Richard Pollack, and Marie-Fran¸coise Roy,
Algorithms in Real Algebraic Geometry, 2nd ed., Algorithms
and Computation in Mathematics, vol. 10, Springer-Verlag,
Berlin, 2006. MR2248869
[3] Jacob E. Goodman, Afne open subsets of algebraic varieties
and ample divisors, Ann. of Math. (2) 89 (1969), 160–183,
DOI 10.2307/1970814. MR242843
[4] Jacob E. Goodman, When is a set of lines in space con-
vex?, Notices Amer. Math. Soc. 45 (1998), no. 2, 222–232.
MR1601812
1052 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
[5] Jacob E. Goodman, The Mathematician, FriesenPress, Al-
tona, Canada, 2023.
[6] Jacob E. Goodman and Joseph O’Rourke (eds.), Hand-
book of Discrete and Computational Geometry, 2nd ed.,
Discrete Mathematics and its Applications (Boca Ra-
ton), Chapman & Hall/CRC, Boca Raton, FL, 2004, DOI
10.1201/9781420035315. MR2082993
[7] Jacob E. Goodman and Richard Pollack, Proof of Grün-
baum’s conjecture on the stretchability of certain arrange-
ments of pseudolines, J. Combin. Theory Ser. A 29 (1980),
no. 3, 385–390, DOI 10.1016/0097-3165(80)90038-2.
MR600606
[8] Jacob E. Goodman and Richard Pollack, Upper bounds for
congurations and polytopes in , Discrete Comput. Geom.
1(1986), no. 3, 219–227, DOI 10.1007/BF02187696.
MR861891
[9] Jacob E. Goodman and Richard Pollack, There are asymp-
totically far fewer polytopes than we thought, Bull. Amer.
Math. Soc. (N.S.) 14 (1986), no. 1, 127–129, DOI
10.1090/S0273-0979-1986-15415-7. MR818067
[10] Jacob E. Goodman and Richard Pollack, Hadwiger’s
transversal theorem in higher dimensions, J. Amer. Math.
Soc. 1(1988), no. 2, 301–309, DOI 10.2307/1990918.
MR928260
[11] Jacob E. Goodman and Richard Pollack, Allowable se-
quences and order types in discrete and computational geometry,
New Trends in Discrete and Computational Geometry, Al-
gorithms Combin., vol. 10, Springer, Berlin, 1993, pp. 103–
134, DOI 10.1007/978-3-642-58043-7_6. MR1228041
[12] Jacob E. Goodman and Richard Pollack, Founda-
tions of a theory of convexity on afne Grassmann man-
ifolds, Mathematika 42 (1995), no. 2, 305–328, DOI
10.1112/S0025579300014613. MR1376730
[13] Jacob E. Goodman, Richard Pollack, and Rephael
Wenger, Geometric transversal theory, New Trends in Dis-
crete and Computational Geometry, Algorithms Com-
bin., vol. 10, Springer, Berlin, 1993, pp. 163–198, DOI
10.1007/978-3-642-58043-7_8. MR1228043
[14] Branko Grünbaum, Arrangements and Spreads, Confer-
ence Board of the Mathematical Sciences Regional Confer-
ence Series in Mathematics, No. 10, American Mathemati-
cal Society, Providence, RI, 1972. MR307027
[15] J´anos Pach, Rom Pinchasi, and Micha Sharir, Solution
of Scott’s problem on the number of directions determined
by a point set in 3-space, Discrete Comput. Geom. 38
(2007), no. 2, 399–441, DOI 10.1007/s00454-007-1344-5.
MR2343314
[16] Richard Pollack and Rephael Wenger, Necessary and suf-
cient conditions for hyperplane transversals, Combinator-
ica 10 (1990), no. 3, 307–311, DOI 10.1007/BF02122783.
MR1092546
[17] Lisa Sauermann, On the speed of algebraically dened graph
classes, Adv. Math. 380 (2021), Paper No. 107593, 55, DOI
10.1016/j.aim.2021.107593. MR4205109
[18] Peter Ungar, noncollinear points determine at least
directions, J. Combin. Theory Ser. A 33 (1982), no. 3, 343–
347, DOI 10.1016/0097-3165(82)90045-0. MR676751
[19] Hugh E. Warren, Lower bounds for approximation by non-
linear manifolds, Trans. Amer. Math. Soc. 133 (1968), 167–
178, DOI 10.2307/1994937. MR226281
J ´anos Pach Micha Sharir
Noga Alon Andreas Holmsen
Credits
Figure 1 is courtesy of Archives of the Mathematisches
Forschungsinstitut Oberwolfach.
Figure 2, Figure 3, Figure 5, and photo of J´anos Pach are cour-
tesy of J´anos Pach.
Figure 4 is courtesy of Emo Welzl.
Figure 6 and photo of Andreas Holmsen are courtesy of An-
dreas Holmsen.
Photo of Micha Sharir is courtesy of Micha Sharir.
Photo of Noga Alon is courtesy of Noga Alon.
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1053
BOOK REVIEW
The Mathematician
Reviewed by Thomas Garrity
The Mathematician
FriesenPress, 2023, 276 pp.
By Jacob E. Goodman
One summer day in 2008, a
younger colleague of mine from
the Williams Chemistry Depart-
ment looked somewhat frazzled.
When asked why, she explained
that her tenure packet was due the
following Friday, and that she was
desperately trying to nish two
more papers, so that on her vita the papers could be listed
as “submitted.” This made perfect sense to me. Only in
December 2008 when the good news came that this col-
league did indeed get tenure (a decision that I considered
a no-brainer), did I briey rethink our summer conversa-
tion. First, there is no way that having two more papers
listed as submitted would push anyone over the tenure bar.
But also, what if this colleague had only submitted these
papers a few days later, after the tenure packet was due,
but still listed them as submitted? Who would have ever
known? As far as cutting corners go, it would not have
been a huge deal. But at the time, my chemistry colleague
never even considered this as an option.
Suppose, though, we were not in the summer of 2008
in Massachusetts but instead in 1937 Germany, and the
young academic was not a chemist but a young mathemati-
cian who is socially awkward and self-centered with a Jew-
ish sounding name (and possibly with actual but hidden
Thomas Garrity is the Webster Atwell Class of 1921 Professor of Mathematics
at Williams College. His email address is tgarrity@williams.edu.
Communicated by Notices Book Review Editor Emily Olson.
For permission to reprint this article, please contact:
reprint-permission@ams.org.
DOI: https://doi.org/10.1090/noti3002
Jewish roots). What would be mere “cutting corners” to-
day could have had life and death consequences then. We
are in the world of this novel.
This novel is an historical mystery. Like most mysteries,
part of the charm is not the mystery itself but the mileu in
which the mystery occurs. The author, the late Jacob Good-
man (see the memorial tribute in this issue), was most def-
initely a serious research mathematician, and hence this
mystery is set in the world of academic mathematics. More
specically, the mystery shifts in the rst half between the
Columbia University math department of 1967 and Ger-
man academe of the 1930s while the second half shifts
between the math worlds of the 1930s and 1990s.
The mystery is about the sudden disappearance in 1967
of Claus Eisenstadt, in the evening after a small congrat-
ulatory party in honor of his receiving a type of “lifetime
achievement” award from his long-term school Columbia.
We quickly learn that Eisenstadt arrived at Columbia as
a Jewish refugee from Germany during the early years of
World War II, but before Pearl Harbor. His reputation
stemmed from two papers written right before his depar-
ture from Germany, papers that were critical in the devel-
opment of algebraic topology. These papers were signif-
icantly different then his few earlier papers, which were
simply not of the same caliber. In the ensuing years, Eisen-
stadt publishes a few more signicant papers, usually with
a coauthor, but overall his career does not live up to the
promise hinted at in the two topology papers.
We also quickly learn that Eisenstadt is quite eccentric,
and not the lovable, charming type. In fact, in the very rst
paragraph of the novel, at the small party in his honor, we
see him smashing the camera of a young journalism stu-
dent who had the audacity of both trying to take his pic-
ture and to interview him about his award for the student
paper. And the very next day, Eisenstadt was gone.
As would be expected, this type of behavior leads to
intense interest and gossip in the Columbia math depart-
ment. In particular, Judy Carter, a second-year graduate
1054 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
Book Review
student, becomes almost obsessed (within reason) with
understanding who Eisenstadt really was; where he went;
and why he vanished.
Judy to a large extent represents a typical mathemati-
cian. She was the star math student as an undergraduate
at a small school in the Midwest. On arrival at graduate
school, she quickly nds herself at home in the cultural
and intellectual hub of the Upper West Side of Manhat-
tan around Columbia. By her second year, she starts to
hear the siren song of the Grothendieck revolution in al-
gebraic geometry. In the second part of the book, when
we shift from 1967 to the midnineties, we see her as a suc-
cessful midcareer research mathematician, long tenured at
City University, with the standard sort of informal network
of colleagues spread throughout the world. And it is sug-
gested her success in mathematics is linked to certain ob-
sessive tendencies, as she still wants to crack the mystery
of Eisenstadt.
We also get to see the mathematical journey of Eisen-
stadt in 1930s Germany, from his undergraduate days to
what would now be called postdoc years. There are small
intense seminars and interactions with “big” names such
as Felix Hausdorff, Emil Artin (his thesis advisor), and
Emmy Noether. There was a continual emphasis by his
mentors and professors on the creation of new ideas, lead-
ing to understandable pressure on Eisenstadt to show that
not only could he synthesize existing mathematics (even at
a high level) but he could also discover new mathematics.
This pressure is still present today. In fact, I seem to re-
call the graduate school insult (biting because of how true
it often was) that someone was merely a “good student,”
with the implication of course that they could learn but
not create mathematics.
There also appear other easily recognizable mathemati-
cal character types. In the 1967 world, there is a Russian-
Jewish tenured math professor at Columbia. He leads a
charmingly intense and decent intellectual life in New York
City, which differs drastically from the life he would have
led in the USSR had his family not escaped when he was
a child. We are also introduced to a graduate student who
its from math topic to math topic and eventually decides
to leave the PhD program. All of us had friends like this
in graduate school. Unlike most though, this person even-
tually becomes a private detective (this is a mystery novel
after all).
There are other issues that naturally arise. For exam-
ple, most readers of the Notices spend their lives in a math
department full of interesting people, some of whom are
quite eccentric. (Unlike the ctional Eisenstadt, almost all
of the math eccentrics that I know are of the endearingly
quirky type). But do any of us really know any of our
colleagues? At the last Williams math department meet-
ing, after reading this novel for the second time, I found
myself looking at the other members of the department,
many of whom I have known for decades, and wondering
how many of them had dark secrets hidden from public
view. (To be clear, I quickly concluded that the answer
was “none”).
I will include here a small warning that the novel con-
tains a few explicit sex scenes that do not really t with the
rest of the book. These few scenes could also potentially
get an American high school teacher in trouble if the book
is recommended to nonadult students.
The novel overall is a great read, one which will hold
your interest from the rst paragraph’s camera smashing
incident to the exciting culmination at the end. I expect
your nonmath friends will enjoy it. But Notices readers will
enjoy it even more, seeing how the mystery unfolds in the
world that we know so well.
Thomas Garrity
Credits
Book cover is courtesy of Naomi Goodman.
Photo of Thomas Garrity is courtesy of L. Pedersen.
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1055
Freeman Hrabowski:
Advocate for Mathematics
and STEM Visionary
Christian Anderson
Introduction
Science, Technology, Engineering, and Mathematics
(STEM) education has been identied as a key factor to
the United States’ national security (Athanasia and Cota,
2022). The country’s ability to produce STEM literate cit-
izens and STEM professionals has received national atten-
tion over the past 7 to 10 decades. On the national land-
scape, Dr. Freeman Hrabowski has emerged as a STEM
visionary, leading STEM education programs and produc-
ing a diverse cadre of STEM professionals. After a notable
tenure as the President of University of Maryland Balti-
more County (UMBC), Dr. Hrabowski created a blueprint
and method to recruit, educate, and graduate STEM pro-
fessionals, many of whom will go on to earn PhDs.
The full impact of the legacy of Dr. Freeman Hrabowski
is yet to be seen. His impact on the STEM community is
still being measured as he continues to promote excellence
in STEM education and advocates for diversity and repre-
sentation. As a child growing up in Birmingham, Alabama,
he advocated for African Americans’ rights in the face of
unfair treatment in the Jim Crow South. As an adult, he
became an advocate for diverse college students with aca-
demic promise to gain access to a rigorous and quality ed-
ucation that focused on STEM.
Christian Anderson is an associate professor of mathematics education at Mor-
gan State University. His email address is christian.anderson@morgan
.edu.
Communicated by Notices Associate Editor Asamoah Nkwanta.
For permission to reprint this article, please contact:
reprint-permission@ams.org.
DOI: https://doi.org/10.1090/noti3016
As we continue to celebrate the accomplishments of Dr.
Hrabowski, we should acknowledge the unique role that
the subject of mathematics played in his life and how his
love for mathematics shaped his approach to creating in-
novative STEM programming, most notably, the Meyer-
hoff Scholars Program. The Meyerhoff Scholars Program is
an undergraduate scholarship program at the University of
Maryland Baltimore County (UMBC) which is designed to
increase diversity among future leaders in STEM who have
a desire to pursue a PhD or combined MD/PhD in a STEM
eld of study. To date, the Meyerhoff Scholars Program
has approximately 1400 alumni with over 300 students
currently enrolled in graduate and professional programs
across the country. Moreover, there are Meyerhoff alumni
who are faculty at prestigious universities (e.g., Harvard) or
working as research scientists at top government agencies
such as the National Institute of Allergy and Infectious Dis-
eases and the National Institute of Health (NIH). Most no-
tably, Meyerhoff Alumna, Kizzmekia Corbett, served as the
leader of the team that developed Moderna’s COVID vac-
cine. UMBC leads the nation among predominately white
institutions (PWI) who produce Black undergraduates that
go on to earn PhDs in the natural sciences, life sciences, en-
gineering, computer science, and mathematics.
As we know, mathematics serves as the gateway to STEM
elds. Unfortunately, mathematics is often viewed as
a difcult subject and is often identied as the reason
students leave STEM programs (Li & Schoenfeld, 2019).
In addition, in many cases, mathematics, is taught in
a way that does not promote creativity or invite diverse
populations. Dr. Hrabowski’s love for mathematics
and insights about the teaching of mathematics enabled
him to train and inspire countless numbers of scientists,
1056 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
engineers, and other STEM professionals. Based on his
own experience learning mathematics, he believes that
mathematics should be taught in the context of the stu-
dents who are learning it. Moreover, Hrabowski envi-
sioned a program where learning takes place collabora-
tively. In his model, students are assigned to cohorts and
taught how to learn and work together. This approach
to learning has been discussed within scholarly literature.
McCallum (2018) identied a similar approach to teach-
ing mathematics known as making-sense, where the man-
ner in which the students experience mathematics is just
as important as their acquisition of the mathematical con-
tent.
Hrabowski’s approach to STEM education with the Mey-
erhoff program created opportunities for students of color
and females to engage in rigorous STEM coursework.
Elements of the making-sense approach are evident in
the Meyerhoff program. Like the approaches described by
Schoenfeld et al. (2018), Hrabowski inspires and ignites
his students’ love of mathematics and science to ascend
to new heights of scholarship and research. Since math-
ematics is the gateway to advanced coursework in K-12
that leads to matriculation as STEM majors in college, the
mathematics community should reexamine the way the
subject is taught. Delvin (2000) argues that mathematics
has four (4) different faces: 1) computation, formal rea-
soning, and problem solving, 2) a way of knowing, 3) a
creative medium, and 4) applications.
In a series of interviews, I met with several alumni of the
Meyerhof Scholars Program to discuss their experiences
in the program and to reect upon the impact that Dr.
Hrabowski had on their lives both personally and profes-
sionally. During my interview with Dr. Hrabowski, he
shared his experiences as a student of mathematics, and
he discussed how those experiences shaped his approach
when designing STEM programs, consulting with STEM
based organizations, and serving as a mentor to STEM pro-
fessionals across the country.
An Advocate for Mathematics
Early exposure to mathematics—“You learn to do, by do-
ing”. In recent years, out of school learning experiences
have been linked to increased student achievement (Neher-
Asylbekov and Wagner, 2022). Providing students with op-
portunities to link content knowledge with real world ex-
periences outside of school allows students to deepen their
conceptual understanding of the concepts being taught
and increases their self-condence to engage in the con-
tent. This was the case with Dr. Hrabowski. His approach
to learning mathematics was rooted in a mathematics-rich
home environment that was created by his parents.
At an early age, Freeman Hrabowski’s love for mathe-
matics was fostered and developed by routines and struc-
tures established in his home. While he was academically
gifted, he developed a work ethic that deepened his love
and appreciation for mathematics. His mother, an English
teacher who was inspired by the New Math movement of
the late 1950’s, believed that strong reading comprehen-
sion skills led to an ability to solve mathematical word
problems. As a child, Hrabowski enjoyed solving prob-
lems that his parents presented to him on a regular basis.
Hrabowski explains, “I found that the more problems I
did, the more I understood concepts and the more fun I
was having. That was happening with the math.. ..”
Additionally, he recalls one of his mother’s fundamen-
tal beliefs when it came to learning mathematics, “you
learn to do, by doing.”
This mindset of engagement and repetition served as
his mental foundation as he progressed through his edu-
cational journey. His continuous engagement with con-
tent (mainly with mathematics) increased his condence
as a student. As he entered new educational spaces, he was
condent in his ability to engage with new learning. This
approach was useful as he entered college at the age of f-
teen.
Mathematics as a passport to new experiences. Lived ex-
periences have a direct impact on the development of a per-
son’s belief system (Pajares, 1992). This is the case with Dr.
Hrabowski. His varied experiences as a child and as a stu-
dent, at all educational levels, shaped his beliefs about the
teaching and learning of mathematics (and other STEM
content areas). Experiencing mathematics in different ed-
ucational settings by different types of people from vari-
ous racial and cultural backgrounds, strengthened his be-
lief that mathematics was universal and was accessible to
all who worked hard to understand it.
As Hrabowski matriculated through his elementary and
secondary education, he excelled in mathematics, distin-
guishing himself from his peers. This led to opportuni-
ties for external learning experiences that allowed him to
see mathematics in action and experience it at a high level.
He spent summers participating in National Science Foun-
dation (NSF) sponsored programs at various college cam-
puses around the country. This provided him with the op-
portunity to see mathematicians engaging in rigorous and
complex mathematical tasks.
These experiences were pivotal for him because he had
limited exposure to STEM professionals of color at his
home school in Birmingham. Being educated in the Jim
Crow South, Hrabowski attended segregated schools. He
notes that the only Black scientist that he heard of as a
child was George Washington Carver. By contrast, in the
NSF summer programs, he had the opportunity to see
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1057
scientists of color. He recalls, “I saw for the rst-time Black
PhDs in STEM, and that’s when I started thinking about the
possibility of a PhD and of teaching math.”
In addition to exposure to rigorous mathematics and
diverse scientists and mathematicians, these summer pro-
grams exposed him to students of different races. Until
then, Hrabowski never shared a classroom with white stu-
dents. He recalls, “...another summer they sent me to Mas-
sachusetts to see what it would be like to be in class with
white children.” Irrespective of the race of the teachers or
the students, Hrabowski’s love of mathematics eclipsed the
environments that he excelled in.
Self-identity in the mathematics space. The concept of
diversity, equity, and inclusion (DEI) has emerged on the
social landscape as a mechanism to ensure that varying
racial, cultural, and gender identity perspectives are rep-
resented in different facets of society, including the STEM
spaces (NSF, 2023). The ability of someone to enter new
spaces with condence in their purpose and qualications
to be in those spaces is rooted in their self-identity. This
was the case with Dr. Hrabowski. His early success with
mathematics combined with his close family relations and
his civil rights activism gave him a strong self-identity that
was rooted in condence in his own mathematics ability.
This was needed as he matriculated through his graduate
programs.
Although Hrabowski was academically strong, he was
aware of the need to advocate for himself to ensure his suc-
cess in the classroom. Establishing positive relationships
with his professors was a key factor in his academic suc-
cess during his undergraduate years. Hrabowski attended
Hampton Institute (now Hampton University) a Histori-
cally Black College / University (HBCU) in Virginia. As an
undergraduate, Hrabowski recalls positive intellectual re-
lationships with two of his African American female pro-
fessors. One of them, he fondly remembered, “.. .she was
such a fascinating intellectual and brought abstract algebra
to life. And I enjoyed the proofs with her, and she prepared
me well. She really did.” These positive experiences would
increase Hrabowski’s self-condence as he continued his
matriculation through higher education.
As a graduate student, Hrabowski attended a predomi-
nantly white university where his environment and experi-
ences were somewhat different from his undergraduate ex-
perience. As an undergraduate, he was welcomed into his
classes by his professors and fellow students. As a gradu-
ate student, he established himself as a serious and capable
student as he developed relationships with his professors.
He remembers, “.. .so when I went to grad school there
was nobody black in my classes, no black professors.” He
soon realized that his level of self-advocacy had to increase
to make himself visible in the mathematics space.
“I did quite well in the math, but nobody would work
with me. Nobody would work with me, and in the rst
classes, when the white male professor would come in
each class, the professor would look right at me, the
only black in the class, and say, this is topology, or
this is set theory. The idea was, You’re probably in the
wrong class. And so by the third time, I said, this is
‘numerical analysis, right?’ But he came in, so let him
know I’m supposed to be in here. . ..and I tell you that
because I needed to let people know I knew where I
was, and why I was there. .. .what I learned from the
experience, though, was since the students, all male or
white. .. .I needed to be aggressive in getting help from
faculty members. I did well, but I had to, really, and
they were not. They just kept saying, ‘Go work with the
students’, and I said, ‘they won’t work with me’. And
so, I did work with the students, and I did better than
most. But I tell you that because by the time I nished
my master’s, I had nobody to talk to. So, I did this
combination of higher education administration and
statistics in the social sciences for my PhD.”
Dr. Hrabowski’s social isolation did not have a negative
impact on his ability to be academically successful. It did,
however, provide him with ideas and insight regarding
the elements of a successful STEM education program that
would educate and train a diverse group of scholars. Sup-
porting and developing a positive self-identity was a key
element that he identied. The creation of a support sys-
tem that contained academic and social support became
a foundational element in the STEM education programs
that he developed.
STEM Visionary
The under-representation of students of color and those
students who identify as female in the STEM elds has
been well documented in the scholarly literature (NSF,
2023). Hrabowski’s ability to navigate the mathematics
space as a Black man is remarkable and has served as
the impetus for his approach to teaching and promoting
unique STEM educational programming.
When he reected on his experiences as a graduate stu-
dent, Hrabowski shared,
“. . .I tell you that because it shaped my thinking about
preparing people for careers in STEM, and one of the
points was collaboration and group work. As I began
my career at UMBC, I knew I wanted to spend my
life producing PhDs of all races with an emphasis on
the students of color, but of all races, and women and
men.”
The ability to create a system that produces a diverse
group of STEM scholars was not an easy task. Identifying
1058 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
students with academic promise was only the beginning.
Creating a university infrastructure to support these stu-
dents was a necessary component as well. The Meyerhoff
Scholars program was the embodiment of Dr. Hrabowski’s
vision for STEM education.
Developing the Model
Dr. Hrabowski’s establishment of the Meyerhoff Scholars
Program provided him with the mechanism to develop
and rene his approach to producing a diverse group of
PhDs in STEM elds. Using his own personal and profes-
sional experiences as a college administrator, the Meyer-
hof Scholars Program was centered around the following
concepts:
1. Building a strong community among the students.
2. Creating a culture of high expectations for the students
and faculty.
3. Understanding that it takes a scientist, mathematician,
or engineer to produce a scientist, mathematician, or
an engineer.
Building a strong community among students. The con-
cept of collaboration was a recurring theme that emerged
during Dr. Hrabowski’s experiences as a student. The abil-
ity to learn by working with others is one of the key pieces
of the Meyerhoff program. Providing students with the op-
portunity to authentically engage in rigorous STEM con-
tent with collaboration embedded in the curriculum cre-
ated a sense of comradery within the program. To build
a strong sense of community, Hrabowski elicited feedback
from program alumni for strategies to strengthen relation-
ships among the students. As a result, the summer in-
duction program implemented a radical approach to en-
couraging the students to get to know one another. This
approach required students to limit their use of their cell
phones during the week. Hrabowski recalls making this de-
cision with the assistance of his senior students from the
Meyerhoff program. He stated:
“. . .It was the students as they graduated from the pro-
gram. We asked them one summer, what could we do
to make the program better? And they said, If you want
the students to get to know each other better, take the
phones away during the week, so they have to depend
on each other rather than on their high school friends
and their mothers, and we did it. And it did make a
difference.”
As a result, students in the program learned to commu-
nicate and collaborate with one another as they matric-
ulated through the program. This tactic also helped stu-
dents to develop personal and professional networks that
took them beyond the classroom.
Creating a culture of high expectations for the students
and faculty. High expectations have been a core value for
Dr. Hrabowski his entire life, and he promoted that as he
established the Meyerhoff Scholars program. He believed
that engaging students in rigorous content and authentic
research was key to their academic development. More-
over, Dr. Hrabowski created an environment where faculty
were expected to engage their students in all phases of the
research process. As a result, many graduates of the Meyer-
hoff program have gone on to become major participants
in current STEM research.
Understanding that it takes a scientist, mathematician,
or engineer to produce a scientist, mathematician, or
an engineer. Dr. Hrabowski believed that scientists, re-
searchers, engineers, and other STEM industry profession-
als had insights into unique and practical applications of
the concepts that were taught in classes. To this end, the
Meyerhoffs were exposed formally and informally to STEM
industry professionals throughout their matriculation in
the program. Exposure to these professionals served as in-
spiration and motivation to the students in the Meyerhoff
program.
Reflections from Meyerhoff Scholars Alumni
Dr. Hrabowski’s model for the Meyerhoff Scholars Pro-
gram provided the framework for many of the scholars to
be successful in the program. I had an opportunity to inter-
view several Meyerhoffs past and present to learn about the
impact of the program on their personal and professional
lives. During our discussion, three themes emerged as the
Meyerhoffs shared their experiences while in the program.
Theme 1: Developing and fostering relationships. As
the Meyerhoffs reected on their time in the program, they
recalled the positive messages that they received from the
program’s faculty and staff, especially the encouragement
that they received from Dr. Hrabowski himself. Although,
he was the President of the university, he was accessible to
the students attending the university and to the students in
the Meyerhoff program. The ability to develop meaningful
relationships helped to create the culture of the Meyerhoff
program, like the relationships that Dr. Hrabowski formed
when he was an undergraduate at Hampton University. He
created those same types of relationships with the students
in the Meyerhoff program. A program alumnus remem-
bers one of their rst interactions with Dr. Hrabowski.
“He was so nice and energetic! He asked me my name
and where I was from, and what I planned to major in.
It was like he really was trying to get know me. . .From
that point on, every time he saw me on campus, he
called me by name and he checked in with me.”
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1059
Another Alumnus reects on their relationship with Dr.
Hrabowski.
“Dr. Hrabowski really took a chance and believed in
me. When I originally applied, I didn’t get in because
my SAT scores weren’t that good. When I met with
him, he told me that I would have to work hard, and if
I did, I would be really successful.. . and I did! I went
on to earn my MD/PhD.”
In addition to developing relationships with several
Meyerhoff scholars himself, Dr. Hrabowski created a cul-
ture that fostered relationships among the scholars them-
selves. Many of the program alumni were able to develop
lifelong friendships with their fellow classmates. They re-
ected on their rst time meeting one another at the sum-
mer orientation for the Meyerhoff program. The orienta-
tion was an intense 4–6-week program where the new co-
hort of scholars took classes that emphasized study skills,
teamwork, collaboration, and group accountability. The
program was structured for the new scholars to “buy-in”
to the tenants of the program. One of the program schol-
ars remembers her rst week in the summer program.
“I was really nervous when I rst got there because I
didn’t really know anyone. I recognized a couple of
people that I met at one of the earlier receptions for
admitted students, but that’s it. Dr. Hrabowski did
a good job making us feel comfortable, and he shared
his vision of us and the success that awaits us. He
made me (us) feel really special. . . . Yeah, my room-
mate from freshman year, we were bridesmaids in each
other’s weddings.”
During their time on campus, the Meyerhoffs operated
on campus as a pseudo-family. While many of them
were active in other campus organizations and clubs, they
participated in several academic and social activities as a
group. They attended class together, they studied together,
and they socialized around academics together as well.
Theme 2: A culture of academic excellence. The com-
radery developed among the Meyerhoffs created a culture
of achievement in the program as well. Academic ex-
cellence was an expectation. While competition existed
within the program, the concepts of teamwork and shared
success prevailed. One Meyerhoff explained her experi-
ences in the program in this way.
“We all had to do well. When we were in the summer
program, everybody was responsible for everyone’s suc-
cess. If one person didn’t understand a concept, it was
the responsibility of the group to help them learn it. No
one was allowed to be left behind.”
One of the goals/expectations of the Meyerhoff pro-
gram is that graduates would go on to graduate school.
As a result, grade point averages (GPA) were of great
importance to the students. This heightened empha-
sis on GPAs resulted in an increased focus on their aca-
demic performance. Meyerhoffs were encouraged to reach
out and establish relationships with their professors to
strengthen and build academic networks on campus just as
Dr. Hrabowski had done as a student. In many cases, this
resulted in opportunities for the Meyerhoffs (and other stu-
dents) to engage in authentic research with their profes-
sors. These research opportunities allowed the students to
deepen their understanding of the content that they were
learning in class, which in turn strengthened the culture of
the Meyerhoff Program.
Reflections for the Mathematics Community
and the Future of STEM Diversity
Mathematics can serve as an access point for advanced
academic studies, and it can, unfortunately, act as a
gatekeeper for those advanced academic studies as well.
The academic career of Dr. Hrabowski is an example.
One crowning achievement is the establishment of the
Freeman Hrabowski Scholars Program sponsored by the
Howard Hughes Medical Institute (HHMI) in 2022. The
Hrabowski Scholars acts as an extension of the Meyerhoff
Scholars program because its goal is to support early ca-
reer faculty. HHMI (2022) describes the selection as this,
“Faculty members are selected for their potential to be-
come leaders in their elds and to create diverse and inclu-
sive lab environments in which everyone can thrive”. Se-
lected scholars receive a salary and a research budget that
includes equipment for a 5-year period with an opportu-
nity to renew for a second 5-year period.
As I concluded my conversation with Dr. Hrabowski,
I asked him if he had any parting recommendations for
the mathematics community. He shared the following
thought,
“I want the mathematicians to be the ambassadors for
the discipline, and to inspire others to keep learning.. . .
We as mathematicians must be those who excite and in-
spire others. We have the capacity to do that, and we
must do it with pride. Let’s do it with pride! That’s my
message. We must speak with pride about mathemat-
ics.”
Dr. Hrabowski’s legacy is still unfolding. His impact
on STEM education will not be measured by the countless
awards that he has received and will continue to receive. It
will be measured by the countless numbers of STEM pro-
fessionals that he has mentored, inspired, and produced
as well as the manner in which these scientists contribute
to their respective elds of study.
1060 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
References
[1] G. Athanasia and J. Cota, The U.S. Should Strengthen STEM
Education to Remain Globally Competitive, Perspectives on
Education, CSIS, 2022.
[2] J. Li and A. H. Schoenfeld, Problematizing teaching and
learning mathematics as “given” in STEM education, Interna-
tional Journal of STEM Education (2019), https://doi
.org/10.1186/s40594-019-0197-9.
[3] W. McCallum, Sense-making and making sense, 2018,
https://blogs.ams.org/matheducation/2018.
[4] National Center for Science and Engineering Statistics
(NCSES), Diversity in STEM: Women, Minorities, and Persons
with Disabilities, Special Report NSF 23-315, National Sci-
ence, Alexandria, VA, 2023.
[5] S. Neher-Asylbekov and I. Wagner, Effects of Out-of-School
STEM Learning Environments on Student Interest: A Critical
Systematic Literature Review, Journal of STEM Education Re-
search 6(2022), no. 1, 1–44.
[6] M. F. Pajares, Teachers’ Beliefs and educational research:
Cleaning up a messy construct, Review of Educational Re-
search 62 (1992), no. 3, 307–332.
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[7] V. Richardson, The role of attitudes and beliefs in learning to
teach, in J. Sikula (Ed.), Handbook of Research on Teacher
Education, Simon and Shuster, New York, NY, 1996.
Christian Anderson
Credits
Photo of Christian Anderson is courtesy of Christian Ander-
son.
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1061
WHAT IS. . .
a Parking Function?
J. Carlos Mart´ınez Mori
Consider a one-way street with numbered
parking spots, denoted . A sequence of cars
enters the street one at a time, each with a preferred spot.
Upon its arrival, car , denoted , drives to its pre-
ferred spot . If spot is unoccupied,
car is lucky and parks there. Otherwise, car displaces
further down the street until it nds the rst unoccupied
spot in which to park, if such a spot exists. If no such spot
exists, car reneges the search process unable to park. Let
be the -tuple encoding the cars’
parking preferences. If all cars are able to park, then is
said to be a parking function of length .
Parking functions were rst implicitly studied by
Pyke [Pyk59] in his study of Poisson processes and later on
by Konheim and Weiss [KW66] in their study of hashing
with linear probing. Parking functions can be seen as func-
tions in the sense that the “parking experiment” narrated
above assigns, to each -tuple of parking preferences, a sin-
gle -tuple encoding its parking outcome. If the -tuple of
preferences is indeed a parking function, its outcome can
be treated as a permutation of written in one-line no-
tation. For example, as depicted in Figure 1, is a
parking function of length with outcome .
Classical enumerative results about parking functions
foreshadow their rich mathematical structure. There are
(1)
J. Carlos Mart´ınez Mori is a Schmidt Science Fellow and a President’s Postdoc-
toral Fellow at the Georgia Institute of Technology. His email address is jcmm
@gatech.edu.
The arXiv version of this article includes a Spanish translation: https://
arxiv.org/abs/2404.15372.
Communicated by Notices Associate Editor Emilie Purvine.
For permission to reprint this article, please contact:
reprint-permission@ams.org.
DOI: https://doi.org/10.1090/noti3004
Figure 1. The parking function has outcome .
Cars and are lucky and park in their preferred spots
and , whereas cars and displace further down the street
before ultimately parking in spots and , respectively.
parking functions of length (OEIS A000272)—this
is Cayley’s formula [Cay89] for the number of labeled trees
on vertices. This count was established independently
by Pyke [Pyk59] and Konheim and Weiss [KW66], and has
been recovered bijectively by numerous authors; refer to
Yan [Yan15, Section 13.2] and the references therein.
An elegant proof of (1), due to Pollak as credited by
Riordan [Rio69], is as follows. Consider a circular one-
way street with numbered parking spots, denoted
. Say, it is a roundabout with no exits and
a single entry between spots and . There are cars at-
tempting to park, so that each -tuple
encodes a possibility for the cars’ parking prefer-
ences. How many of these -tuples are in fact parking func-
tions of length ? Since there are spots but only
cars, there will always be one spot left unoccupied; which
exact spot this is depends on the preferences. Note that the
preference tuples form a group under component-wise
addition modulo , and consider the subgroup
generated by the all-ones preference tuple . As
depicted in Figure 2, the cosets
for form equivalence classes of order , each
of which contains exactly one parking function, namely
1062 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
What is. . .
Figure 2. Pollak’s argument for (1) using the example in
Figure 1. The tuples ,,,,
and form an equivalence class of order . Their
outcomes are depicted as layers overlaying a circular
one-way road with spots . The only parking function
in this class is because it is the only tuple that leaves
spot unoccupied in its outcome, as depicted in the
outermost layer of the illustration.
whichever tuple leaves spot unoccupied in its out-
come. Therefore, there are
parking functions of length .
Now, consider an arbitrary -tuple
. Can one tell whether it is a parking function of length
without explicitly considering its parking outcome? A
classical inequality-based characterization of parking func-
tions allows precisely this. Let be the
weakly increasing rearrangement of . Then, the following
holds:
is a parking func-
tion of length for all
.(2)
This characterization can be veried as follows. If there
exists some for which , then at least cars
attempt to park in the last spots . This is
more cars than spots available, so at least one of these cars
will be unable to park. Conversely, if some car is unable
to park, then there exists an earliest spot left unoccupied,
which can only be the case if .
Note that (2) has the following remarkable implication:
parking functions are invariant under the action of the
symmetric group , which permutes their subscripts. In
particular, the set of parking functions is obtained from
the rearrangements of weakly increasing parking functions.
For example, ,,, and
are all parking functions of length , each with a different
parking outcome, whereas no rearrangement of
can possibly be a parking function of length .
Figure 3. The Dyck path of length , illustrated
with bold dashed red lines, maps to the weakly increasing
parking function of length . For
example, the fourth step in the path is the sixth step overall,
as illustrated with a solid bold red line. There are a total of
two steps before it, so .
Parking functions are also closely related to the Catalan
numbers (OEIS A000108), which are given by the recur-
rence relation
(3)
for with . In particular, the set of weakly
increasing parking functions of length is enumerated
by (3). To verify this, suppose you construct a weakly
increasing parking function of length , denoted
, with your choice of for
the greatest index satisfying . For any such choice of
, the -tuple must be a parking func-
tion of length , the -tuple
must be a parking function of length ,
and by induction there are possibilities.
As a consequence, weakly increasing parking functions
are in bijection with the wide variety of Catalan objects; re-
fer to Stanley [Sta15] for a survey. For example, weakly in-
creasing parking functions of length are in bijection with
Dyck paths of length ; these are lattice paths from
to using only north steps and east steps ,
denoted “N” and “E” respectively, and which do not cross
below the main diagonal. Armstrong, Loehr, and Warring-
ton [ALW16, Section 2.2] describe the following bijection:
given a Dyck path of length , obtain a weakly increasing
parking function of length by letting
be one plus the total number of E steps appearing be-
fore the th N step for all . Figure 3 illustrates this
construction using the weakly increasing parking function
; the weakly increasing rearrangement of the ex-
ample in Figure 1.
Note that if is a weakly in-
creasing parking function, then the th car (with pref-
erence ) parks in the th spot . Therefore, the total
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1063
What is. . .
displacement of a weakly increasing parking function
is given by
(4)
This statistic has further combinatorial interpretations. For
example, the number of full squares between a Dyck path
and the main diagonal, which in the case of Figure 3 is ,
is the same as the total displacement of its corresponding
weakly increasing parking function. Lastly, note that the
dependency of (4) on reduces to the summation of its
terms. This implies that the total displacement of a park-
ing function is also preserved under rearrangements!
Despite rst appearing in the literature more than
six decades ago, the combinatorics of parking func-
tions remains a vibrant area of research. Recent work
ranges from the study of discrete statistics such as dis-
placement [KY23, EHKMM23] or the number of lucky
cars [GS04, SV24, SY23], the introduction of new variants
and/or generalizations on the classical “parking experi-
ment” rule (refer to Carlson et al. [CCH21] for an accessi-
ble tour of endless possibilities in the style of “choose your
own adventure”), polyhedral aspects [AW22, HLVM24],
and surprising connections to seemingly unrelated ob-
jects [AAH23, HKMM24], to mention just a few. In re-
cent work, my collaborators and I use a subset of park-
ing functions we call unit Fubini rankings, and in partic-
ular their outcome map, to characterize and enumerate
the Boolean intervals of rank in the weak order poset
of [EHKMM24]. Figure 4 illustrates our construction.
References
[AAH23] Yasmin Aguillon, Dylan Alvarenga, Pamela E. Har-
ris, Surya Kotapati, J. Carlos Mart´ınez Mori, Casandra D.
Monroe, Zia Saylor, Camelle Tieu, and Dwight Anderson
Williams II, On parking functions and the tower of Hanoi,
Amer. Math. Monthly 130 (2023), no. 7, 618–624, DOI
10.1080/00029890.2023.2206311. MR4623327
[AW22] Aruzhan Amanbayeva and Danielle Wang, The con-
vex hull of parking functions of length , Enumer. Comb.
Appl. 2(2022), no. 2, Paper No. S2R10, 10, DOI
10.54550/eca2022v2s2r10. MR4459984
[ALW16] Drew Armstrong, Nicholas A. Loehr, and Gregory S.
Warrington, Rational parking functions and Catalan numbers,
Ann. Comb. 20 (2016), no. 1, 21–58, DOI 10.1007/s00026-
015-0293-6. MR3461934
[CCH21] Joshua Carlson, Alex Christensen, Pamela E.
Harris, Zakiya Jones, and Andr´es Ramos Rodr´ıguez,
Parking functions: choose your own adventure, Col-
lege Math. J. 52 (2021), no. 4, 254–264, DOI
10.1080/07468342.2021.1943115. MR4309295
[Cay89] Arthur Cayley, A theorem on trees, Quarterly Journal
of Pure and Applied Mathematics 23 (1889), 376–378.
[EHKMM23] Jennifer Elder, Pamela E. Harris, Jan
Kretschmann, and J. Carlos Mart´ınez Mori, Cost-sharing in
parking games, arXiv preprint arXiv:2309.12265 (2023).
Figure 4. Weak order poset of with a Boolean interval of
rank highlighted, with minimal element and maximal
element written in one-line notation. The outcome of
, a parking function of length , is
and corresponds to the minimal element of the interval. The
outcome of , another parking function of length ,
is again and corresponds to the minimal element
of the interval. In fact, is a “unit Fubini ranking
with distinct ranks” and corresponds to a Boolean interval of
rank (i.e., a cube), whereas is a “unit
Fubini ranking with distinct ranks” and corresponds to a
Boolean interval of rank (i.e., a node). Refer to Elder
et al. [EHKMM24] for details.
[EHKMM24] Jennifer Elder, Pamela E. Harris, Jan
Kretschmann, and J. Carlos Mart´ınez Mori, Parking
functions, fubini rankings, and Boolean intervals in the weak
order of , To appear in Journal of Combinatorics (2024).
[GS04] Ira M. Gessel and Seunghyun Seo, A renement of Cay-
ley’s formula for trees, Electron. J. Combin. 11 (2004/06),
no. 2, Research Paper 27, 23, DOI 10.37236/1884.
MR2224940
[HLVM24] Mitsuki Hanada, John Lentfer, and Andr´es
R. Vindas-Mel´endez, Generalized Parking Function Poly-
topes, Ann. Comb. 28 (2024), no. 2, 575–613, DOI
10.1007/s00026-023-00671-1. MR4747488
[HKMM24] Pamela E. Harris, Jan Kretschmann, and J. Car-
los Mart´ınez Mori, Lucky Cars and the Quicksort Algorithm,
Amer. Math. Monthly 131 (2024), no. 5, 417–423, DOI
10.1080/00029890.2024.2309103. MR4739576
1064 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
[KY23] Richard Kenyon and Mei Yin, Parking functions:
from combinatorics to probability, Methodol. Comput. Appl.
Probab. 25 (2023), no. 1, Paper No. 32, 30, DOI
10.1007/s11009-023-10022-5. MR4549917
[KW66] Alan G. Konheim and Benjamin Weiss, An occupancy
discipline and applications, SIAM Journal on Applied Mathe-
matics 14 (1966), no. 6, 1266–1274.
[Pyk59] Ronald Pyke, The supremum and inmum of the Pois-
son process, Ann. Math. Statist. 30 (1959), 568–576, DOI
10.1214/aoms/1177706269. MR107315
[Rio69] John Riordan, Ballots and trees, J. Combinatorial The-
ory 6(1969), 408–411. MR234843
[SV24] Anton´ın Slav´ık and Marie Vestenick´a, Lucky
cars: expected values and generating functions, Amer.
Math. Monthly 131 (2024), no. 4, 343–348, DOI
10.1080/00029890.2023.2293616. MR4723560
[Sta15] Richard P. Stanley, Catalan numbers, Cam-
bridge University Press, New York, 2015, DOI
10.1017/CBO9781139871495. MR3467982
[SY23] Richard P. Stanley and Mei Yin, Some enumerative prop-
erties of parking functions, arXiv preprint arXiv:2306.08681
(2023).
[Yan15] Catherine H. Yan, Parking functions, Handbook of
enumerative combinatorics, Discrete Math. Appl. (Boca
Raton), CRC Press, Boca Raton, FL, 2015, pp. 835–893.
MR3409354
J. Carlos
Mart´ınez Mori
Credits
All gures are courtesy of J. Carlos Mart´ınez Mori.
Author photo is courtesy of Nicole Semp´ertegui.
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1065
Suppor fo you esearc
APPLY FOR AMS FELLOWSHIPS
Application period: July 15–September 30, 2024
Learn more and apply:
www.ams.org/ams-fellowships
Joan and Joseph Birman Fellowship
for Women Scholars
A midcareer research fellowship specially
designed to fi t the unique needs of women.
The most likely awardee is a midcareer
woman, based at a US academic institution,
with a well-established research record in
a core area of mathematics.
For more information see:
www.ams.org/birman-fellow
AMS Claytor-Gilmer Fellowship
To further excellence in mathematics research
and generate wider and sustained participation
by Black mathematicians. The most likely
awardee is a midcareer Black mathematician
based at a US institution whose achievements
demonstrate signifi cant potential for further
contributions to mathematics.
For more information see:
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Awarded annually to an outstanding
mathematician to help further their career
in research.
Eligible mathematicians have held a doctoral
degree for 3–12 years, and currently hold a
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comparable position at a US institution.
For more information see:
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portfolio of a mathematician who specializes in
the areas of real analysis, complex analysis, or
partial differential equations.
Eligible mathematicians are at US institutions,
have not received tenure or comparable, and
have not held signifi cant fellowship support.
For more information see:
www.ams.org/bergman-fellow
BOOKSHELF
New and Noteworthy Titles on our Bookshelf
September 2024
Cover is courtesy of George Csicsery/Zala Films.
Journeys of Black
Mathematicians:
Forging Resilience
Directed by George Csicsery
In February 2024, the Notices
published the article “Making
Journeys of Black Mathematicians,”
which highlighted the recently re-
leased documentary lm. The
article, by Csicsery himself, out-
lined the motivation for the lm
and the process with which the
lm was made. Here, I plan to highlight aspects of the
lm that I hope will entice you to seek it out.
There is a lot to learn from hearing the stories and
journeys of others. While we all face uncertainty and
difculties, there is no doubt that Black mathematicians
have faced unique challenges. The stories presented in the
documentary span from the civil rights movement of the
1960s and integration to the current lack of representation
among mathematics teachers across the United States. The
interviews feature a wide range of mathematicians, includ-
ing students, postdocs, and professors, both employed and
retired. The lm also discusses the important role HBCUs
(Historically Black Colleges and Universities) have played
in holding their students to a high standard and the power
of community within the National Academy of Mathemati-
cians (NAM).
The lm can either be watched through an individual
rental or an institutional purchase. I screened the lm
at my institution, and my undergraduate students in at-
tendance offered a variety of reactions. One student in
computer science declared a mathematics minor the next
day. Future teachers remarked on the importance of hav-
ing high expectations for all students and how crucial it is
for every student to hear they have capabilities and should
This Bookshelf was prepared by Notices Associate Editor Emily J. Olson.
Appearance of a book in the Notices Bookshelf does not represent an endorse-
ment by the Notices or by the AMS.
Suggestions can be sent to notices-booklist@lists.ams.org.
DOI: https://doi.org/10.1090/noti2995
follow their dreams. Ultimately, this documentary helped
me reect on what it means to be a mathematician while
standing out from the stereotypical role of what is expected
of mathematicians. The director has indicated there will
be a sequel, which I hope will continue the stories started
in Forging Resilience and expand upon some of the impor-
tant themes of belonging and progress.
Cover is courtesy of Basic Books.
The Waltz of Reason
Basic Books, 2023, 448 pp.
By Karl Sigmund
Some of the greatest thinkers of
all time have pondered deep and
often insightful questions which
are both mathematical and philo-
sophical in nature. The quintes-
sential example might be “What
is a number?” Karl Sigmund, the
author of this new book, states
that many mathematicians and
philosophers might reply to such a query by remarking
that “what is?” questions make little sense.
This book is a dance between math and philosophy,
and when one seems to take the lead on a question over
the other throughout history, we see “that the two elds
have wonderful ways of stimulating and often surprising
each other.” The text takes the reader on a tour through
concepts and historical activities and gures. It contains
pictures of mathematicians and philosophers and illustra-
tions that describe mathematical concepts and proofs, but
not many equations. The book is divided into four parts,
which focus on number, algorithm, axiom, and proof;
chance, probability, and the continuum; morality, eco-
nomics, social contracts, politics, and law; and language
and understanding. It is not exclusively chronological, as
it occasionally needs to refer to a person or event in ear-
lier times to explain a more recent topic of inquiry. The
book includes many historical gures from both mathe-
matics and philosophy and also indicates that sometimes
it is hard to tell the difference between the two. The Waltz
of Reason is not a technical book, but a book to be enjoyed
by anyone interested in the interplay between math and
philosophy.
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1067
THANK YOU
The Next Generation Fund is a permanently endowed fund supporting doctoral
students and early career mathematicians. Thanks to the generosity of AMS donors,
a growing number of early career mathematicians receive funding for travel grants,
collaboration support, mentoring, and more.
To learn more and make a gift, go to
www.ams.org/nextgen
SUPPORT
Want to learn more?
Visit www.ams.org/giving
or contact
development@ams.org
401.455.4111
THE NEXT GENERATION FUND
Label Bias: A Pervasive
and Invisibilized Problem
Yunyi Li, Maria De-Arteaga,
and Maytal Saar-Tsechansky
As machine learning (ML) systems are increasingly em-
ployed to automate various aspects of our lives and to as-
sist in human decision-making, it is imperative that the
accompanying issues of fairness, equity, and ethics con-
cerning these AI artifacts are thoughtfully considered and
appropriately addressed. While there has been a notable
surge in research focused on the development of fairness-
aware algorithms in recent years, a majority of these en-
deavors typically rely on a fundamental assumption: the
training labels utilized to train these systems are accurate
and unbiased. This assumption, while convenient, does
not hold true in many critical practical scenarios. In this
article, we argue that overlooking label bias while devel-
oping algorithms to address inductive bias runs the risk
of invisibilizing and exacerbating existing societal biases.
We do so by rst presenting the concept of label bias
and reviewing the literature that characterizes or concep-
tually discusses label bias in important domains includ-
ing healthcare, predictive policing, and content modera-
tion. Subsequently, we explain how confusion matrix-
based measures of “fairness” used to mitigate inductive
bias can overlook label bias. We then outline several note-
worthy risks that may stem from this limitation. We con-
clude by proposing paths forward, emphasizing the impor-
Yunyi Li is a PhD candidate at the University of Texas at Austin. Her email
address is yunyi.li@mccombs.utexas.edu.
Maria De-Arteaga is an assistant professor at the University of Texas at Austin.
Her email address is dearteaga@mccombs.utexas.edu.
Maytal Saar-Tsechansky is a professor at the University of Texas at Austin. Her
email address is maytal.saar-tsechansky@mccombs.utexas.edu.
Communicated by Notices Associate Editor Richard Levine.
For permission to reprint this article, please contact:
reprint-permission@ams.org.
DOI: https://doi.org/10.1090/noti2941
tance of clearly discerning the types of bias that fairness-
aware algorithms aim to address, and underscoring the
need for developing algorithms that mitigate label bias.
1. Pervasiveness of Label Bias
With advancements in AI technology and its ability to pro-
cess vast amounts of data, ML systems are increasingly
relied upon to make accurate predictions and assist in
decision-making. This ability to process large amounts of
data allows ML systems to uncover patterns in the data that
link various features to outcomes, a task that was tradition-
ally performed by human decision-makers. We refer to the
outcome or the target variable’s value that ML systems are
trained to predict as labels. Human-generated labels can be
collected by recording historical human decisions, or by
actively asking experts or crowd-sourced annotators to la-
bel the data as instructed. Labels used to train ML models
may also correspond to observed outcomes that do not di-
rectly correspond to a human decision. For example, when
hiring a salesperson, the observed outcome may be sales
volume; and when developing algorithms for healthcare,
there may be a medical outcome of interest. Regardless of
the origin of the label, given a set of input features and
label pairs, supervised ML methods aspire to induce mod-
els that offer consistent predictions for instances with un-
known outcomes and achieve good performance over the
population. This process is commonly known as “induc-
tion” in the ML domain.
As ML and AI become ubiquitous in our daily life, re-
searchers started to notice that AI technology might cause
social harm. In 2013, Sweeney [Swe13] found racial bias
in the targeting of online advertisements concerning arrest
records. This bias was associated with names that are cul-
turally belonging to Black or white individuals. The re-
search highlighted how individuals with names that are
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1069
perceived as Black-sounding may be more likely to en-
counter online ads suggesting they have arrest records,
even if they do not [Swe13]. The evidence presented in
this work emphasized the need to design technologies
with consideration of societal consequences, particularly
those related to structural racism. Researchers have found
evidence of bias across multiple other ML technologies.
For example, investigations into facial analysis software
revealed that commercial facial classication algorithms
exhibit lower accuracy when it comes to darker-skinned
women [KBK12, BG18]. Such ndings, while centered
on gender classication, have raised concerns regarding re-
lated technologies, such as facial analysis tools used by
law enforcement. Law enforcement agencies utilize fa-
cial recognition technology to match suspects’ photos with
mugshots and driver’s license images [Gar16]. As of 2016,
it is estimated that over 117 million American adults—
nearly half of the population—have their photos in facial
recognition networks used by the police [Gar16]. As a
result of biases in facial recognition, there are concerns
that the deployment of such tools may disproportionately
harm marginalized communities, including African Amer-
icans [Gar16].
Algorithmic bias can arise from various sources. One
signicant factor is the representation issue. For example,
there might be an underrepresentation of minority groups
in the training data. Underrepresentation occurs when cer-
tain demographic groups are not adequately represented
or are disproportionately scarce in the dataset. The bias
arises because ML algorithms learn patterns from the data
they are trained on. If the training data is not diverse
enough and lacks representation from minority groups,
the algorithm may not learn the correct relationship be-
tween predictive covariates (features) and labels for those
groups. Instead, the relationships it learns become domi-
nated by the patterns it observed for the majority group(s).
The landmark “Gender Shades” research [BG18] described
how gender classication systems can be biased due to
the representation issue in the training data. If a gender
classication algorithm is trained primarily on images of
lighter-skinned individuals, it is likely to only perform well
in identifying and classifying faces of people with similar
skin tones [BG18]. Another example of representation is-
sue is the selective labels problem [LKL17]1, which refers
to a situation in which the observed labels in a dataset are
not assigned randomly or uniformly across all instances
but are inuenced by the choices or decisions of human
decision-makers [LKL17]. In other words, the process
of labeling data is selective and depends on certain con-
1We note that while this is sometimes referred to as a label bias problem, it is in
nature a bias caused by a representation issue, which is different from the label
bias we will introduce in this article.
ditions or criteria set by decision-makers. For example, if
a model is trained to decide whom to grant bail, the avail-
able training data includes information about individuals
who were granted bail in the past, but it may not include
data about those who were not [LKL17].
Another critical source of algorithmic bias, which we
focus on in this paper, is label bias. Let
denote an ob-
served label used to train an ML system. Such a label is
usually readily accessible in a set of training examples, but
it may not always correspond to the ground truth or gold
standard of interest, which we denote . Since we are
concerned with bias affecting demographic subgroups, let
be a variable denoting group membership. While, in
general, does not need to be binary, we assume it is
binary for simplicity in our explanation. Label bias refers
to a “systematic disparity between the ground truth labels
intended to train an AI system and the observed labels,
such that the relationship underlying the mismatch dif-
fers across groups” [LDAST22]. The way the relationship
differs may take different forms. For example, [LDAST22]
proposes measuring gaps across different groups in the ac-
curacy of observed labels
with respect to latent ground
truth labels . Under [LDAST22]’s denition, label bias
occurs if there is a group of relevance such that,
Naturally, other measures may be more relevant to de-
termine label bias in different contexts. It may be the case
that
, but the
types of errors of observed labels
with respect to latent
ground truth labels are different for different groups.
There are two types of errors that could occur: false posi-
tive error ( when
) and false negative error
(when
). For example, let correspond to
gender and to women, then one may say that
exhibits
label bias with respect to if either or both of the follow-
ing equations are satised, no matter what the accuracy
rates are
It is crucial to highlight that under this denition, iden-
tifying the existence of label bias requires the availability of
ground truth labels for the training data used to train a su-
pervised learning system. As a result, the presence of label
bias relies on the specic predictive task that the supervised
learning system aims to accomplish. Moreover, since the
ground truth label is not available in many important and
consequential contexts, it is not always possible to directly
perform this assessment.
Having formally introduced the notion of label bias, we
now provide a review of label bias and its implications
1070 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
within three prominent domains: healthcare, predictive
policing, and content moderation.
1.1. Label bias in healthcare. In 2019, Obermeyer et
al. [OPVM19] examined racial disparities in an algorithm
used to prioritize patients for high-risk care management
programs. The algorithm was designed to identify patients
with complex health needs who could benet from such
programs. However, researchers found that the predictive
algorithm, trained to predict healthcare spending but used
to forecast patients’ health risk scores, exhibited algorith-
mic bias. It consistently underestimated the severity of
health needs for Black patients compared to their white
counterparts.
The racial bias in this widely used algorithm for dis-
tributing healthcare resources meant that Black patients
with equivalent risk levels assigned by the algorithm
tended to have more severe health conditions compared
to white patients who received similar scores. Con-
sequently, the algorithm disproportionately prioritized
healthier white patients over more ill Black patients.
The primary issue identied in Obermeyer et al.’s
study [OPVM19] is label bias, wherein healthcare costs
were utilized as the proxy label for health needs, yield-
ing a systematically inaccurate measure of the underly-
ing healthcare needs of patients. Historical inequities in
the US have led to Black patients incurring lower costs
than white patients with similar healthcare needs. Con-
sequently, the algorithm favored healthier white patients
over sicker Black patients, resulting in a signicant reduc-
tion in the number of Black patients identied for health
management programs.
This serves as an excellent example of one of the sources
of label bias referred to as the construct gap, wherein a
mismatch occurs between the theoretical construct of in-
terest (health risks in the example above) and the ob-
served label used to train an ML system (healthcare spend-
ing) [LDAST22]. This discrepancy often arises due to
the practical accessibility of one over the other. For in-
stance, in the healthcare example that we outlined above,
nancial incentives lead to detailed data collection for in-
surance claims, which are then repurposed for other ML
tasks [LDAST22]. Besides, the complexity of high-level ob-
jectives sometimes necessitates the use of proxies, posing
potential risks of label bias in ML systems, as certain out-
comes may be more evenly distributed in the population
than others [PB19].
1.2. Label bias in predictive policing. Predictive polic-
ing algorithms, which are increasingly used by law enforce-
ment for proactive crime prevention, rely on data analysis
and ML to predict potential criminal activities. Lum and
Isaac (2016) [LI16] found that predictive policing of drug
crimes leads to a disproportionate focus on historically
over-policed communities. One way of understanding the
bias in police-recorded data is label bias, which arises due
to different error types for different neighborhoods, with
false negatives more prevalent in higher-income or white
communities and false positives more prevalent in lower-
income or Black communities.
Such predictive policing algorithms trained using data
that exhibit label bias may perpetuate and even amplify ex-
isting biases in the data, leading to ineffective or discrimi-
natory practices. Specically, informed by the algorithm’s
predictions, police departments may focus their attention
on areas predicted to be high-risk, leading to more arrests
in those neighborhoods. These new arrests then feed back
into the predictive policing algorithm, reinforcing the be-
lief that those areas are high-crime zones, perpetuating a
harmful feedback loop [LI16].
Moreover, while companies developing such systems
have often claimed that their predictive policing systems
rely on victim reports, which they argue are not inuenced
by biased police-recorded data, new research has shown
that this alternative source of data is also prone to label
bias. Akpinar et al. [ADAC21] highlights the existence of
differential victim crime reporting rates across neighbor-
hoods, and shows that this disparity can result in the misal-
location of policing resources when using predictive mod-
els [ADAC21].
1.3. Label bias in content moderation. Content moder-
ation algorithms used by social media platforms, such as
Twitter and Facebook, are shaping the online content we
consume every day. However, algorithmic bias has been
observed in these systems. For example, biases affecting
posts written in African American English (AAE) have been
studied, and researchers have revealed that AAE tweets are
agged as offensive by automated hate speech detection
algorithms at a rate up to two times higher than other
tweets [SCG19]. The presence of racial bias in these sys-
tems is attributed to label bias: Sap et. al (2019) found un-
expected correlations between surface markers of African-
American English and ratings of toxicity in widely used
hate speech datasets used to train hate speech detection
algorithms. As the training labels systematically contain
more false positives with respect to ground truth for AAE
posts, the automatic hate speech algorithms trained using
such data are more likely to falsely identify an AAE tweet
as containing hate speech.
This bias in the algorithm’s performance can have detri-
mental consequences, further marginalizing historically
disadvantaged communities. The misidentication of
hate speech from minority communities may result in
the unjust suppression of legitimate speech, and it may
hinder efforts to address genuine issues faced by these
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1071
Figure 1. Invisibilization of label bias in measures of “fairness” is exemplified through three confusion matrices: (a), which
compares observed labels
and algorithmic predictions
; (b), which contrasts ground truth labels and algorithmic
predictions
; and (c), which compares the ground truth labels and observed labels
. When label bias exists, resulting in a
higher false positive rate for group as illustrated in figure (c), the current fairness measures relying on confusion matrix (a)
would fail to effectively capture the concerning error rate disparity evident in the confusion matrix (b), which was our intended
matrix for fairness measures.
communities. This can be understood as a participatory
injustice [NDAF22], as it makes it harder for some com-
munities to share their perspectives and participate in con-
versations. Thus, the lack of fairness and accuracy in hate
speech detection algorithms not only erodes user trust in
the platforms but also raises serious concerns about issues
of justice and who is able to make use of and participate
in social media platforms.
The label bias in datasets used to train hate speech
detection algorithms often arises due to human labeling
bias [LDAST22]. Identifying hate speech is a challenging
task, as it heavily relies on the specic context and dialect.
For instance, certain derogatory terms and phrases may be
reappropriated by the communities that these terms have
historically targeted, which means that a term may hold
different meanings depending on the identity of the per-
son using it, while remaining harmful and offensive when
used by outsiders. In the example above, when collect-
ing training labels from crowd-sourced annotators, who
are mostly non-AAE speakers and mostly not sensitive to
dialect, may mistakenly label an AAE post as containing
hate speech when it is not [SCG19].
Bias in human labeling is a widespread phenomenon,
regardless of whether the labels are gathered from crowds
or domain experts. Human cognitive biases and social
stereotypes can creep into the labels collected from hu-
man annotators in a subtle way. These biases, when com-
bined with the underrepresentation of annotators from mi-
nority groups, can result in labeled data that mirrors soci-
etal prejudices and stereotypes. For example, Davani et
al. [DAKD23] investigates the impact of social stereotypes
on hate speech detection, revealing that aggregated annota-
tions in curated datasets reect normative stereotypes, and
these biases contribute to systematic errors in hate speech
classiers.
2. Invisibilization of Label Bias in Measures
of “Fairness”
The risks posed by algorithmic bias have led to a surge in
research on algorithmic fairness, which aims to mitigate al-
gorithmic bias and enable the development of ML systems
that do not exhibit disparate performance across groups.
Increasingly, these methodologies are being adopted in
practice[BBB21] and thereby potentially impact humans,
organizations, and societies.
A cornerstone of algorithmic fairness work is the devel-
opment of measures of fairness–quantitative metrics that
seek to measure predictive bias. Such measures are then
used to diagnose algorithmic bias, as well as to mitigate it,
which often involves enforcing constraints that ensure a
predictive algorithm meets a given performance based on
a fairness measure. Perhaps the most popular family of al-
gorithmic fairness measures are those that measure group
fairness, which assesses the disparities in a metric of inter-
est across groups [MPB21].
The simplest group fairness measure is demographic par-
ity, which measures gaps in selection rates across groups.
For instance, in the context of healthcare needs, achieving
1072 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
demographic parity corresponds to the same rate of pa-
tients being predicted to have complex health needs across
racial groups. While demographic parity has the advan-
tage of not relying on observed labels, and thus is unaf-
fected by issues of label bias, its shortcomings quickly be-
come evident: what happens if health needs are different
across groups? For this reason, equalizing measures that
stem from the confusion matrix, which reects erroneous
predictions, is frequently preferred [MPB21]. The confu-
sion matrix, also referred to as the error matrix, is central
to ML and algorithmic fairness. It stands as a foundational
element for assessing the performance of predictions gen-
erated by ML classication models. This matrix, typically
presented as a two-by-two arrangement (for a binary classi-
cation task), facilitates a comprehensive evaluation of the
model’s predictive accuracy and potential biases. Essen-
tially, the confusion matrix enables the inspection of the
types of errors that a predictive model incurs, and serves as
a comparative framework between the observed labels and
the predictions generated by an algorithm, as depicted in
Figure 1 (a). For each group, the matrix discerns between
various types of accuracies and errors, systematically cate-
gorizing them into distinct categories:
1. True Positives (TP): Instances accurately predicted as
positives by the algorithm, congruent with the ob-
served labels.
2. True Negatives (TN): Instances correctly predicted as
negatives by the algorithm, congruent with the ob-
served labels.
3. False Positives (FP): Instances that are observed nega-
tives, but are erroneously classied as positives by the
algorithm.
4. False Negatives (FN): Instances that are observed posi-
tives, but are ierroneously classied as negatives by the
algorithm.
Most fairness measures depend on the confusion ma-
trix. For example, a commonly used fairness measure,
“equalized opportunity”, seeks to match the true positive
rate across groups [HPS16]. Similarly, “equalized odds”
matches both the true positive rate and the false positive
rate across groups [HPS16]. Different types of errors re-
ected by the confusion matrix can have different impli-
cations from a fairness perspective. For example, a high-
prole investigation on racial bias in recidivism prediction
systems showed that a commercial algorithm used to assist
judges in bail decisions was more likely to incur false neg-
ative errors for white defendants while incurring false posi-
tive errors for Black defendants [ALMK16]. This implies a
systematic underestimation of recidivism risk for a specic
group (eg. white defendants) and a systematic overestima-
tion of recidivism risk for another group (eg. Black defen-
dants). Such disparities serve to uncover the potential bias
inherent in the algorithmic decision-making process.
In an ideal scenario, measures of algorithmic bias
should compare algorithmic predictions with ground
truth labels , resulting in a confusion matrix demon-
strated in Figure 1 (b). Yet, the pervasiveness of label bias
in many important domains often introduces a systemati-
cal disparity between the utilized observed labels for eval-
uation and the actual ground truth labels (please refer the
denition of label bias in Section 1), resulting in a confu-
sion matrix as depicted in Figure 1 (c). In the presence of
label bias, fairness measures relying on observed labels, as
depicted in confusion matrix 1 (a) can be misleading.
To illustrate the risks of overlooking label bias and re-
lying on observed labels in the measure of “fairness,” let
us consider the context of algorithmic hate speech detec-
tion. Let denote the group of posts written in AAE
and represent posts written in non-AAE (eg. Standard
American English), with the task being the identication
of hate speech within a post. In scenarios involving crowd-
sourced annotators, posts written in AAE are more likely to
be incorrectly labeled as containing hate speech than posts
written in non-AAE, leading to an elevated False Positive
Rate (FPR) for in Figure 1 (c). Consequently, the algo-
rithm may learn this biased pattern and be more prone to
erroneously ag AAE posts as containing hate speech (as
seen in Figure 1 (b)). However, a challenge in assessing
algorithmic bias emerges: both the algorithmic prediction
and the biased annotation
exhibit a high rate of false
positives for group . If the algorithm exhibits high ac-
curacy, both algorithmic predictions and biased annota-
tions incorrectly ag the same set of non-hateful instances
as containing hate speech. Consequently, evaluating al-
gorithmic prediction
based on observed label
might
fail to identify the concerning pattern of a higher FPR for
group . The interplay among observed label
, algorith-
mic prediction
, and ground truth label is illustrated
in Figure 2.
At its core, the invisibilization of label bias stems from a
conation of inductive bias, which refers to bias that is intro-
duced during training an ML algorithm, and the broader
phenomenon of algorithmic bias. When addressing the con-
cern of perpetuating and amplifying societal biases (algo-
rithmic bias), it is important to acknowledge that such bias
may be encoded in the observed label, and thus measures
that assume the observed label to be accurate and unbi-
ased are ill-posed to tackle the problem at hand. More pre-
cisely, consider breaking down a standard machine learn-
ing pipeline into three successive phases: 1) problem for-
mulation and data preparation; 2) learning/training and
evaluation; and 3) deployment. The potential for algo-
rithmic bias to emerge spans all these stages. However,
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1073
Figure 2. The interplay among observed label
, algorithmic prediction
, and ground truth label .
most of the proposed algorithms that aim to mitigate in-
ductive bias pertain exclusively to bias arising during the
learning phase, by assuming no bias exists in the other two
phases. Label bias, which appears at the problem formula-
tion and data preparation phase, signies a data-oriented
challenge that can cause harmful downstream effects for
every following stage. If we inappropriately apply strate-
gies intended to alleviate inductive bias in elds where la-
bel bias is a core issue, the efcacy of mitigating algorith-
mic bias might be compromised or might even be counter-
productive.
3. Risks of Overlooking Label Bias
In this section, we discuss that in scenarios when label bias
exists, naive implementation of algorithms aimed at reduc-
ing inductive bias and data representation issues may not
be fully effective, and in fact, could potentially have ad-
verse outcomes. Specically, failing to account for label
bias may result in the incorrect automated identication
of disadvantaged groups, misguide model selection, and
mislead data collection. Even in an ideal scenario with
perfect induction accuracy and comprehensive representa-
tion of the entire population in training data, the induced
model can still learn biased patterns from the label bias.
Rather than being a true reection of the decision rule, this
perfect-performing induction relying on the biased label
may serve as a vehicle for the propagation of the societal
biases intricately embedded into the label.
3.1. Misidentify disadvantaged group. Some studies
suggest automatically detecting disadvantaged groups
through algorithmic fairness metrics assessment
[AAK20, AAT22]. These studies then focus on address-
ing bias by targeting improvements in algorithm perfor-
mance specically for the disadvantaged group, as dynam-
ically indicated by disparities in error rates. For instance,
Anahideh et al. [AAT22] rely on the identication of in-
stances that contribute most effectively to reducing dispar-
ities in a given metric. This approach rst identies the
disadvantaged group based on a fairness metric and then
seeks to enhance model performance for the group with
the worst model performance.
However, approaches rooted in data-driven analysis to
discern the disadvantaged group often overlook contextual
elements and historical injustices. This oversight becomes
particularly concerning when historical biases lead to a
biased labeling process. In such cases, there exists a po-
tential for incorrectly identifying the marginalized group,
which can lead to counterproductive efforts in bias mitiga-
tion [LDAST22].
3.2. Misguide model selection. In addition to the po-
tential consequences of misidentifying the disadvantaged
group, an equally signicant concern associated with ne-
glecting label bias is the potential misguidance in the pro-
cess of model selection. Consider a scenario in which mul-
tiple ML models are being evaluated, each vying for selec-
tion as the best-performing model. These algorithms ex-
hibit comparable levels of accuracy in their predictions,
while fairness metrics play a critical role in this decision-
making process. These fairness metrics are used to gauge
the performance of each algorithm in terms of bias mit-
igation. If label bias exists but is disregarded, the entire
process of model selection becomes vulnerable to distor-
tion. Label bias can signicantly impact the apparent per-
formance of algorithms, potentially favoring the major-
ity group, and further marginalizing the already disadvan-
taged group. Selecting the top-performing model through
evaluations reliant on the biased label can inadvertently
1074 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
favor models replicating the same errors present in the bi-
ased training label [LDAST22].
3.3. Mislead data collection. A recent body of research
proposes to address algorithmic bias through data collec-
tion. Early research in algorithmic fairness often attributed
biased predictions to representation issues of the training
dataset. While this is an important problem that leads to
algorithmic bias, as discussed in Section 1, it is only one
of many potential sources of bias.
Due to the substantial expenses associated with data la-
beling, active learning learning techniques are commonly
employed to facilitate the data collection process. These
techniques can be tailored to identify the most useful train-
ing instances for label acquisition, which are subsequently
used to train machine learning models. The design of
appropriate heuristics, often in the form of utility func-
tions, depends on the specic objectives of the nal pre-
dictive model. Recent work has proposed fairness-aware
algorithms for data collection [AAT22, AAK20]. For in-
stance, Anahideh et al. (2022) [AAT22] propose “fair ac-
tive learning” (FAL), an active learning algorithm that pro-
poses a utility function for selecting prospective training in-
stances that aims to improve fairness metrics in addition to
overall model accuracy. FAL selects instances for labeling
based on both uncertainty-based Shannon entropy and ex-
pected improvement in fairness. In their experiments, FAL
demonstrated a signicant reduction in model bias while
maintaining accuracy, as evaluated using the observed la-
bel [AAT22].
The advancement of fairness-aware active learning
methodologies for data acquisition introduces a promis-
ing and dynamic strategy. Nevertheless, overlooking po-
tential label bias can reduce the efcacy of these methods,
and in some cases, may even exacerbate the existing bias
they aim to mitigate. Li et al. (2022) [LDAST22] showed
that incorporating active learning techniques, such as FAL,
for additional data collection can exacerbate bias. The rea-
son behind this is intuitive: if the label used for calculating
the utility scores exhibit bias, the utility score becomes a
biased measure. Furthermore, acquiring more instances
with label bias may reinforce the model’s reliance on bi-
ased patterns rather than mitigating the intended bias.
4. Conclusion
Label bias is a pervasive issue in the domain of machine
learning. We have presented concrete examples where al-
gorithmic bias emerges due to label bias, particularly in
important domains such as healthcare, predictive policing,
and content moderation. However, signicant efforts to
address algorithmic bias still overlook the presence of la-
bel bias and rely on “fairness” measures that assume the
observed label perfectly aligns with the ground truth la-
bel. When approaches designed to mitigate inductive bias
are used in contexts where label bias is a core issue, inad-
vertent consequences may transpire. These consequences
include misidentifying disadvantaged groups, misguiding
model selection, and misleading data collection. We high-
light the importance of distinctly identifying the various
forms of bias that fairness-aware algorithms endeavor to
mitigate, and underscore the necessity of developing algo-
rithms that explicitly aim to address label bias.
4.1. Paths forward. While awareness of label bias in ML
systems continues to grow, existing methods aimed at mit-
igating this bias are still inadequate. This highlights the
need for further research and development in this area. We
outline multiple paths forward to tackle the problem.
A sociotechnical perspective. In some cases, label bias arises
due to the difculty of precisely collecting the ground truth
labels for training an ML system, which closely relates
to a signicant challenge: problem formulation. Prob-
lem formulation in the context of ML refers to the pro-
cess of dening and shaping a specic question or task
that can be addressed using computational techniques and
predictive models. It involves translating high-level objec-
tives, goals, or real-world challenges into well-dened and
tractable problems that can be answered using ML algo-
rithms [PB19].
The target variable selected for prediction is central to
problem formulation. A problem formulation process
that considers the historical inequalities and societal bi-
ases in the choice of target variable can help alleviate the
algorithmic bias and associated consequences [PB19]. For
example, in the context of predicting health needs, re-
searchers were able to mitigate algorithmic bias by chang-
ing the target label: rather than predicting health spend-
ing, they created a more holistic variable based on cost and
health information [OPVM19].
Crucially, algorithmic audits must also adopt a socio-
technical perspective. Solely focusing on fairness measures
without considering if the label used to create these mea-
sures may contain bias risks invisibilizing harms.
Improving measurement. Throughout this article, we have
focused on group fairness measures. This choice was made
given the popularity of such measures, which is arguably
due to the feasibility of applying them in practice. How-
ever, alternative measures of fairness have been proposed.
For example, some causal notions of fairness aim to assess
how predictions would change under different counterfac-
tuals [MPB21]. Fundamental issues on the validity of cer-
tain counterfactuals have been raised, and practical issues
have largely prevented such measures from being used in
practice, since we rarely have knowledge of the causal re-
lationship between different covariates. However, many
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1075
of these notions do not necessitate the absence of label
bias. Further research on alternative measures of fairness
may thus enable better assessments of algorithmic bias
that stems from label bias.
Improving mitigation. Moreover, for bias in human-gener-
ated labels, identifying the set of instances that have been
mislabeled in the training dataset can be an effective way
of preventing the propagation of societal bias through ML
systems [LDAS23]. For instance, Li et al. (2023) [LDAS23]
propose a pruning method—Decoupled Condent Learn-
ing (DeCoLe)—designed to mitigate label bias. DeCoLe
identies instances likely to be mislabeled based on group-
and class-conditional label uncertainty and prunes them
to create a training dataset with less bias. DeCoLe focuses
on addressing a particular form of label bias, where errors
in the label are conditioned on class and group. There is
a need for additional algorithmic approaches to tackle dif-
ferent forms of label bias.
In conclusion, addressing label bias in machine learn-
ing requires a multifaceted approach. Firstly, a proactive
strategy involves avoiding label bias from the start by care-
fully formulating the machine learning task. This neces-
sitates a keen awareness of potential biases and their im-
plications throughout the problem denition stage. Sec-
ondly, when considering bias or fairness mitigation meth-
ods, it is crucial to critically evaluate the assumptions un-
derlying these techniques. Many existing methods assume
label accuracy, which may lead to suboptimal outcomes
in real-world scenarios where mislabeling is prevalent. By
being mindful of these assumptions, researchers and prac-
titioners can make more informed choices in implement-
ing mitigation strategies. Finally, to advance the state-of-
the-art, there is a need for the development of novel and
improved bias measures as well as label mitigation meth-
ods. This involves exploring innovative approaches that go
beyond existing paradigms and contribute to the ongoing
evolution of techniques for ensuring fairness and reducing
bias in machine learning systems.
ACKNOWLEDGMENT. This research was supported in
part by NIH grant R01NS124642 and by Good Systems,
a UT Austin Grand Challenge to develop responsible AI
technologies.
References
[AAK20] Jacob Abernethy, Pranjal Awasthi, Matthäus
Kleindessner, Jamie Morgenstern, Chris Russell, and Jie
Zhang, Active sampling for min-max fairness, arXiv preprint
arXiv:2006.06879 (2020).
[AAT22] Hadis Anahideh, Abolfazl Asudeh, and Saravanan
Thirumuruganathan, Fair active learning, Expert Systems
with Applications 199 (2022), 116981.
[ADAC21] Nil-Jana Akpinar, Maria De-Arteaga, and Alexan-
dra Chouldechova, The effect of differential victim crime re-
porting on predictive policing systems, Proceedings of the 2021
ACM Conference on Fairness, Accountability, and Trans-
parency, 2021, pp. 838–849.
[ALMK16] Julia Angwin, Jeff Larson, Surya Mattu, and Lau-
ren Kirchner, Machine bias (2016).
[BBB21] Chlo´e Bakalar, Renata Barreto, Stevie Bergman, Mi-
randa Bogen, Bobbie Chern, Sam Corbett-Davies, Melissa
Hall, Isabel Kloumann, Michelle Lam, Joaquin Qui˜nonero
Candela, and others, Fairness on the ground: Applying al-
gorithmic fairness approaches to production systems, arXiv
preprint arXiv:2103.06172 (2021).
[BG18] Joy Buolamwini and Timnit Gebru, Gender shades: In-
tersectional accuracy disparities in commercial gender classi-
cation, Conference on fairness, accountability and trans-
parency, 2018, pp. 77–91.
[DAKD23] Aida Mostafazadeh Davani, Mohammad Atari,
Brendan Kennedy, and Morteza Dehghani, Hate speech clas-
siers learn normative social stereotypes, Transactions of the
Association for Computational Linguistics 11 (2023), 300–
319.
[Gar16] Clare Garvie, The perpetual line-up: Unregulated police
face recognition in america, Georgetown Law, Center on Pri-
vacy & Technology, 2016.
[HPS16] Moritz Hardt, Eric Price, and Nati Srebro, Equality of
opportunity in supervised learning, Advances in neural infor-
mation processing systems 29 (2016).
[KBK12] Brendan F Klare, Mark J Burge, Joshua C Klontz,
Richard W Vorder Bruegge, and Anil K Jain, Face recognition
performance: Role of demographic information, IEEE Trans-
actions on Information Forensics and Security 7(2012),
no. 6, 1789–1801.
[LDAS23] Yunyi Li, Maria De-Arteaga, and Maytal Saar-
Tsechansky, Mitigating label bias via decoupled condent
learning, arXiv preprint arXiv:2307.08945 (2023).
[LDAST22] Yunyi Li, Maria De-Arteaga, and Maytal Saar-
Tsechansky, When more data lead us astray: Active data acqui-
sition in the presence of label bias, Proceedings of the AAAI
Conference on Human Computation and Crowdsourcing,
2022, pp. 133–146.
[LI16] Kristian Lum and William Isaac, To predict and serve?,
Signicance 13 (2016), no. 5, 14–19.
[LKL17] Himabindu Lakkaraju, Jon Kleinberg, Jure
Leskovec, Jens Ludwig, and Sendhil Mullainathan, The
selective labels problem: Evaluating algorithmic predictions
in the presence of unobservables, Proceedings of the 23rd
ACM SIGKDD International Conference on Knowledge
Discovery and Data Mining, 2017, pp. 275–284.
[MPB21] Shira Mitchell, Eric Potash, Solon Barocas, Alexan-
der D’Amour, and Kristian Lum, Algorithmic fairness: choices,
assumptions, and denitions, Annu. Rev. Stat. Appl. 8(2021),
141–163, DOI 10.1146/annurev-statistics-042720-125902.
MR4243544
1076 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
[NDAF22] Terrence Neumann, Maria De-Arteaga, and Sina
Fazelpour, Justice in misinformation detection systems: An
analysis of algorithms, stakeholders, and potential harms, Pro-
ceedings of the 2022 ACM Conference on Fairness, Ac-
countability, and Transparency, 2022, pp. 1504–1515.
[OPVM19] Ziad Obermeyer, Brian Powers, Christine Vogeli,
and Sendhil Mullainathan, Dissecting racial bias in an algo-
rithm used to manage the health of populations, Science 366
(2019), no. 6464, 447–453.
[PB19] Samir Passi and Solon Barocas, Problem formulation
and fairness, Proceedings of the conference on fairness, ac-
countability, and transparency, 2019, pp. 39–48.
[SCG19] Maarten Sap, Dallas Card, Saadia Gabriel, Yejin
Choi, and Noah A Smith, The risk of racial bias in hate
speech detection, Proceedings of the 57th Annual Meeting
of the Association for Computational Linguistics, 2019,
pp. 1668–1678.
[Swe13] Latanya Sweeney, Discrimination in online ad delivery,
Communications of the ACM 56 (2013), no. 5, 44–54.
Yunyi Li Maria De-Arteaga
Maytal
Saar-Tsechansky
Credits
Figures 1 and 2 are courtesy of the authors.
Photo of Yunyi Li is courtesy of Jian Teng.
Photo of Maria De-Arteaga is courtesy of the University of
Texas at Austin.
Photo of Maytal Saar-Tsechansky is courtesy of Lauren Gerson,
McCombs School of Business, The University of Texas at
Austin.
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1077
AMS
CAREER
FAIR
Meet and recruit mathematical talent at the
Recruiters:
• Represent your company or organization
at the largest mathematics gathering in
the world
• Connect with hundreds of potential
candidates in the mathematical sciences
• Introduce career opportunities to
exceptional talent
Candidates:
• Meet with employers in Business,
Entrepreneurship, Government, Industry,
and Nonprofit (BEGIN)
• Discover how your mathematical training
makes you a strong candidate
• Explore a variety of career paths
Check the Joint Mathematics Meetings registration site for
updated dates and times: www.jointmathematicsmeetings.org
2025
AT
Learn more:
www.ams.org/career-fair
Double-Anonymous Peer
Review in Mathematics:
Implementation for American
Mathematical Society Journals
Dan Abramovich, Henry Cohn, David Futer,
and Robert Harington
In March 2022, the American Mathematical Society (AMS)
launched double-anonymous peer review across its jour-
nal program, beginning with Proceedings of the American
Mathematical Society and Representation Theory. In Feb-
ruary 2024 implementation expanded to the Transactions
and Memoirs of the American Mathematical Society. Further
rollout across AMS journals is ongoing. The goal of the
double-anonymous peer review policy is to reduce implicit
bias in peer review, including bias along gender, racial,
and geographical lines, along with seniority bias.
Peer review in mathematics has traditionally centered
on single-anonymous peer review. In this model, re-
viewers are aware of the identities of an article’s au-
thor(s), but the reviewer(s) remain anonymous to authors.
In double-anonymous peer review (formerly known as
Dan Abramovich serves as managing editor of the Transactions and Memoirs
of the AMS. He is L. Herbert Ballou University Professor of Mathematics at
Brown University. His email address is Dan_Abramovich@Brown.edu.
Henry Cohn is a principal researcher at Microsoft, and an adjunct professor of
mathematics at MIT. His email address is cohn@math.mit.edu.
David Futer serves as managing editor of the Proceedings of the AMS. He is
a professor of mathematics at Temple University. His email address is david
.futer@temple.edu.
Robert Harington is the chief publishing ofcer at the American Mathematical
Society. His email address is rmh@ams.org.
For permission to reprint this article, please contact:
reprint-permission@ams.org.
DOI: https://doi.org/10.1090/noti3013
double-blind), both reviewers and authors are anonymous
to each other.
Peer review is central to scholarship and is deployed
across all academic disciplines. As Melinda Baldwin dis-
cusses in her excellent article [2] (see similar discussion in
[1]):
The most widely accepted story about peer re-
view’s origin credits Henry Oldenburg with in-
venting it for the seventeenth-century Philosophical
Transactions of the Royal Society, creating the impres-
sion that refereeing has been an unchanging part
of science for over three hundred years. However,
new historical work is beginning to shed more
light on peer review’s development—and the real
story is far more complicated than the neat tale of
Oldenburg inventing refereeing out of whole cloth
during the Scientic Revolution.
Most existing histories of peer review have fo-
cused on the emergence of the scientic referee
during the nineteenth century or on the inner
workings of referee systems at particular journals.
Those studies have shown that refereeing was not
initially thought of as a process that bestowed
scientic credibility and that many high-prole
journals and grant organizations had unsystem-
atic (or nonexistent) refereeing processes well into
the twentieth century.
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1079
Across much of the social sciences and humanities,
double-anonymous peer review is the standard system
of peer review. It is also becoming increasingly com-
mon in the natural sciences. In physics, the Institute of
Physics Publishing (IOPP) launched double-anonymous
peer review across all of its sixty-one journals in 2021
[3]. In astronomy, NASA practices double-anonymous
peer review in its grant applications. In certain sub-
disciplines of computer science, including cryptography,
double-anonymous peer review is quite common.
Why did the AMS move to implement double-
anonymous peer review across its journals? The moti-
vation for the AMS Council to adopt this change is so
the referee’s rst impressions of a paper are not domi-
nated by its list of authors and their afliations. Instead,
double-anonymous refereeing aims to focus attention on
the mathematics. It may still be possible to infer who
wrote a submission, but it is hoped that double-anonymity
lowers the likelihood of implicit bias and therefore sup-
ports inclusivity and diversity across mathematics.
The AMS Council approved the transition to double-
anonymous peer review as a policy in January 2021.
The AMS President, Ruth Charney, subsequently formed
the Double-Anonymous Refereeing Committee, on which
we served, charged with discussing how an implemen-
tation could work. During implementation discussions,
it became clear that in mathematics, this form of peer
review needed to be implemented with a light touch.
While authors are required to submit manuscripts with-
out author names or afliations, they are not required to
anonymize references, acknowledgments, or funding in-
formation, or to make other edits. Journal editors con-
tinue to have access to author and reviewer identities in
double-anonymous peer review. In particular, editors can
use this information in order to avoid conicts of interest.
The new policy is guided by the AMS’s strong belief in
open dissemination of mathematics. There are no new re-
strictions on how authors choose to disseminate and pub-
licize their work—for example, by giving talks, posting
preprints that include author names, and discussing with
colleagues.
Referees are asked not to go out of their way to try to
identify authors, but the AMS policy accepts that referees
will sometimes already know who wrote the submission.
For instance, a referee may already know the paper if they
have seen it posted to the arXiv preprint server. Even in
those cases, double-anonymous refereeing is a statement
of principle about how submissions should be evaluated.
Authors submitting articles to AMS journals employ-
ing double-anonymous peer review are tasked with the
following tasks that represent a light touch to double-
anonymous peer review.
At article submission, authors are asked to submit a ver-
sion which does not include their names or afliations in
the preamble, headers, or footnotes of the paper. Authors
may also choose to reword other instances in the paper
that would tend to identify them, but this is not required.
For example, a phrase like “we showed” or “the second au-
thor showed” could be replaced by “Smith showed,” refer-
ring to the author in the third person. If a paper is accepted,
such wording can be adjusted prior to publication. At ini-
tial submission, AMS staff screens for author names in the
paper itself, in the running heads, and in the afliations
list.
A typical communication used in requests and re-
minders typically reads: “The PDF must not contain any
information that can identify the author(s). In particu-
lar, there must be no authors’ names nor authors’ aflia-
tions/email addresses listed in the paper, nor any links to
their identities in any way.” In addition, identiable ver-
sions are never released to referees.
Implementation of this light-touch approach to double-
anonymous peer review is underway. As the AMS rolls this
peer review model out across the journal portfolio, it is
important to gather data and assess the effects of double-
anonymous peer review on authors, reviewers, and editors.
The AMS is in the early stages of developing an
organization-wide approach to demographic data collec-
tion for both our membership and publishing/research
communities. In order to monitor the existence of many
types of bias in our research community this data will need
to be collected both at submission of a manuscript and
post decision. In the meantime, we can choose to look at
geographical data and institutional data.
The data to be collected will help the AMS see to what
extent the goals of the policy are achieved. But, rst and
foremost, we need to know that no serious harm is done.
Concerns raised by journal editors included: (i) Will peo-
ple continue to agree to provide expert opinions and write
detailed referee reports? (ii) Will authors continue to sub-
mit papers appropriate to the journal’s portfolio? (iii) Will
colleagues continue to agree to serve as editors? A previous
introduction of a double-anonymous policy in 1975 was
abandoned in 1980 on these counts [4].
After double-anonymous peer review was rolled out to
Proceedings and Representation Theory, we were able to ex-
amine the rst two of these concerns.
Both Proceedings and Representation Theory transitioned
to double-anonymous peer review in 2022. As illustrated
in Figure 1, double-anonymous peer review does not seem
to have had any impact that sets them apart from our other
journals.
1080 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
Figure 1.
Figure 2.
Figure 2 is a preliminary look at reviewer denial data for
the two AMS journals utilizing double-anonymous peer re-
view, versus all other AMS journals.
This data indicates that reviewers are not turning down
an opportunity to review for the AMS due to journals em-
bracing double-anonymous peer review.
For the third concern (editors’ willingness to serve), the
data sample is too small to conduct statistics. Sufce it to
say that the AMS Editorial Boards Committee did not face
difculties in lling editorial vacancies in the two journals
in which the policy was implemented.
References
[1] Melinda Baldwin, In referees we trust?, Physics Today
70 (2017), no. 2, https://doi.org/10.1063/PT.3
.3463.
[2] Melinda Baldwin, Scientic Autonomy, Public Accountabil-
ity, and the Rise of “Peer Review” in the Cold War United
States, Isis 109 (2018), no. 3, https://www.journals
.uchicago.edu/doi/full/10.1086/700070.
[3] Rachael Harper, IOP Publishing commits to adopt-
ing double-anonymous peer review for all journals
(2020), https://ioppublishing.org/news/iop
-publishing-commits-to-adopting-double-blind
-peer-review-for-all-journals/.
[4] Everett Pitcher, A history of the second fty years, Amer-
ican Mathematical Society, 1939–1988, Vol. I, Ameri-
can Mathematical Society, Providence, RI, 1988, DOI
10.1007/bf01017168. MR1002190
Dan Abramovich Henry Cohn
David Futer Robert Harington
Credits
Figures 1 and 2 are courtesy of the AMS.
Photo of Dan Abramovich is courtesy of Deidre Confar.
Photo of Henry Cohn is courtesy of MFO. CC BY-Sa 2.0 DE.
Photo of David Futer is courtesy of the AMS.
Photo of Robert Harington is courtesy of the AMS.
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1081
WASHINGTON UPDATE
The AMS Marks Twenty Years
of Sending Mathematicians
to Congress
Karen Saxe
In the fall of 2005, mathematician David Weinreich joined
a few dozen other scientists—each sponsored by a scien-
tic society—bringing their expertise to Congress. David
was the rst Congressional Fellow sponsored by the AMS,
though the program began in 1973, with a class of seven
Fellows.
AMS Congressional Fellows are part of a larger program
bringing scientic expertise to the US government; the 30–
35 Congressional Fellows are joined by about 250–275
who work in executive branch agencies. There is also one
Fellow placed in the judicial branch. Fellows bring scien-
tic expertise to government decision-makers. The ques-
tion of where Congress gets their science and tech informa-
tion is a great one and has a complex answer. The rst an-
swer is that members of Congress have science policy advi-
sors, and some have strong scientic backgrounds. From
1974–1995, the Ofce of Technology Assessment (OTA)
served as a primary support, providing Congress with ob-
jective analyses of science and technology issues.1This
Karen Saxe is senior vice president, Government Relations, at the AMS. Her
email address is kxs@ams.org.
For permission to reprint this article, please contact:
reprint-permission@ams.org.
DOI: https://doi.org/10.1090/noti3003
1https://crsreports.congress.gov/product/details?prodcode=
R46327
ofce inuenced legislation and fostered relationships be-
tween Congress and the scientic and technological com-
munity. Congressional Fellows help ll the gap left by the
OTA’s disbandment.
After the fellowship year, AMS Congressional Fellows
follow different career paths. Almost all AMS Fellows have
come from academia. A few returned to academia, but
most have not. The AMS Congressional Fellowship has
proved transformational for the careers of individual math-
ematicians for twenty years.
2005–2006 Fellow David Weinreich focused on a wide
range of issues during his year working for Representa-
tive Robert Andrews from southwestern New Jersey, issues
from agriculture to water resources. One success during
his fellowship was a provision of law that prevented log-
ging in Alaska’s Tongass National Forest. His fellowship
was followed by full-time employment in the US House
of Representatives. For four years he was the Legislative
Director for Representative Bob Etheridge of North Car-
olina, followed by helping establish the ofce for rst-year
Representative Hansen Clarke of Detroit as Policy Direc-
tor. In 2011, he left Congress and founded a consulting
rm, the Weinreich Strategic Group, and he is very active
as the Director of Policy and Government Relations of
1082 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
Washington Update
STM.2STM is the leading association of scholarly publish-
ers and its members publish, roughly, two-thirds of all
published papers in science, technology, medicine, social
sciences, and humanities. “The fellowship not only trans-
formed my career trajectory, but enabled me to bring my
analytical skills and knowledge of the research commu-
nity to inform public policy,” David said. “It also led
to many productive conversations with fellow mathemati-
cians about funding policy and how to effect change in the
government.”
2019–2020 Fellow Lucia Simonelli is a senior climate re-
searcher at Giving Green. Following her year working for
Rhode Island’s Senator Sheldon Whitehouse on climate
and energy policy, she transitioned to full-time work in cli-
mate. “The AMS Congressional Fellowship enabled me to
witness the importance of integrating scientists into poli-
cymaking and reect on how the skills derived from math-
ematical training can transfer to a broader context. The
experience helped restore my respect for, and faith in the
government, and it taught me the power and importance
of creating a strong network—not for self-gain, but more
as a collective that can work together in various ways to
support the advancement of common goals and causes.”
2021–2022 Fellow AJ Stewart, who spent his fellowship
year working on economic policy for Georgia’s Senator
Raphael Warnock, is now a policy advisor at the US De-
partment of the Treasury. There, he investigates national
security issues stemming from foreign investment in the
United States. “I was always good at math and even though
it took me a while to nd my way towards becoming a
mathematician, once I did I was hooked. However, I as-
sumed that there was only one way to be a mathemati-
cian, by performing research at a university. This fellow-
ship opened my eyes to how mathematics is applied across
government in amazingly unique ways. Being able to tan-
gibly apply mathematics every day and see the effects of
that work has changed my whole view of what it means to
perform mathematics.”
This short article highlights the post-fellowship paths
of only three of the amazing AMS Congressional Fellow
alumni group.
Read more about all alumni fellows, and nd out
how you can join this wonderful group (applications
due February 1), at: https://www.ams.org/government
/government/ams-congressional-fellowship.
2The International Association of Scientic, Technical and Medical Publishers,
known as STM, has more than 140 members across the globe including all
the major commercial publishers, professional society publishers, and university
presses. The AMS is a member.
Karen Saxe
Credits
Photo of Karen Saxe is courtesy of Macalester College/David
Turner.
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1083
Joint Prizes
JPBM Communications Award
This award is given each year to reward and encourage com-
municators who, on a sustained basis, bring mathematical
ideas and information to nonmathematical audiences.
About this award. This award was established by the Joint
Policy Board for Mathematics (JPBM) in 1988. JPBM is
a collaborative effort of the American Mathematical Soci-
ety, the Mathematical Association of America, the Society
for Industrial and Applied Mathematics, and the American
Statistical Association.
Up to two awards of US$2,000 are made annually. Both
mathematicians and nonmathematicians are eligible.
Next prize. January 2025
Nomination period. Open
Nomination procedure. Nominations should be submit-
ted on MathPrograms.org. Note: Nominations collected
before September 15 in year N will be considered for an
award in year N+2.
Information on how to nominate can be found here:
https://www.ams.org/jpbm-comm-award.
Fellowships and Programs
Joan and Joseph Birman Fellowship
for Women Scholars
The Joan and Joseph Birman Fellowship for Women Schol-
ars is a midcareer research fellowship specially designed to
t the unique needs of women. This program is made pos-
sible by a generous gift from Joan and Joseph Birman. One
award will be made for the 2024–2025 academic year in
the amount of US$50,000. AMS membership will also be
offered to the recipient for the duration of the fellowship.
About this fellowship. The fellowship seeks to address
the paucity of women at the highest levels of research in
mathematics by giving exceptionally talented women extra
research support during their midcareer years. The most
likely awardee will be a midcareer woman whose achieve-
ments demonstrate signicant potential for further con-
tributions to mathematics. Applications will be accepted
from mathematicians currently holding a tenured, tenure-
track, postdoctoral, or comparable (at the discretion of the
selection committee) position at a US institution.
The fellowship will be directed toward those for whom
the award will make a real difference in the development
of their research career. Candidates must have a statement
regarding the applicant’s overall program of research, past
and planned, that is meaningful to mathematicians who
are not specialists. The statement should be no more than
three pages, including bibliographical references. Special
circumstances (such as time taken off for care of children
or other family members) may be taken into consideration
in making the award. Awardees may use the fellowship in
any way that most effectively enables their research—for
instance, for release time, participation in special research
programs, travel support, childcare, etc. The award is is-
sued through the recipient’s institution, and no part of it
may be utilized for indirect costs.
Application period. Applications will be collected via
MathPrograms.org July 15, 2024–September 30, 2024
(11:59 p.m. ET). Find more information at https://www
.ams.org/birman-fellow. For questions, contact the
Programs Department at fellowships@ams.org.
Centennial Research Fellowship
The AMS Centennial Fellowship Program makes an award
annually to an outstanding mathematician to help further
their career in research. One award will be made for the
2024–2025 academic year in the amount of US$50,000.
Acceptance of the fellowship cannot be postponed. AMS
membership will also be offered to the recipient for the
duration of the fellowship.
1084 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
About this fellowship. Eligibility: The eligibility rules are
as follows:
The primary selection criterion for the Centennial Fel-
lowship is the excellence of the candidate’s research.
•Preference will be given to candidates who have not
had extensive fellowship support in the past.
•Recipients may not hold the Centennial Fellowship
concurrently with another research fellowship such
as a Sloan, NSF Postdoctoral fellowship, or CAREER
award.
•Under normal circumstances, the fellowship cannot
be deferred.
•A recipient of the fellowship shall have held his or her
doctoral degree for at least three years and not more
than twelve years at the inception of the award (that is,
received between September 1, 2013, and September
1, 2022).
•Applications will be accepted from mathematicians
currently holding a tenured, tenure-track, postdoc-
toral, or comparable (at the discretion of the selection
committee) position at a US institution.
Applications should include a detailed research plan
for the fellowship period that is contextualized by the re-
search statement. The plan should include a description of
how the fellowship will support the applicant’s success. It
should be no more than one page. The selection commit-
tee will consider the plan in addition to the quality of the
candidate’s research and will try to award the fellowship
to those for whom the award would make a real difference
in the development of their research careers. Work in all
areas of mathematics, including interdisciplinary work, is
eligible.
Application period. Applications will be collected via
MathPrograms.org July 15, 2024–September 30, 2024
(11:59 p.m. ET). Find more information at https://www
.ams.org/centfellow. For questions, contact the Pro-
grams Department at fellowships@ams.org.
Claytor-Gilmer Fellowship
The AMS established the Claytor-Gilmer Fellowship to fur-
ther excellence in mathematics research and to help gener-
ate wider and sustained participation by Black mathemati-
cians. One award will be made for the 2024–2025 aca-
demic year in the amount of US$50,000. AMS member-
ship will also be offered to the recipient for the duration
of the fellowship.
About this fellowship. Awardees may use the fellowship
in any way that most effectively enables their research—for
instance, for release time, participation in special research
programs, travel support, childcare, etc. The award is is-
sued through the recipient’s institution, and no part of it
may be utilized for indirect costs. Given the aims of the fel-
lowship, the most likely awardee will be a midcareer Black
mathematician whose achievements demonstrate signi-
cant potential for further contributions to mathematics.
Applications will be accepted from mathematicians cur-
rently holding a tenured, tenure-track, postdoctoral, or
comparable (at the discretion of the selection committee)
position at a US institution.
Application period. Applications will be collected via
MathPrograms.org July 15, 2024–September 30, 2024
(11:59 p.m. ET). Find more information at https://www
.ams.org/claytor-gilmer. For questions, contact the
Programs Department at fellowships@ams.org.
Stefan Bergman Fellowship
The Stefan Bergman Fellowship was established in 2023
with the proceeds of the Stefan Bergman Trust to support
the advancement of the research portfolio of a mathemati-
cian who specializes in the areas of real analysis, complex
analysis, or partial differential equations. One award will
be made for the 2024–2025 academic year in the amount
of US$25,000. AMS membership will also be offered to
the recipient for the duration of the fellowship.
About this fellowship. Applications will be accepted
from mathematicians at a US institution who have not re-
ceived tenure or comparable (at the discretion of the selec-
tion committee) and have not held signicant fellowship
support.
Awardees may use the fellowship in any way that most
effectively enables their research—for instance, for release
time, participation in special research programs, travel sup-
port, childcare, etc. The award is issued through the re-
cipient’s institution, and no part of it may be utilized for
indirect costs.
Application period. Applications will be collected via
MathPrograms.org July 15, 2024–September 30, 2024
(11:59 p.m. ET). Find more information at https://www
.ams.org/bergman-fellow. For questions, contact the
Programs Department at fellowships@ams.org.
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1085
Executive Director Report
Lucy Maddock, Interim Executive Director
DOI: https://doi.org/10.1090/noti2992
Both transition and growth marked 2023 for the American
Mathematical Society (AMS). After seven years as AMS ex-
ecutive director (ED), Dr. Catherine Roberts, resigned in
March, and the next several months were dedicated to tran-
sition planning. In May, the AMS Board of Trustees (BT)
appointed me to serve as interim ED, in addition to my on-
going position as chief nancial ofcer (CFO). Meanwhile,
an Executive Director Search Committee began a nine-month,
nationwide search. In January 2024, the BT announced
the appointment of Dr. John Meier, provost and David
M. and Linda Roth Professor of Mathematics of Lafayette
College, as the new ED. He will begin a ve-year term on
July 1, 2024, and we are extremely excited to welcome him
aboard.
Like elsewhere, change has been a consistent theme
at the AMS for the past four years. As the world navi-
gated challenges of the COVID-19 pandemic, many com-
panies experienced increased employee turnover in a pe-
riod known as the “Great Resignation.” The AMS was no
exception; however, employee numbers have since stabi-
lized. Particularly noteworthy is that 2023 marked the year
with the lowest AMS employee turnover since 2018, and
we expect this stabilization to continue in 2024.
In August, the BT approved a new organizational struc-
ture for the Society upon the recommendation of RW
Jones, a strategic consulting rm focused on education.
As a result, seven AMS divisions were transformed into
three, allowing for greater collaboration and a streamlined
reporting hierarchy. Additionally, Ashley Northington,
MPA, the senior vice president and managing director of
RW Jones, was brought on as the AMS’s interim chief ex-
ternal relations ofcer. Despite these ongoing transitions,
staff have successfully adapted and remain steady in sup-
porting the AMS mission of advancing research and con-
necting the mathematical community.
1086 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
In January, the rst in-person reimagined Joint Mathe-
matics Meetings (JMM) (and rst in-person annual meet-
ing since January 2020) took place in Boston, with sixteen
partner societies and 5,140 participants. The Meetings De-
partment worked closely with partners to arrange their pro-
grams and exhibits, including soliciting their input for the
grand opening reception. Meeting in-person again was an
enormous success, and seventeen partners were secured for
2024. The AMS also held seven sectional meetings in At-
lanta; Cincinnati; Fresno, California; Buffalo, New York;
Omaha, Nebraska; and Mobile, Alabama. The spring East-
ern Sectional was held virtually.
Growth and retention in membership were notable
achievements in 2023. There were 6,769 new member en-
rollments, a record number and 1,400 more than in 2022.
This growth includes 2,250 new dues-paying members. In-
creased efforts in membership retention paid off with 952
more members renewing in 2023 than in 2022. This will
continue to be an area of focus.
Meanwhile, the AMS continued to expand its services
and resources for the mathematical community. Two new
programs were launched: the Stefan Bergman Fellowship
and the AMS-Simons Research Enhancement Grants for
PUI Faculty, and ve new prizes were established: the
Ivo and Renata Babuˇska Thesis Prize (awarded in 2024),
the Elias M. Stein Prize for New Perspectives in Analysis
(awarded in 2024), the Elias M. Stein Prize for Transfor-
mative Exposition (to be awarded in 2025), the Elias M.
Stein Mentoring Award (to be awarded in 2026), and the I.
Martin Isaacs Prize for Excellence in Mathematical Writing
(to be awarded in 2025).
Additionally, the Ofce of Equity, Diversity, and Inclu-
sion and the Division of Meetings and Professional Ser-
vices (now both part of External Affairs) secured NSF sup-
plemental funding for a new AMS partnership with the In-
clusive Graduate Education Network (IGEN), an alliance
of disciplinary societies, research centers, and other orga-
nizations dedicated to advancing equity in STEM gradu-
ate education. The proposed IGEN Mathematics Initia-
tive (IGEN-Math) is a one-year national capacity-building
project. It’s team will work closely with internal and ex-
ternal stakeholders, including IGEN Alliance partner orga-
nizations and the IGEN-Math Advisory Group, to engage
the mathematics community in developing the framework
for a centralized hub of bridge programs in mathematics
intended to improve equity and inclusion in mathematics
graduate education.
As part of the AMS’s ongoing accessibility initiative,
MathViewer (accessible HTML) was expanded to include
all primary journals and added to ePub production work-
ows, beginning with a retrospective conversion of the
Graduate Studies in Mathematics series. The MathViewer
journal article output increased from 275 articles to 939.
The AMS is now in full compliance with existing open ac-
cess (OA) journal mandates through zero-embargo Green
OA, Diamond OA, and Gold OA (on the B journals). In ad-
dition, the new user interface of MathSciNet was released
in June, the rst major revision since 2006. The updated
interface incorporates a modern look, greater use of the
database to help users rene their searches, and improved
accessibility, especially for users with vision impairments
or ne motor control limitations.
Each year, speakers bring science directly to Capitol Hill
via congressional briengs organized by the Ofce of Gov-
ernment Relations (OGR). These speakers offer stories of
how federal investment in basic research in math and sci-
ence pays off for American taxpayers and helps the nation
remain a world leader in innovation. Beginning in 2023,
each brieng highlighted work connected to one of the
National Science Foundation (NSF)-funded Mathematical
Sciences Institutes; last year’s was in partnership with the
Institute for Pure and Applied Mathematics (IPAM). The
AMS helped organize and host the following briengs
with various coalitions: “Investing to Win: The Essential
Role of Federally Funded Research,” “Federally Funded Re-
search and the Advent of Articial Intelligence: A TFAI De-
constructing Event,” “The National Imperative to Develop
STEM Talent: Why the Investment in Education Matters,”
and “STEM 101.”
The Ofce of Government Relations also made more
than 100 visits to Congressional and Executive Branch of-
ces in 2023, both with AMS leadership and in partnership
with various coalitions. In addition, it supported the work
of the Advisory Group on Articial Intelligence (AI) and
the Mathematical Community, which is charged with fo-
cusing on issues at the forefront of developments in AI,
including: the role of mathematics in the development
and deployment of articial intelligence, the use of AI in
publications, education, and research, and its impact on
research in mathematics and our community.
As OGR built relationships in Washington, DC, our
Communications and Marketing Department focused
on building community and conveying the AMS brand
through the organization’s messaging. Together with Cre-
ative Services, they implemented a marketing and promo-
tion campaign planning process to help departments meet
their messaging objectives and to advance AMS strategic
priorities. This process created a cohesive visual and mes-
saging brand for all the pieces of a campaign, including so-
cial media, advertising, meeting materials, brochures, web-
site graphics, and email. Additionally, by planning, creat-
ing, and implementing content for each type of platform,
there was a 234-percent increase in users of AMS social
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1087
platforms and a 221-percent upsurge in impressions, the
number of times AMS content was seen.
Led by the Secretary’s Ofce, the AMS is conducting a
comprehensive analysis of the bylaws to ensure that they
align with the current legal landscape, that they serve the
organization as well as possible, and that they reect the
AMS’s values. As part of this analysis, legal counsel was
asked to assist in bringing the bylaws into compliance with
ambient law and to make additional suggestions for im-
provement. A Bylaws Review Taskforce was created, and
its recommendations will be brought to Council in Janu-
ary 2025. The Secretary’s Ofce also worked with Informa-
tion Services at the AMS to create a search portal for mem-
bers of the Committee on Committees and Nominating
Committee to better illuminate past membership in AMS
committees. This tool will also help in recruiting election
candidates and committee members, and the hope is that
it can eventually be expanded.
In the area of planned giving, the AMS saw several estate
gifts of more than one million dollars, following a decade
of preparation and stewardship. With guidance and en-
couragement from the CFO and the Development Com-
mittee, the Development Department revised the gift ac-
ceptance policy and the AMS Book Fund to provide both
more exible and more enduring support for the AMS.
Also, the hiring of a development communications ofcer
has increased the AMS’s capacity to apply for grants from
foundations and government agencies.
In other highlights, beginning in 2022 and continuing
into 2023, Human Resources designed and implemented
a formal internship program for graduate and undergrad-
uate students. The AMS will look to partner with local
colleges to provide recurring opportunities for both work-
study students as well as traditional internship opportuni-
ties. This internship program allows students to become
involved with the organization and connect with other
members of the mathematical community.
The AMS applied for the Employee Tax Retention Credit
(ERTC) under the Coronavirus Aid, Relief, and Economic
Security Act (CARES Act). This is a payroll tax credit aimed
at employers impacted by the COVID-19 pandemic. As a
result, the AMS will receive $2,727,926 in future credits.
Additionally, some 1,100 statements and 800 payments
went out to authors in mid-April for 2023 sales and roy-
alties. More than $350,000 in royalty payments were pro-
cessed. Finance Division staff continue to attempt to con-
tact some 500 authors who are missing tax or other docu-
mentation needed to process payment.
The Computer Sciences Division participated in two
critical projects, one to become compliant with version
four of the Payment Card Industry’s Data Security Stan-
dards (PCI-DSS) and a second to select and implement a
software package for collection and remittance of sales tax
in all states in which the AMS has nexus. In addition, more
applications were integrated into Single Sign-On with mul-
tifactor authentication, and the network rewalls in Prov-
idence were upgraded to next-generation models with en-
hanced threat detection.
Although 2023 was marked by both signicant tran-
sitions and notable accomplishments, the AMS is well-
positioned to build upon its successes. Employee turnover
stabilized, and the launch of a new internal organizational
structure created a more collaborative environment for
staff. The Society organized and held its rst in-person
JMM since January 2020 and saw achievements in accessi-
bility, outreach, development, campaign planning, mem-
bership, programs offered to the math community, and
more. Despite the challenges of 2023, the AMS thrived,
and I look forward to what is ahead for the organization
under Dr. Meier’s leadership.
Lucy Maddock
Interim Executive Director
May 2024
1088 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
105 Notices of the AmericAN mAthemAticAl society Volume 66, Number 1
Special Section
2019 Election
FROM THE AMS SECRETARY
Ballots
AMS members will receive email with instructions for vot-
ing online by August 20, or a paper ballot by September 20.
If you do not receive this information by that date, please
contact the AMS (preferably before October 1) to request
a ballot. Send email to ballot@ams.org or call the AMS
at 800-321-4267 (within the US or Canada) or 401-455-
4000 (worldwide). The deadline for receipt of ballots is
November 2, 2018.
Write-in Votes
It is suggested that names for write-in votes be accompa-
nied by the institution or web address of the individual for
whom the vote is cast.
Replacement Ballots
A member who has not received a ballot by September 20,
2018, or who has received a ballot but has accidentally
spoiled it, may write to ballot@ams.org or Secretary of
the AMS, 201 Charles Street, Providence, RI 02904-2213,
USA, asking for a second ballot. The request should in-
clude the individual’s member code and the address to
which the replacement ballot should be sent. Immediately
upon receipt of the request a second ballot, indistin-
guishable from the original, will be sent by first class or
airmail. However, the deadline for receipt of ballots will
not be extended.
Biographies of Candidates
The next several pages contain biographical information
about all candidates. All candidates were given the oppor-
tunity to provide a statement of not more than 200 words
to appear at the end of their biographical information.
Photos were supplied by the candidates.
Description of Offices
The vice president and the members at large of the
Council serve for three years on the Council. That body
determines all scientific policy of the Society, creates and
oversees numerous committees, appoints the treasurers
and members of the Secretariat, makes nominations of
candidates for future elections, and determines the chief
editors of several key editorial boards. Typically, each of
these new members of the Council will also serve on one
of the Society’s five policy committees. Current and past
members of the Council may be found here: www.ams
.org/comm-all.html#COUNCIL.
Vice President
(one to be elected)
Sara Billey
Abigail Thompson
Board of Trustees
(one to be elected)
Matthew Ando
Rick Miranda
Member at Large
of the Council
(five to be elected)
Dan Freed
Fernando Q. Gouvêa
Christopher D. Hacon
Daniel Krashen
Susan Loepp
Lenhard Ng
Kasso A. Okoudjou
Maria Cristina Pereyra
Hal Schenck
Melanie Matchett Wood
Nominating Committee
(three to be elected)
Sami H. Assaf
Ricardo Cortez
Rebecca Garcia
Yongbin Ruan
David Savitt
Deane Yang
Editorial Boards Committee
(two to be elected)
Ian Agol
David Marker
James McKernan
Terence Tao
List of Candidates–2019 Election
Update in
master page
MP: a-ballot-opener
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2024 Election
Special Section
List of Candidates
Vice President
(one to be elected)
Minerva Cordero
Malabika Pramanik
Board of Trustees
(one to be elected)
Henry Cohn
Brooke E. Shipley
Member at Large of the Council
(ve to be elected)
Alejandra Alvarado
Benjamin Antieau
Dawei Chen
Emily Clader
Carla Cotwright
Dan Isaksen
Yvonne Lai
Christopher J. Leininger
Adriana Salerno
Pham Huu Tiep
Nominating Committee
(three to be elected)
David Fisher
Aimee S. A. Johnson
Lily Signe Khadjavi
Kasso A. Okoudjou
Gigliola Staflani
Jared Wunsch
Editorial Boards Committee
(two to be elected)
Ivan Corwin
Irene Fonseca
Christopher Hacon
Michael J. Larsen
Online Ballots
AMS members will receive email with instructions for vot-
ing online by August 12. If you do not receive this infor-
mation by that date, please contact the AMS (preferably be-
fore October 1). Send email to election@ams.org. The
deadline for receipt of online ballots is November 1, 2024.
Starting in 2024, the AMS Election will be conducted fully on-
line without the ability to receive a paper ballot.
Write-in Votes
It is suggested that names for write-in votes be accompa-
nied by the institution or web address of the individual
for whom the vote is cast.
Biographies of Candidates
The next several pages contain biographical information
about all candidates. All candidates were given the oppor-
tunity to provide a statement of not more than 200 words
(400 words for presidential candidates) to appear at the
end of their biographical information. Photos were sup-
plied by the candidates.
Description of Offices
The vice president and the members at large of the Coun-
cil serve for three years on the Council. That body deter-
mines all scientic policy of the Society, creates and over-
sees numerous committees, appoints the treasurers and
members of the Secretariat, makes nominations of candi-
dates for future elections, and determines the chief editors
of several key editorial boards. Typically, each of these new
members of the Council will also serve on one of the Soci-
ety’s six policy committees. Current and past members of
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1089
Election Special Section
the Council may be found here: https://www.ams.org
/comm-all.html#COUNCIL.
The Board of Trustees, of whom you will be electing
one member for a ve-year term, has complete duciary
responsibility for the Society. Among other activities, the
trustees determine the annual budget of the Society, prices
of journals, salaries of employees, dues (in cooperation
with the Council), registration fees for meetings, and in-
vestment policy for the Society’s reserves. The person you
elect will likely serve as chair of the Board of Trustees dur-
ing the fourth year of the term. Current and past members
of the Board of Trustees, as well as the full charge for a
trustee, may be found here: https://www.ams.org/comm
-all.html#BT.
The candidates for vice president, members at large,
and trustee were suggested to the Council either by the
Nominating Committee or by petition from members.
While the Council has the nal nominating responsibil-
ity, the groundwork is laid by the Nominating Committee.
The candidates for election to the Nominating Committee
were nominated by the current president, Bryna Kra. The
three elected will serve three-year terms. The main work of
the Nominating Committee takes place during the annual
meeting of the Society, during which it meets over three
days. The Committee then reports its suggestions to the
spring Council, which makes the nal nominations. Cur-
rent and past members of the Nominating Committee, as
well as the full charge, may be found here: https://www
.ams.org/comm-all.html#NOMCOM.
The Editorial Boards Committee is responsible for the
stafng of the editorial boards of the Society. Members
are elected for three-year terms from a list of candidates
named by the president. The Editorial Boards Committee
makes recommendations for almost all editorial boards
of the Society. Managing editors of Communications of
the AMS, Journal of the AMS, Mathematics of Computation,
Proceedings of the AMS, and Transactions of the AMS; and
chairs of the Colloquium, Mathematical Surveys and Mono-
graphs, and Mathematical Reviews editorial committees are
ofcially appointed by the Council upon recommenda-
tion by the Editorial Boards Committee. In virtually all
other cases, the editors are appointed by the president,
again upon recommendation by the Editorial Boards Com-
mittee. Current and past members of the Editorial Boards
Committee, as well as the full charge, may be found here:
https://www.ams.org/comm-all.html#EBC.
Elections to the Nominating Committee and the Edi-
torial Boards Committee are conducted by the method of
approval voting. In the approval voting method, you can
vote for as many or as few of the candidates as you wish.
The candidates with the greatest number of the votes win
the election.
A Note from AMS Secretary Boris Hasselblatt
The choices you make in these elections impact the direc-
tion the Society takes in its publications, conferences, pro-
grams, and policies. On behalf of the other ofcers and
Council members, I urge you to take a few minutes to re-
view the candidates’ biographies, ll out your ballot, and
submit it. The Society belongs to its members and by vot-
ing, you will inuence its policies and priorities.
Also, I invite you to consider other ways of participating
in Society activities. The Nominating Committee, the Edi-
torial Boards Committee, and the Committee on Commit-
tees are always interested in learning of members who are
willing to serve the Society in various capacities. Names
are always welcome, particularly when accompanied by a
few words detailing the person’s background and interests.
Self-nominations are probably the most useful. Recom-
mendations can be transmitted through an online form
(https://www.ams.org/committee-nominate) or sent
directly to the secretary: secretary@ams.org or Ofce of
the Secretary, American Mathematical Society, 201 Charles
Street, Providence, RI 02904-2213 USA.
1090 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
105 Notices of the AmericAN mAthemAticAl society Volume 66, Number 1
Special Section
2019 Election
FROM THE AMS SECRETARY
Ballots
AMS members will receive email with instructions for vot-
ing online by August 20, or a paper ballot by September 20.
If you do not receive this information by that date, please
contact the AMS (preferably before October 1) to request
a ballot. Send email to ballot@ams.org or call the AMS
at 800-321-4267 (within the US or Canada) or 401-455-
4000 (worldwide). The deadline for receipt of ballots is
November 2, 2018.
Write-in Votes
It is suggested that names for write-in votes be accompa-
nied by the institution or web address of the individual for
whom the vote is cast.
Replacement Ballots
A member who has not received a ballot by September 20,
2018, or who has received a ballot but has accidentally
spoiled it, may write to ballot@ams.org or Secretary of
the AMS, 201 Charles Street, Providence, RI 02904-2213,
USA, asking for a second ballot. The request should in-
clude the individual’s member code and the address to
which the replacement ballot should be sent. Immediately
upon receipt of the request a second ballot, indistin-
guishable from the original, will be sent by first class or
airmail. However, the deadline for receipt of ballots will
not be extended.
Biographies of Candidates
The next several pages contain biographical information
about all candidates. All candidates were given the oppor-
tunity to provide a statement of not more than 200 words
to appear at the end of their biographical information.
Photos were supplied by the candidates.
Description of Offices
The vice president and the members at large of the
Council serve for three years on the Council. That body
determines all scientific policy of the Society, creates and
oversees numerous committees, appoints the treasurers
and members of the Secretariat, makes nominations of
candidates for future elections, and determines the chief
editors of several key editorial boards. Typically, each of
these new members of the Council will also serve on one
of the Society’s five policy committees. Current and past
members of the Council may be found here: www.ams
.org/comm-all.html#COUNCIL.
Vice President
(one to be elected)
Sara Billey
Abigail Thompson
Board of Trustees
(one to be elected)
Matthew Ando
Rick Miranda
Member at Large
of the Council
(five to be elected)
Dan Freed
Fernando Q. Gouvêa
Christopher D. Hacon
Daniel Krashen
Susan Loepp
Lenhard Ng
Kasso A. Okoudjou
Maria Cristina Pereyra
Hal Schenck
Melanie Matchett Wood
Nominating Committee
(three to be elected)
Sami H. Assaf
Ricardo Cortez
Rebecca Garcia
Yongbin Ruan
David Savitt
Deane Yang
Editorial Boards Committee
(two to be elected)
Ian Agol
David Marker
James McKernan
Terence Tao
List of Candidates–2019 Election
Update in
master page
MP: a-ballot-opener
PS: NEWS SUBHEAD
CS: BCS: I
PS: TEXT NO INDENT
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INDENT
1 PICA
PS: AHEAD
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master page
2024 Election
Candidate Biographies
Vice President
Minerva Cordero
Professor of Mathematics and
Vice Provost for Faculty Success,
The University of Texas at Ar-
lington
PhD: University of Iowa,
1989.
Selected publications or
other forms of scholarship:
1. with J. A. Mendoza ´
Alvarez,
Considerations for increasing
participation of minoritized
ethnic and racial groups in
mathematics, Notices Amer. Math. Soc.68 (2021), no.
2, 235–239, MR4202342; 2. with M. Mast, Valuing and
supporting work in mathematics education: An adminis-
trative perspective, Chapter 3 in Mathematics education: A
spectrum of work in mathematical sciences departments, AWM
Series, Springer, 2016; 3. with L. Chen, Fractional dimen-
sional semield planes, Note Mat. 32 (2012), no. 2, 57–
61, MR3071793; 4. with V. Jha, Fractional dimensions in
semields of odd order, Des. Codes Cryptogr.61 (2011),
no. 2, 197–221, MR2826957; 5. Semield planes of order
p4and kernel GF(p2), J. Geom.83 (2005), no. 1-2, 5–9,
MR2193222.
Selected addresses or public presentations: Non-
associative algebraic structures in cryptography, National
Conference of the Society for Advancing Chicanos and
Native Americans in Science, SACNAS, Long Beach, CA,
October 2016; Discovering mathematics in everything
we do, Smithsonian Museum, Futures That Inspire Hall,
March 2022; Advocating For Diversity in STEM, Susan G.
Komen’s Big Data For Breast Cancer initiative, Breast Can-
cer Hackathon Challenge, Women in Computational Bi-
ology, UT Southwestern, Dallas, TX, March 2023; Semi-
elds and other nite algebraic structures in coding algo-
rithms, UT Arlington MAA Student Chapter 20th Anniver-
sary Celebration, March 2023; Advocating For Diversity in
Mathematics from a Personal and Professional Perspective,
EDGE25, Celebrating 25 years of EDGE, Bryn Mawr Col-
lege, October 2023.
Synergistic activities: M. Cordero, T. Jorgensen, and B.
Shipman, Designing contracts and honors thesis projects
in mathematics, Chapter 13 in The other culture: Science
and mathematics education in honors, Ed. E. B. Buckner
and K. Garbutt, National Collegiate Honors Council, Lin-
coln, 2012; Chair of the Mathematical Sciences Research
Institute (MSRI) Human Resources Advisory Committee,
Berkeley, California, 2013–2015; M. Cordero, J. Epper-
son, and T. Jorgensen, Linking mathematics research to
secondary school classrooms: The GK-12 MAVS program,
Proceedings EDULEARN14: 6th International Conference on
Education and New Learning Technologies, Barcelona, Spain,
2014, pp. 4715–4723; Member of Harvard’s Pipelines into
Biostatistics Advisory Board, 2015–present; P. Harris et al.,
Editors, Testimonios: Stories of Latinx and Hispanic Mathe-
maticians, A co-publication of the American Mathematical
Society and the Mathematical Association of America, Sep-
tember 2021, 59–70.
Candidate statement: I am honored to be nominated
and considered for the role of Vice President of the Ameri-
can Mathematical Society (AMS). As an advocate for math-
ematics and mathematicians, I have long admired the AMS
and its profound commitment to advancing the eld of
mathematics on both national and international fronts.
Since its founding in 1888 the AMS has been a beacon
of excellence in promoting mathematical inquiry, commu-
nication, and application. Through its diverse array of pub-
lications, meetings, advocacy efforts, and programs, the
AMS tirelessly strives to fulll its mission. The mission of
the AMS resonates deeply with my own values and aspira-
tions as a mathematician and educator. I am committed
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1091
Election Candidate Biographies
to upholding and furthering these ideals, advocating for
the continued advancement of mathematics and the em-
powerment of mathematicians from diverse backgrounds.
If entrusted with the role of Vice President, I pledge to
work tirelessly in service to the AMS community, collab-
orating with fellow members to chart a course that pro-
motes inclusivity, excellence, and innovation in the mathe-
matical sciences. Together, we will continue to build upon
the rich legacy of the AMS, ensuring that mathematics re-
mains a vibrant and indispensable force for understanding
the world around us.
Malabika Pramanik
Professor, University of British
Columbia, Vancouver
PhD: University of Califor-
nia, Berkeley, 2001.
AMS ofces and commit-
tees: Transactions and Mem-
oirs Editorial Committee, 2011–
2019; Western Section Program
Committee, 2022–2024; Chair,
Fellows Program Selection
Committee, 2022–2025.
Selected publications or
other forms of scholarship: 1. with I. Łaba, Maximal op-
erators and differentiation theorems for sparse sets, Duke
Math. J. 158 (2011), no. 3, 347–411, MR2805064; 2. with
T. Collins and A. Greenleaf, A multi-dimensional resolu-
tion of singularities with applications to analysis, Amer. J.
Math.135 (2013), no. 5, 1179–1252, MR3117305; 3. with
J. Kim, L2bounds for a maximal directional Hilbert trans-
form, Anal. PDE 15 (2022), no. 3, 753–794, MR4442840;
4. with Y. Liang, Fourier dimension and avoidance of lin-
ear patterns, Adv. Math. 399 (2022), Paper No. 108252,
50 pp., MR4384610; 5. On some properties of sparse sets:
a survey, ICM—International Congress of Mathematicians.
Vol. IV. Sections 5–8, 3224–3248, 2023, MR4680359.
Selected addresses or public presentations: Plenary
lecture, AMS Spring Sectional Meeting, USA, April 2021;
Fourier Analysis @ 200, International Centre for Mathe-
matical Sciences (ICMS), June 2022; Invited address, 44th
Summer Symposium in Real Analysis, Paris, June 2022; In-
ternational Congress of Mathematicians (virtual), Analy-
sis session, July 2022; Barnett Public Lecture, University of
Cincinnati, April 2023.
Synergistic activities: Lead organizer, ”Diversity in
Mathematics,” an undergraduate summer school for
women and gender minorities, funded by Pacic Insti-
tute for Mathematical Sciences (PIMS), Fields Institute,
Centre de Recherches Math´ematiques (CRM), 2016–2020;
Lead organizer, ”Diversity in Mathematics,” a math camp
for high school students, funded by Natural Science and
Engineering Research Council of Canada (NSERC) Pro-
moScience, 2016–2020; Scientic co-director, Canadian
Mathematical Society National Meeting, Vancouver, De-
cember 2018; Co-organizer, ”Fourier Restriction Online,”
virtual program during COVID, December 2020–March
2021; Speaker, participant, and member, Indian Women
in Mathematics, 2020–present; Sadosky Award Commit-
tee, Association for Women in Mathematics (AWM), 2022–
2024; Co-chair, Travel Grants Sub-Committee, Interna-
tional Congress of Mathematicians, International Math-
ematical Union-Centre for Developing Countries (IMU-
CDC), 2026.
Additional experience/qualications you bring to the
position: Vice President for the Pacic Region, Execu-
tive Committee of the Canadian Mathematical Society,
2017–2019; Canadian Journal of Mathematics and Canadian
Mathematics Bulletin, 2019–present; Inaugural Fellow of
the Canadian Mathematical Society, 2019; Scientic di-
rector, Banff International Research Station (BIRS), 2020–
2025; Fellow of the American Mathematical Society, 2021;
Editor-in-Chief, Research in Mathematical Sciences, Springer,
2022–present; Review panelist for Natural Science and En-
gineering Research Council of Canada (NSERC), 2022–
2024, and National Science Foundation (NSF), 2008–
2011, 2022–2023.
Candidate statement: The AMS is a vibrant commu-
nity alive with ideas, initiatives, and activities. Through its
meetings, publications, travel grants, MathSciNet, Math-
ematics Research Communities, and more, it has some-
thing to offer to every mathematician, regardless of their
background, interests, or career stage. Its impact spans re-
search, teaching, pedagogy, art, industry, and government
policy. It connects mathematics and the wider society
where its members live and work. In a world increasingly
fraught with differences, it celebrates the wealth of exper-
tise, perspectives, and lived experiences within the mathe-
matical community.
The AMS has been a backdrop of my academic journey
for the last three decades. My rst paper as a graduate stu-
dent was in an AMS journal. Over the years, I have been
deeply inspired by the talent that AMS nurtures, and fortu-
nate to serve on its committees and editorial boards, while
experiencing rsthand its impact around the world. I am
honored by this opportunity to give back to the commu-
nity that has been a constant source of professional sup-
port for me. If elected, I will continue to build an inclu-
sive environment in mathematics and raise awareness of
the new opportunities and challenges for mathematicians
in a rapidly changing world.
1092 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
Election Candidate Biographies
Board of Trustees
Henry Cohn
Senior Principal Researcher and
Adjunct Professor of Mathemat-
ics, Microsoft Research New
England and MIT
PhD: Harvard University,
2000.
AMS ofces and commit-
tees:Journal of the AMS, Asso-
ciate Editor, 2012–2016; Rob-
bins Prize Committee, 2012;
Fellows Selection Committee,
2015–2017; AMS Council,
2016–2021; Committee on Publications, 2016–2018;
Committee on Committees, 2017–2020; Executive Com-
mittee, 2018–2021; Long Range Planning Committee,
2019–2020; Committee on Equity, Diversity, and Inclu-
sion, 2020–2021; Doubly Anonymous Refereeing Com-
mittee, 2021; Conant Prize Committee, 2022–2024; Liai-
son Committee with the American Association for the Ad-
vancement of Science, 2022–2024; Committee on Educa-
tion, 2024–2027.
Selected publications or other forms of scholarship:
1. with J. Blasiak, T. Church, J. A. Grochow, E. Naslund,
W. F. Sawin, and C. Umans, On cap sets and the group-
theoretic approach to matrix multiplication, Discrete Anal.
(2017), Paper No. 3, 27 pp., MR3631613; 2. with A. Ku-
mar, S. D. Miller, D. Radchenko, and M. Viazovska, The
sphere packing problem in dimension 24, Ann. of Math.
(2) 185 (2017), no. 3, 1017–1033, MR3664817; 3. with F.
Gon¸calves, An optimal uncertainty principle in twelve di-
mensions via modular forms, Invent. Math. 217 (2019),
no. 3, 799–831, MR3989254; 4. with C. Borgs, J. T.
Chayes, and S. Ganguly, Consistent nonparametric estima-
tion for heavy-tailed sparse graphs, Ann. Statist. 49 (2021),
no. 4, 1904–1930, MR4319235; 5. with N. Afkhami-Jeddi,
T. Hartman, and A. Tajdini, Free partition functions and an
averaged holographic duality, J. High Energy Phys. (2021),
no. 1, Paper No. 130, 42 pp., MR4257711.
Selected addresses or public presentations: Lecture in
Combinatorics Section, International Congress of Math-
ematicians, August 2010; AMS Arnold Ross Lecture, Salt
Lake City, November 2014; Math Encounters talk, Na-
tional Museum of Mathematics, September 2018; Conant
Lecture, Worcester Polytechnic Institute, November 2018;
Invited Address, joint AMS/VMS meeting, Quy Nhon, Viet-
nam, June 2019.
Synergistic activities: OurCS Conference for Under-
graduate Women in Computer Science, research project
mentor, October 2007, March 2011, and October 2013;
PROMYS Foundation, Trustee, 2011–present; Advisory
Council for National Museum of Mathematics, 2011–
present; Advisory Board for Building Computational
Thinkers project, Boston Museum of Science, 2013–2016;
Program in Mathematics for Young Scientists (PROMYS),
teaching a daily class to high school students for six weeks
each summer, 2015–present.
Additional experience/qualications you bring to the
position: My combination of academic and industrial ex-
perience gives me a perspective I hope would be useful for
the Board of Trustees.
Candidate statement: I am grateful to be a candidate
for the Board of Trustees. If elected, I will do my best to en-
sure that the business affairs of the AMS are conducted ef-
ciently and transparently on behalf of the members, and
that the Society remains on a solid nancial footing. It’s
important for the health of the mathematical community
that the AMS both continue its important work in support-
ing fundamental research in mathematics and broaden the
scope of its activities to include those who have been over-
looked or excluded in the past. These goals reinforce each
other, and neither could be achieved in isolation. Mak-
ing progress requires thoughtful decisions by the Board of
Trustees, and I would be honored to assist in this process.
Brooke E. Shipley
Professor, Department of Math-
ematics, Statistics, and Com-
puter Science (MSCS), Univer-
sity of Illinois at Chicago
PhD: Massachusetts Institute
of Technology, 1995.
AMS ofces and commit-
tees: Academic Freedom,
Tenure, and Employment Se-
curity, 2004–2007; AMS-IMS-
SIAM Committee on Summer
Research Conferences in the
Mathematical Sciences, 2004–2007; Proceedings Editorial
Committee of the AMS, 2009–2013; AMS-Simons Travel
Grants Committee, 2013–2016; Committee on Publica-
tions, 2018–2021; AMS Council, Member at Large, 2018–
2021; Search Committee for the Notices Chief Editor,
2020–2021; Doubly Anonymous Refereeing Committee,
2021–2022; Committee on the Profession, 2024–2027.
Selected publications or other forms of scholarship:
1. with M. Hovey and J. Smith, Symmetric spectra, J. Amer.
Math. Soc.13 (2000), no. 1, 149–208, MR1695653; 2.
with D. Dugger, K-theory and derived equivalences, Duke
Math. J.124 (2004), no. 3, 587–617, MR2085176; 3.
𝐻ℤ-algebra spectra are differential graded algebras, Amer.
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1093
Election Candidate Biographies
J. Math.129 (2007), no. 2, 351–379, MR2306038; 4.
with K. Hess, M. Kedziorek, and E. Riehl, A necessary
and sufcient condition for induced model structures, J.
Topol. 10 (2017), no. 2, 324–369, MR3653314; 5. with
J. P. C. Greenlees, An algebraic model for rational torus-
equivariant spectra, J. Topol.11 (2018), no. 3, 666–719,
MR3830880.
Selected addresses or public presentations: Invited
Address, AMS Sectional Meeting, Boulder, CO, 2003; Wolf-
son Lecture Series, Manchester, England, 2006; Lecture Se-
ries, Workshop on Algebraic Topology, MSRI, 2013; Ple-
nary Speaker, Nebraska Conference for Undergraduate
Women in Mathematics, Lincoln, NE, 2017; Panelist, Crit-
ical Issues in Mathematics Education, MSRI, Berkeley, CA,
2022.
Synergistic activities: NSF ADVANCE Co-PI, UIC,
Women in Science and Engineering System Transforma-
tion (WISEST), 2009–2012; Interim Director, WISEST,
2012–2013; AWM Committee on Committees, 2013–
2016; Executive Advisory Board, Department of Education
HSI-STEM program, UIC Latinos Gaining Access to Net-
works for Advancement in Science (L@S GANAS), 2017–
2020; Co-Director, UIC Young Scholar’s Program (four-
week summer program open to all Chicago high school
students), 2020–present; Graduate Research Opportu-
nities Workshop (GROW) Steering Committee, 2020–
present; AWM Research Networks Committee, 2021–2024.
Additional experience/qualications you bring to the
position: NSF Postdoctoral Research Fellow, 1995; NSF
Career Award, 2002; Sloan Research Fellow, 2002; AWM
Noether Lecture Selection Committee, 2009–2012; ELATE
Fellow, Drexel University, 2014–2015; Head, UIC Depart-
ment of Mathematics, Statistics, and Computer Science
(MSCS), 2014–2022; AMS Fellow, 2015; Co-Chair, UIC
Faculty Equity Committee, 2017–2024; NSF Institute for
Mathematical and Statistical Innovation (IMSI), Co-PI and
Board of Advisors, 2020–present; Senior Berwick Prize,
London Mathematical Society, joint with John Greenlees,
2022.
Candidate statement: I am honored to be considered
as a candidate for Trustee of the AMS. Of the many expe-
riences and roles listed above, two main roles are espe-
cially relevant to nancial stewardship and to furthering
the multifaceted mission of the AMS. For eight years (2014
to 2022), I was Head of a department that encompasses
pure and applied mathematics, statistics, mathematical
computer science, and mathematical education at one of
the most diverse research universities in the country. Since
2020, I have served as Co-PI and member of the Board of
Advisors for the NSF Institute for Mathematical and Sta-
tistical Innovation (IMSI). Scientic activity at IMSI is fo-
cused on applications of the mathematical sciences, em-
phasizing questions of importance to society at large. The
two other pillars of IMSI’s mission are improving and facil-
itating communication and broadening access and partic-
ipation. Throughout my career, I have been dedicated to
serving the broader mathematical community, and I look
forward to the opportunity to continue to do so as Trustee
for the AMS.
Member at Large
Alejandra Alvarado
Professor of Mathematics, East-
ern Illinois University
PhD: Arizona State Univer-
sity, 2009.
AMS ofces and commit-
tees: Central Section Program
Committee, 2022–2024; MAA-
SIAM-AMS Hrabowski-Gates-
Tapia-McBay Lecture Selection,
2023–2025.
Selected publications or
other forms of scholarship:
1. with J.-J. Delorme, On the Diophantine equation
x4+y4+z4+t4=w2,J. Integer Seq.17 (2014), no. 11, Article
14.11.5, 14 pp., MR3291083; 2. with C. R. Price, Academic
preparation for business, industry, and government posi-
tions, Assoc. Women Math. Ser., 18, Springer, 2019, 79–
87, MR4061882.
Selected addresses or public presentations: Arith-
metic Progressions on Curves, Conference on Strengthen-
ing Community in Research Mathematics, invited speaker,
Pomona College, Claremont, CA, 2023; EDGE: The Early
Years, invited panelist for Mobilizing the Power of Diver-
sity: Celebrating 25 Years of EDGE, 2023.
Synergistic activities: EDGE Co-director, summer
2016; CSU Channel Island faculty mentor, summer REU,
2017.
Additional experience/qualications you bring to the
position: Government employee, 2018–2020; AWM Exec-
utive Committee Clerk, 2022–present.
Candidate statement: Thank you for considering me
for Member at Large.
If elected, I hope that I can contribute towards making
mathematics a welcoming community and bring aware-
ness to its contribution not just in the sciences but to mod-
ern society.
I am concerned that mathematics research is not being
valued and appreciated as it could be, such as the pro-
posed cuts of West Virginia University graduate programs,
1094 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
Election Candidate Biographies
and the loss of our master’s programs in mathematics and
mathematics education. Math is the backbone of many of
the sciences.
I appreciate that the AMS offers travel grants for PUI fac-
ulty to attend JMM. As a recipient of the AMS-Simons grant,
this is especially important as I am at a PUI.
My experience in government has allowed me to see
rsthand the applications of mathematics and given me
unique insight on faculty transition to (and from) nonaca-
demic positions.
I am committed to the increase and advancement of
women and underserved students in the mathematical sci-
ences. I hope that my actions and experience will serve as
evidence of my candidacy.
Benjamin Antieau
Professor, Northwestern Uni-
versity
PhD: UIC, 2010.
AMS ofces and commit-
tees: Associate Editor, Journal of
the AMS, 2023–2027.
Selected publications or
other forms of scholarship: 1.
with A. Mathew, M. Morrow,
and T. Nikolaus, On the Beilin-
son ber square, Duke Math.
J. 171 (2022), no. 18, 3707–
3806, MR4516307; 2. with B. Bhatt and A. Mathew, Coun-
terexamples to Hochschild-Kostant-Rosenberg in charac-
teristic p,Forum Math. Sigma 9(2021), Paper No. e49,
26, MR4277271; 3. with T. Nikolaus, Cartier modules and
cyclotomic spectra, J. Amer. Math. Soc.34 (2021), no.
1, 1–78, MR4188814; 4. Periodic cyclic homology and
derived de Rham cohomology, Ann. K-Theory 4(2019),
no. 3, 505–519, MR4043467; 5. with D. Gepner and J.
Heller, K-theoretic obstructions to bounded t-structures,
Invent. Math. 216 (2019), no. 1, 241–300, MR3935042.
Selected addresses or public presentations: The nilpo-
tency of v1in the K-theory of ℤ/pn, Workshop on p-adic
Hodge theory and applications, Clay Math Institute, 2022;
The cyclotomic t-structure (two talks), Drinfeld seminar,
online, 2022; Derived algebraic geometry (nine lectures),
MSRI summer school, 2023; Integral models for spaces,
Conference on generalized Lie algebras in derived alge-
braic geometry, Utrecht, 2023; Motivic ltrations (ve lec-
tures), Masterclass on THH and zeta values, Copenhagen,
2023.
Synergistic activities: Co-founder (with David Dumas)
of the Math Computing Laboratory at UIC, which in-
troduced undergraduate students to visualization and ex-
ploration techniques in mathematics through term-long
projects; Co-organizer of many schools and workshops,
such as the electronic Algebraic K-Theory Seminar and Vi-
tamin K1.
Candidate statement: I am interested in research for
its own sake as well as its broad use in many other human
endeavors. Increasingly, I am also interested in computer
algorithms for understanding highly complex objects aris-
ing in algebraic and arithmetic geometry. Finally, I have
also always been interested in pedagogy and the role of
the university in the development of science and scien-
tists. The AMS is an important companion to us in each
of these realms: it is crucial in helping individual mathe-
maticians ourish, inside and outside of the academy, in
advising national leaders on science policy, and in help-
ing to craft academic policy in research and teaching. I
am excited about the prospect of serving the AMS, and the
mathematical community at large, as a Member at Large
of the Council of the AMS and being a part of this work.
I hope that by doing so, I can help further participation
in a subject beloved to me and also help to inuence the
organization’s response to various opportunities and chal-
lenges facing the (mathematical) world at present.
Dawei Chen
Professor of Mathematics, Bos-
ton College
PhD: Harvard University,
2008.
Selected publications or
other forms of scholarship: 1.
with I. Coskun, Extremal effec-
tive divisors ℳ1,n,Math. Ann.
359 (2014), no. 3-4, 891–908,
MR3231020; 2. with M. Möller
and D. Zagier, Quasimodular-
ity and large genus limits of
Siegel-Veech constants, J. Amer. Math. Soc. 31 (2018),
no. 4, 1059–1163, MR3836563; 3. with M. Bainbridge, Q.
Gendron, S. Grushevsky, and M. Möller, Compactication
of strata of Abelian differentials, Duke Math. J. 167 (2018),
no. 12, 2347–2416, MR3848392; 4. with M. Möller, A.
Sauvaget, and D. Zagier, Masur-Veech volumes and inter-
section theory on moduli spaces of Abelian differentials,
Invent. Math. 222 (2020), no. 1, 283–373, MR4145791;
5. with M. Möller and A. Sauvaget, Masur-Veech volumes
and intersection theory: the principal strata of quadratic
differentials, Duke Math. J.172 (2023), no. 9, 1735–1779,
MR4608330.
Selected addresses or public presentations: IAS-PCMI
Summer School on Moduli Spaces of Riemann Surfaces,
Park City, UT, Invited Talk, 2011; Algebraic Geometry
Northeastern Series, Stony Brook, NY, Plenary Speaker,
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1095
Election Candidate Biographies
2011; The Abel Symposium, Lofoten, Norway, Plenary
Speaker, 2017; Algebraic Geometry Fall School, Berlin,
Germany, Minicourse, 2019; Texas Geometry and Topol-
ogy Conference, Fort Worth, TX, Plenary Speaker, 2023.
Synergistic activities: Co-organizer for over 20 con-
ferences, such as Algebraic Geometry Northeastern Series
(AGNES), workshops at BIRS, HMI, ICERM, and an online
seminar series Moduli Across the Pandemic (MAP); Men-
tor for the 2015 Algebraic Geometry Summer Research In-
stitute Bootcamp and the 2023 AGNES Summer School;
Mentor for over 20 undergraduate, graduate, and postdoc-
toral research and independent study projects; Notices of
the AMS Early Career Section article author (“Developing
Relationships with Experts”); MathSciNet reviewer (over
110 reviews) and zbMath reviewer (over 150 reviews); NSF
panelist (4 times); Proposal reviewer for the European
Research Council, French National Research Agency, and
Chile National Fund for Scientic and Technological De-
velopment; Letter writer and reference for over 60 peo-
ple; Referee for over 40 journals; Boston College (BC)
University Core Renewal Committee member, 2014–2017;
BC Mathematics Graduate Program Director, 2016–2018,
2020–2021; BC Undergraduate Educational Policy Com-
mittee member, 2020–2023.
Additional experience/qualications you bring to the
position: Clay Liftoff Fellow, 2008; NSF standard grants,
2011–2014, 2020–2023, 2023–2026; NSF CAREER Award,
2014–2020; AIM SQuaRE, 2017–2019; IAS von Neumann
Fellow, 2019; Simons Fellow, 2024.
Candidate statement: I am honored to be nominated
to run for the position of Member at Large of the AMS
Council. Today, the mathematical community faces many
new challenges and diverse voices. Nevertheless, creat-
ing an environment of equality and inclusivity, promot-
ing diversity in education and research, and nurturing the
growth of young mathematicians are shared ideals and
goals that I have always been dedicated to. For example,
during the pandemic, to provide young scholars with op-
portunities to showcase their research, I organized a series
of online conferences where all speakers were graduate stu-
dents or postdoctoral scholars, with half of them being
women or from other underrepresented groups. I also con-
tributed an article to Notices of the AMS, introducing how
early-career researchers can develop connections with ex-
perts in their elds. Furthermore, I volunteered for STEM
activities at local schools, demonstrating surface geometry
to elementary school students using donuts and pretzels.
AMS is a big family, and I aspire to listen to and assist the
needs of community members just like caring for family
members, and to nurture the growth of the new generation
of mathematicians as if they were my own children. I am
committed to giving my best effort towards this endeavor.
Emily Clader
Associate Professor, San Fran-
cisco State University
PhD: University of Michi-
gan, 2014.
Selected publications or
other forms of scholarship:
1. Landau–Ginzburg/Calabi–
Yau correspondence for the
complete intersections 𝑋3,3and
𝑋2,2,2,2,Adv. Math.307 (2017),
1–52, MR3590512; 2. Editor,
with Y. Ruan, B-model Gromov–
Witten theory, Birkhäuser, 2018, MR3967072; 3. with S.
Grushevsky, F. Janda, and D. Zakharov, Powers of the theta
divisor and relations in the tautological ring, Int. Math.
Res. Not. (2018), no. 24, 7725–7754, MR3892277; 4.
Why twelve tones? The mathematics of musical tuning,
Math. Intelligencer 40 (2018), no. 3, 32–36, MR3851071;
5. with F. Janda and Y. Ruan, Higher-genus quasimap wall-
crossing in the gauged linear sigma model, Duke Math. J.
170 (2021), no. 4, 697–773, MR4280089.
Selected addresses or public presentations: ”Dou-
ble ramication cycles and tautological relations,” West-
ern Algebraic Geometry Symposium (WAGS), October
2016; ”Wall-crossing in Gromov–Witten theory,” Alge-
braic Geometry Northeastern Series (AGNES), October
2017; ”Wall-crossing in Gromov–Witten theory,” plenary
lecture, Texas Algebraic Geometry Symposium, April 2018;
”The moduli space of curves and its tautological ring,”
colloquium, UC Berkeley, February 2020; ”Permutohe-
dral complexes and curves with cyclic action,” colloquium,
Brown University, April 2022.
Synergistic activities: Mentor of eleven master’s theses
(six by women) and three undergraduate research groups
(ve of ten students women, seven of ten Black or Lat-
inx), San Francisco State University, Fall 2016–present;
Faculty advisor, Mathematistas (student group for gender
equity in mathematics), San Francisco State University,
2017–present; Volunteer, San Quentin Math Circle (at San
Quentin State Prison), Fall 2018; Invited speaker, San Fran-
cisco Nerd Nite (public lecture for an audience of about
200 people), Spring 2019; Author of numerous exposi-
tory texts, including a forthcoming undergraduate-level
algebraic geometry textbook Algebra and Geometry (with
Dustin Ross) and a forthcoming invited article ”Curve-
Counting and Mirror Symmetry” in the Notices of the AMS;
Faculty advisor and founder, Math Department PhD Ap-
plication Group (to mentor students through the PhD
application process), San Francisco State University, Fall
2022–present; Co-developer, Math Department course
on Quantitative Reasoning for Civic Engagement, San
1096 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
Election Candidate Biographies
Francisco State University, Fall 2023–present; Research
project mentor at four graduate workshops (Women in Al-
gebraic Geometry, ICERM, 2020; AGNES Summer School
on Intersection Theory on Moduli Spaces, Brown Uni-
versity, 2023; Workshop on Combinatorics of Moduli of
Curves, BIRS, 2024; Women in Algebraic Geometry 2, IAS,
2024).
Additional experience/qualications you bring to
the position: NSF DMS grant, 2018–2021; Partic-
ipant, UndocuAlly training (to better serve undocu-
mented students), San Francisco State University, Spring
2019; Chair, San Francisco State University Math De-
partment Curriculum Committee, Fall 2020–present;
Co-organizer, San Francisco State University Algebra-
Geometry-Combinatorics Seminar, 2020–present; NSF
panelist, 2021; NSF CAREER grant, 2022–2027.
Candidate statement: I am honored to be nominated
as a candidate for the position of Member at Large of the
Council of the AMS. I am committed to thinking deeply
and concretely about how mathematics can be made ac-
cessible to the broadest possible community, by (among
other things) creating texts and teaching materials that
strive for effective communication, by mentoring students
of many backgrounds through key transitions, and by
choosing seminar and conference speakers from a broad
range of demographic communities, subject areas, and ca-
reer stages.
If elected to serve, I would bring to the position my ex-
perience as a faculty member at a primarily undergraduate
institution whose student body is diverse in myriad ways.
San Francisco State University in general, and its Math De-
partment in particular, has a long history of attempting
to infuse its curriculum with equity, inclusivity, and social
relevance. My rsthand experience with both the successes
and the shortfalls of this mission equip me to bring cre-
ative ideas but also a critical eye to the crucial conversa-
tions that the AMS leads, and by which our community is
shaped.
Carla Cotwright
Supervisory Applied Research
Mathematician, Department of
Defense
PhD: The University of Mis-
sissippi, 2006.
AMS ofces and commit-
tees: Committee on Science
Policy, 2022–2025 (Chair,
2023–2025); Committee on
Education, 2023–2025; Liaison
Committee with the American
Association for the Advance-
ment of Science, 2024–2026.
Selected publications or other forms of scholarship:
1. A Mathematician’s Journey to Public Service, in: S.
D’Agostino, S. Bryant, A. Buchmann, M. Guinn, and L.
Harris (eds.), A Celebration of the EDGE Program’s Impact
on the Mathematics Community and Beyond, Association for
Women in Mathematics Series, vol. 18, Springer, 2019,
MR4061899.
Selected addresses or public presentations: MAA Dis-
tinguished Lecturer Series, on the intersection of Math &
Policy, October 2023.
Synergistic activities: Throughout my career I have en-
deavored to give back to my personal community as well
as the math community through invited talks and panels
at various diversity-focused events online and in person.
I consistently mentor budding mathematicians, helping
them navigate the challenges of undergraduate and grad-
uate school. Most recently I was asked to serve at the next
Innite Possibilities Conference. I have also inuenced
mathematics awareness through my service with the AMS
CSP and COE.
Additional experience/qualications you bring to the
position: Former AMS Congressional Fellow; Former
Tenure-Track Assistant Professor.
Candidate statement: I believe it’s important to have
diversity throughout the ranks of the AMS leadership. This
diversity of thought and experience serves to greatly en-
rich the international community of the AMS membership.
The face of math continues to evolve along with opportu-
nities to engage and communicate, building new and cul-
tivating ongoing research collaborations. In addition, the
opportunity is great to reintroduce and reinvigorate math-
ematical science foundations and bridge interdisciplinary
sciences gaps for the expansion of emerging technologies
including articial intelligence (AI) and machine learning
(ML) within the math community.
I hope to have the opportunity to continue to work with
AMS in this distinguished capacity. Thank you for your
consideration.
Dan Isaksen
Professor of Mathematics,
Wayne State University
PhD: University of Chicago,
1999.
AMS ofces and commit-
tees: Central Section Program
Committee, 2024–2026.
Selected publications or
other forms of scholarship: 1.
with D. Dugger, The Hopf con-
dition for bilinear forms over
arbitrary elds, Ann. of Math.
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1097
Election Candidate Biographies
(2) 165 (2007), no. 3, 943–964, MR2335798; 2. with
G. Wang and Z. Xu, Stable homotopy groups of spheres,
Proc. Natl. Acad. Sci. USA 117 (2020), no. 40, 24757–
24763, MR4250190; 3. with G. Wang and Z. Xu, Stable
homotopy groups of spheres and motivic homotopy the-
ory, Proceedings of the International Congress of Mathemati-
cians 4(2022), 2768–2790; 4. with G. Wang and Z. Xu,
Stable homotopy groups of spheres: from dimension 0 to
90, Publ. Math. Inst. Hautes ´
Etudes Sci.137 (2023), 107–
243, MR4588596.
Selected addresses or public presentations: Introduc-
tory Workshop: Algebraic Topology, Mathematical Sci-
ences Research Institute, Berkeley, 2014; Homotopy The-
ory in the Ecliptic, Reed College, 2017; Motivic, Equi-
variant, and Non-Commutative Homotopy Theory, mini-
course presenter, Institut des Hautes ´
Etudes Scientiques,
France, 2020; Advances in Homotopy Theory IV, Beijing
Institute of Mathematical Sciences and Applications, 2023;
Homotopy theory in honor of Paul Goerss, Northwestern
University, 2023.
Synergistic activities: Founder and Organizer, Elec-
tronic Computational Homotopy Theory research com-
munity, 2017–current.
Additional experience/qualications you bring to the
position: Editorial board member, Algebraic and Geomet-
ric Topology, 2010–current; Editorial board member, Ho-
mology, Homotopy and Applications, 2019–current; Editorial
board member, Proceedings of the London Mathematical So-
ciety, 2023–current; Fellow of the American Mathematical
Society, 2024.
Candidate statement: Our society constantly changes,
and mathematical research culture must constantly adapt.
The modes by which people interact and communicate
are currently changing rapidly. Traditionally, departments
of mathematics (and analogous organizations in govern-
ment and industry) were essential to create critical masses
to foster healthy mathematical discussion and interaction.
The rise of videoconferencing and other online communi-
cations tools makes physical proximity less relevant.
For the long-term health of research mathematics, we
should prepare for a future in which the Mathematical
Conversation is carried out primarily online. Rising gen-
erations of mathematicians are and will continue to be u-
ent in this medium, and it is our responsibility to build
infrastructure in which they will thrive.
Online communication tools have the democratizing
potential to include populations that were previously ex-
cluded from the Mathematical Conversation. We should
be intentional about building an online mathematical cul-
ture that furthers mathematical opportunity for everyone
who seeks it.
Should the Society promulgate principles and best prac-
tices for online mathematical opportunities? Does the So-
ciety have a role to play in online conferences? How can
the Society facilitate the increasingly online employment
process? What can the Society do to encourage the inter-
scholastic cooperation that is a key aspect of online activi-
ties?
Yvonne Lai
Professor & Graduate Chair,
Department of Mathematics,
University of Nebraska-Lincoln
PhD: University of Califor-
nia, Davis, 2008.
AMS ofces and commit-
tees: AMS Lecture on Ed-
ucation Selection Committee,
2024–2026; Notices Editorial
Board Committee, 2025–2027.
Selected publications or
other forms of scholarship: 1.
An effective compactness theorem for Coxeter groups,
Geometriae Dedicata 145 (2010), no. 1, 195–217,
MR2600954; 2. Teaching Undergraduate Mathematics, Crit-
ical Issues in Mathematics Education Series, vol. 4, Math-
ematical Sciences Research Institute, Berkeley, CA, 2012;
3. with M. A. Carlson and R. Heaton, Giving reason and
giving purpose, in Mathematics Matters in Education: Essays
in honor of Roger E. Howe, Springer, New York, 2017, 141–
179; 4. as member of writing team, Catalyzing Change in
High School Mathematics: Initiating Critical Conversations,
National Council of Teachers of Mathematics, Reston, VA,
2018, vi+107; 5. with G. Burrill, H. Cohn, D. Sinha, J. Y.
Son, and K. E. Stevenson, Listening for common ground
in high school and early collegiate mathematics, Notices
Amer. Math. Soc. 70 (2023), no. 6, 798–805.
Selected addresses or public presentations: Invited
presentation at workshop on ”Mathematicians and School
Mathematics Education” at the Banff International Re-
search Station, Banff, 2014; Brieng of the US House Com-
mittee on Science, Space, & Technology on “STEM Educa-
tion 101,” Washington, DC, 2023; Invited speaker for Fo-
rum for World Education on “Raising the Bar on Mathe-
matics Education: Lessons from Far and Near,” New York
City, 2024; JMM Invited Address on “Building Bridges in
Mathematics Education” (Project NeXT Lecture on Teach-
ing and Learning), San Francisco, 2024.
Synergistic activities: Explore Math Program/Davis
Math Circle: Co-Founder (2005–2008); Algebra Project:
Invited Young Mathematicians (2007), Lesson Planning
Team Member (2009–2010), co-PI (2021–2025); Math-
ematicians in Mathematics Education: Presenter (2011),
1098 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
Election Candidate Biographies
Organizer (2012), Lead Organizer (2013–2014) with
William McCallum, Deborah Ball, and Roger Howe; MAA
Committee on the Mathematical Education of Teachers:
Member (2016–2022), Chair (2022–2024), created NCTM
liaison for committee, led review of NCTM policy doc-
ument, conducted bi-monthly virtual listening tour of
mathematics faculty across nation invested in the math-
ematical education of teachers; Metric Geometry and Ger-
rymandering Group’s Geometry of Redistricting Summer
School: Instructor (2018); MAA Task Force on the Treat-
ment of Teacher Education: Co-Chair (2021–2023), pro-
duced guiding structure for MAA CUPM Curriculum Guide
Program Area Reports on Teacher Education; Board of Di-
rectors of the Mathematical Foundation of America (ad-
ministers the Canada/USA Mathcamp): Member (2018–
2022), Vice Chair (2023–2025); AMS Committee on Edu-
cation Forum: Evolving Curriculum in High School and
Early Undergraduate Mathematical Sciences Education:
Organizer and Moderator (2022); Nebraska State High
School Mathematics Standards: Post-Secondary Advisor
(2022); AMS Special Session on Mathematics Education,
Standards, Policy, and Politics: Lead Co-Organizer (2023);
Conference Board of the Mathematical Sciences (CBMS)
Task Force on Modernizing Mathematics: Member (2023–
2024), produced consensus document proposed to mem-
bership of CBMS; CBMS Steering Committee for the Math-
ematical Education of Teachers III (MET III): Member
(2024–2026).
Additional experience/qualications you bring to the
position: Service to the profession. Association for Women
in Mathematics (AWM) Educational Columnist; Bay Area
Mathematics Olympiad for Teachers: “Chief Inspirer” to
director Joshua Zucker, 2010; National Association of
Math Circles: Advisory Board Member, 2012–2015; US-
AMO: Grader, 2015–2016; Associate Editor of AMS Blog
on Teaching and Learning, 2018–2021; Associate Editor
and Editorial Board for Problems, Resources, and Issues in
Mathematics Undergraduate Studies, 2020–2024; Associate
Editor of AMS Column on Teaching and Learning, 2022–
2025; Subcommittee of the AMS Committee on Education,
2023–2024.
Memberships. American Educational Research Asso-
ciation (AERA), American Mathematical Society (AMS),
Association for Women in Mathematics (AWM), As-
sociation of Mathematics Teacher Educators (AMTE),
Canadian Mathematics Education Study Group/Groupe
Canadien d’´
Etudes en Didactique des Math´ematiques
(CMESG/GCEDM), Consortium for Mathematics and
its Applications (COMAP), Mathematical Association of
America (MAA), National Association of Mathematicians
(NAM), National Council for Teachers of Mathematics
(NCTM), TODOS Mathematics for All (TODOS).
Grants. PI NSF award DUE-1726744, PI NSF award
DUE-2408993, co-PI NSF award DRL-2101393, co-PI NSF
award DUE-1747937, co-PI NSF award DUE-1439867, PI
NSF award DGE-1445551, co-PI NSF award DUE-1035268,
co-PI NSF award DMS-1135049.
Candidate statement: I am deeply honored to be nomi-
nated to serve on the AMS Council. The AMS has a leading
role in the health of the mathematical sciences community.
If elected, I will work for the AMS to:
(1) Advocate. The AMS depends on graduate education
for its future vitality. I will advocate for graduate studies in
the mathematical sciences, including identifying factors in
success, persisting, and attrition.
(2) Connect. The AMS is energized when it supports and
sees value in a wide array of perspectives and experiences.
I aim for the AMS to make intentional connections across
industry, sciences, and education to strengthen the entire
mathematics community.
(3) Advance. The AMS should advance access to the
mathematical sciences. Too many students and teachers
have too few experiences of mathematics as beautiful, joy-
ful, and powerful. I will work for the AMS to continue and
expand efforts that foster full and equitable participation
in the mathematical sciences.
Throughout my career collaborating with researchers
and teachers, I have been driven to connect communities
and ideas across the mathematical sciences to improve ed-
ucation. If elected, I bring this commitment to the Council
to inuence policy and practice that sustains and nurtures
the mathematical sciences community.
Christopher J. Leininger
Professor, Rice University
PhD: University of Texas,
Austin, 2002.
AMS ofces and commit-
tees: AMS-Simons Travel
Grants Committee, 2015–2018;
Transactions and Memoirs Edito-
rial Committee, 2024–2028.
Selected publications or
other forms of scholarship: 1.
with M. Bestvina, K. Bromberg,
and A. E. Kent, Undistorted
purely pseudo-Anosov groups, J. Reine Angew. Math. 760
(2020), 213–227, MR4069890; 2. with S. Dowdall and I.
Kapovich, Dynamics on free-by-cyclic groups, Geom. Topol.
19 (2015), no. 5, 2801–2899, MR3416115; 3. with M.
Duchin and K. Ra, Length spectra and degeneration of
at metrics, Invent. Math.182 (2010), no. 2, 231–277,
MR2729268; 4. with R. P. Kent, IV, Shadows of map-
ping class groups: capturing convex cocompactness, Geom.
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1099
Election Candidate Biographies
Funct. Anal.18 (2008), no. 4, 1270–1325, MR2465691; 5.
with E. Field, H. Kim, and M. Loving, End-periodic home-
omorphisms and volumes of mapping tori, J. Topol.16
(2023), no. 1, 57–105, MR4532490.
Selected addresses or public presentations: Lawrence,
KS, AMS Sectional Meeting 1081, Invited Address, 2012;
Midway, Utah, Conference in Honor of Mark Feighn, Ple-
nary Lecture, 2018; Luminy, France, Conference in Honor
of Mladen Bestvina, Plenary Lecture, 2019; Columbia Uni-
versity, Conference in Honor of Walter Neumann, Plenary
Lecture, 2022; Monte Verita, Switzerland, Conference in
Honor of Ursula Hamenstädt, Plenary Lecture, 2023.
Synergistic activities: Co-organizer of University of
Warwick EPSRC Symposium, UK, 2017–2018; Associate
editor for New York Journal of Mathematics (2020–2024),
Advances in Mathematics (August 2022–present), Transac-
tions and Memoirs of AMS (February 2024–present); Co-
organizer for Conference: Geometry, Arithmetic, and
Groups, University of Texas, Austin, 2022; Co-organizer
for SLMath Semester program, Topological and Geomet-
ric Structures in Low Dimensions, Spring 2026; Super-
vised/mentored a diverse group of 20 PhD students (7 cur-
rent) and 8 postdocs.
Additional experience/qualications you bring to
the position: University of Illinois, Mathematics Ex-
ecutive Committee, 2012–2014; University of Illinois,
Mathematics Promotion and Tenure Committee, 2017–
2019 (Chair, 2018–2019); University of Illinois, Campus
Research Board (campus-wide research funding board),
2017–2020; Director of Graduate Studies, Rice University,
2021–present; Rice University Graduate Council, 2021–
present.
Candidate Statement: I am committed to advancing
mathematical research, and promoting diversity and inclu-
siveness at all levels. I am happy to serve as a Member at
Large, should I be elected.
Adriana Salerno
Professor, Bates College
PhD: University of Texas at
Austin, 2009.
AMS ofces and commit-
tees: Task Force: AMS In Racial
Discrimination, 2020–2021.
Selected publications or
other forms of scholarship: 1.
with J. H. Silverman, Integral-
ity properties of Bottcher co-
ordinates for one-dimensional
superattracting germs, Ergodic
Theory and Dynamical Systems 40 (2020), no. 1, 248–
271, MR4038034; 2. with C. Doran, T. Kelly, S. Sper-
ber, J. Voight, and U. Whitcher, Hypergeometric decom-
position of symmetric K3 quartic pencils, Research in the
Mathematical Sciences 7(2020), no. 2, Paper No. 7,
81 pp., MR4078177; 3. with L. Schneps, Mould theory
and the double shufe Lie algebra structure, in Periods in
Quantum Field Theory and Arithmetic: Proceedings of the IC-
MAT, Madrid, Spain, September 15–December 19, 2014,
Springer Proceedings in Mathematics and Statistics, 339–
430, 2020, MR4100685; 4. with U. Whitcher, Hasse-Witt
matrices and mirror toric pencils, Advances in Theoretical
and Mathematical Physics 26 (2022), no. 9, 3345–3375;
5. with D. Banerjee and S. Chari, Higher dimensional
origami constructions, Involve 16 (2023), no. 2, 297–312,
MR4597247.
Selected addresses or public presentations: Diagonal
pencils and Hasse-Witt invariants, Newton Institute, Cam-
bridge, UK, 2022; Teaching Math Is Hard (Haimo Award
presentation), MAA MathFest, Tampa, FL, 2023; Arith-
metic, Hypergeometric Functions, and Mirror Symmetry,
Lathisms Cafe Con Leche lecture, online (https://youtu
.be/Dc4tZwZ98eY?si=EC-7CxDo9RrD_2sf), 2023; The
mathematics of secrets (plenary), MAA Golden Section
meeting, University of California, Santa Cruz, CA, 2024;
The mathematics of secrets (plenary), MAA North Central
Section meeting, University of St. Thomas, Minneapolis,
MN, 2024.
Synergistic activities: Public awareness of mathemat-
ics: I have been the writer for several blogs for the AMS.
Among them, PhD + epsilon: An early career mathemati-
cian blogs about her experiences and challenges (http://
blogs.ams.org/phdplus/), a biweekly blog I wrote from
March 2011 to September 2015; Inclusion/Exclusion from
January 2016 to January 2020, about issues of diversity and
inclusion in the mathematical sciences; and the Joint Math
Meetings blog from 2008 to 2019. I was also on the edito-
rial board for MAA FOCUS and the American Mathematical
Monthly, for which I wrote a few articles. I have also writ-
ten book reviews and articles for a general audience for
American Scientist, the Notices of the AMS, and the AWM
Newsletter.
Diversity, equity, and inclusion: I am a graduate of the
AAAS-SACNAS-sponsored Linton-Poodry Summer Leader-
ship Institute (summer 2016), and the HHMI-SACNAS Ad-
vanced Leadership Institute (summer 2018). I was in the
core leadership for an HHMI grant to diversify STEM edu-
cation at Bates. I have mentored women in mathematics
as a co-leader of a project, with Ursula Whitcher, during
Sage Days 50: Women in Sage, in July 2013, then as a co-
leader of a project, with Leila Schneps, during the Women
in Numbers - Europe workshop, in October 2013. I co-
organized the Roots of Unity workshop, a workshop for
women of color in early graduate studies. I was one of
1100 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
Election Candidate Biographies
the writers of the report “Towards a Fully Inclusive Mathe-
matics Profession, Report of the Task Force on Understand-
ing and Documenting the Historical Role of the AMS in
Racial Discrimination,” with Tasha Inniss, Jim Lewis, Irina
Mitrea, Kasso Okoudjou, Francis Su, and Dylan Thurston.
Undergraduate research, applications, and computa-
tion: I have advised students in independent research both
during the academic year and during the summer. I have
advised several senior theses at Bates, two of which led to
publications. Many of these theses have been in number
theory and cryptography, and use a high level of computa-
tion and experimentation, including machine learning. I
have also advised math education theses.
Student-centered learning: I have developed several
inquiry-based learning courses and team-based learning
courses at Bates in a variety of topics, including complex
and real analysis, number theory, calculus, and p-adic anal-
ysis.
Additional experience/qualications you bring to the
position: I have been a rotating Program Ofcer at the NSF
since September 2021.
Candidate statement: Mathematicians are people who
do math, so I believe that advancing mathematics research
necessitates supporting and nurturing the mathematical
community. Nobody should have to leave their identity at
the door, mathematics is not neutral, and we are all better
at mathematics if we create a community where everyone
can ourish.
I love doing math, and I love helping others do math.
That’s a lesson I learned as an undergraduate at the Univer-
sidad Simon Bolivar in my home of Venezuela, just falling
in love with the subject.
After completing my PhD in number theory at UT-
Austin, I served as Professor of Mathematics and Chair
at Bates College. As a liberal arts professor, my commit-
ment to creating and developing a more inclusive and eq-
uitable math community has grown, through innovative
pedagogy, equity work, and communication of mathemat-
ics.
As an NSF Program Ofcer, I have expanded my perspec-
tive on the research community which continues to strive
towards being a more inclusive and inviting place. I would
be honored to join the AMS leadership to help advance
mathematics and the people who love and do mathemat-
ics.
Pham Huu Tiep
Joshua Barlaz Professor, Distin-
guished Professor of Mathemat-
ics, Department of Mathemat-
ics, Rutgers University
PhD: Moscow State Univer-
sity, Moscow, Russia, 1989.
AMS ofces and commit-
tees: Proceedings Editorial
Committee of the AMS, 2011–
2019; Joint AMS-Vietnamese
Mathematical Society, Quy
Nhon, 2017–2019; Mathemati-
cal Reviews Editorial Committee, 2017–2025.
Selected publications or other forms of scholarship:
1. with G. Navarro, A reduction theorem for the Alperin
weight conjecture, Invent. Math.184 (2011), 529–565,
MR2800694; 2. with G. Navarro, Characters of relative p′-
degree over normal subgroups, Annals of Math.178 (2013),
1135–1171, MR3092477; 3. with R. Bezrukavnikov, M.
Liebeck, and A. Shalev, Character bounds for nite groups
of Lie type, Acta Math.221 (2018), 1–57, MR3877017; 4.
with M. Larsen and A. Shalev, Probabilistic Waring prob-
lems for nite simple groups, Annals of Math.190 (2019),
561–608, MR3997129; 5. with R. M. Guralnick and M.
Larsen, Character levels and character bounds for nite
classical groups, Invent. Math.235 (2023), 151–210.
Selected addresses or public presentations: Invited
Speaker, 2012 AMS Spring Western Section Meeting, Hon-
olulu, HI, 2012; Plenary Speaker, Annual Meeting of the
DFG Priority Programme on Representation Theory SPP
1388, Bad Boll, Germany, 2013; Invited Speaker, ICM
2018, Rio de Janeiro, Brazil, 2018; Plenary Speaker, 2019
Canadian Mathematical Society Summer Meeting, Regina,
Canada, 2019; Principal Speaker, Groups St. Andrews,
Newcastle, UK, 2022.
Synergistic activities: Co-organizer, Global/Local Con-
jectures in Representation Theory of Finite Groups, BIRS,
Banff, Canada, March 13–18, 2011; Co-organizer, An In-
troduction to Character Theory and the McKay Conjecture,
Summer graduate school, MSRI, Berkeley, CA, July 11–22,
2016; Co-organizer, New Perspectives in Representation
Theory of Finite Groups, BIRS, Banff, Canada, Oct. 15–
20, 2017; Lead organizer, Group Representation Theory
and Applications, Semester program, MSRI, Berkeley, CA,
Jan. 16–May 25, 2018; Co-organizer, Groups, Representa-
tions, and Applications: New Perspectives, Isaac Newton
Institute for Mathematical Sciences, Cambridge, UK, Jan.
6–March 18, 2020, May 3–July 29, 2022; Co-organizer,
Monodromy and Its Applications, Princeton, NJ, Dec. 7–9,
2023.
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1101
Election Candidate Biographies
Additional experience/qualications you bring to the
position: Fellow of the AMS, Inaugural Class, 2013; Si-
mons Fellow in Mathematics, 2014–2015, 2022–2023;
Clay Senior Scholar, Clay Mathematics Institute, 2016;
ICM 2018, IMU Panel “Machine-Assisted Proofs,” Rio de
Janeiro, Brazil, 2018; Chern Professor, MSRI, Berkeley, CA,
Spring 2018; Kalman Prize for Best Paper, New Zealand
Mathematical Society, 2021; Currently serves on the Edi-
torial Boards of Annals of Mathematics (Associate Editor),
Algebra and Number Theory, J. Pure Applied Algebra, and
Springer Developments in Mathematics.
Candidate statement: I am honored to be nominated
for election as a Member at Large of the Council of the
AMS. If elected, I would like to contribute my efforts to
some of the challenges that we mathematicians are facing,
which include (i) how we can advocate the role of math-
ematics to society and broaden the connections of mathe-
matics to all areas of everyday life; (ii) how we can improve
the quality of mathematics teaching at all levels, starting
from K–12 schools to colleges to PhD programs; and (iii)
how the AMS can broaden and strengthen its relationships
with mathematicians in developing countries.
Nominating Committee
David Fisher
Milton B. Porter Professor of
Mathematics, Rice University
PhD: University of Chicago,
1999.
AMS ofces and commit-
tees: Joan and Joseph Birman
Fellowship Selection Commit-
tee, 2023–2025.
Selected publications or
other forms of scholarship: 1.
with G. Margulis, Local rigidity
of afne actions of higher rank
groups and lattices, Ann. of Math. (2) 170 (2009), no.
1, 67–122, MR2521112; 2. with A. Eskin and K. Whyte,
Coarse differentiation of quasi-isometries I: Spaces not
quasi-isometric to Cayley graphs, Ann. of Math. (2) 176
(2012), no. 1, 221–260, MR2925383; 3. with A. Eskin
and K. Whyte, Coarse differentiation of quasi-isometries
II: Rigidity for Sol and lamplighter groups, Ann. of Math.
(2) 177 (2013), no. 3, 869–910, MR3034290; 4. with
U. Bader, N. Miller, and M. Stover, Arithmeticity, super-
rigidity, and totally geodesic submanifolds, Ann. of Math.
(2) 193 (2021), no. 3, 837–861, MR4250391; 5. with A.
Brown and S. Hurtado, Zimmer’s conjecture: subexponen-
tial growth, measure rigidity, and strong property (T), Ann.
of Math. (2) 196 (2022), no. 3, 891–940, MR4502593.
Selected addresses or public presentations: Harvard,
Invited talk at Clay Mathematics Institute Annual Meeting,
2007; Notre Dame, Invited Address, AMS Sectional Meet-
ing, 2010; Virtual, Vinberg Online Distinguished Lecture
Series, 2022; Virtual, ICM invited address in 3 sections: Dy-
namics, Geometry, Topology, 2022.
Synergistic activities: Edited three books, two with a
special emphasis on high expository quality surveys; Edi-
tor at various points for a total of ve different journals:
Geometry and Topology, Journal of Topology and Analysis, In-
diana University Math Journal, Journal of Lie Theory, Geome-
triae Dedicata; Organized math-related lm series at Indi-
ana University Cinema; Organized 42 conferences, sum-
mer schools, and workshops, including a special trimester
at IHP in Spring 2024.
Additional experience/qualications you bring to the
position: I work in an area of mathematics that crosses
boundaries between several areas of analysis, dynamics,
geometry, and topology. This leads to having a broad and
deep network across many elds of mathematics, which
will help with the task of nding mathematicians willing
to do the important work of the AMS.
Candidate statement: We live in a difcult time for
mathematics, science, and higher education. Public trust
in science and higher education are low. The AMS needs
exceptional leadership to help mathematics research and
education continue to thrive at all levels and for all peo-
ple in the United States and abroad. I expect support for
research and education to be challenged at both the state
and federal levels and for diversity programs to continue
to face particularly strong challenges. We need to be pre-
pared to meet these problems adroitly and effectively. The
Nominating Committee plays a key role in selecting the
leadership that will be on the front lines; I would be very
happy to help nd people who are ready to ll those roles.
Aimee S. A. Johnson
Professor, Swarthmore College
PhD: University of Mary-
land, College Park, 1990.
AMS ofces and commit-
tees: AMS-Simons PUI Re-
search Grants Committee,
2024–2026.
Selected publications or
other forms of scholarship: 1.
with K. Madden, Putting the
pieces together: understanding
Robinson’s nonperiodic tilings,
College Math. J. (1997), no. 3, 172–181, MR1444004;
1102 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
Election Candidate Biographies
2. with A. Sahin, Isometric extensions of zero entropy ℤ𝑑
loosely Bernoulli transformations, Trans. Amer. Math. Soc.
352 (2000), no. 3, 1329–1343, MR1670842; 3. with K.
Madden and A. Sahin, Discovering discrete dynamical sys-
tems, Classr. Res. Mater. Ser., MAA, MR3677179; 4. with
D. McClendon, Topological speedups of ℤ𝑑-actions, Dyn.
Syste.37 (2022), no. 2, 222–261, MR4430568; 5. with V.
Cyr, B. Kra, and A. Sahin, The complexity threshold for
the emergence of Kakutani inequivalence, Israel J. Math.
(2022), no. 1, 271–300, MR4555896.
Selected addresses or public presentations: University
of Wroclaw Wandering Seminar, March 2017; Little School
Dynamics, Feb. 2021; Expanding Dynamics virtual Con-
ference Series, May 2021; Brigham Young University Col-
loquium, March 2022; Special session in Ergodic Theory,
Symbolic Dynamics, and Related Topics, AMS Joint Meet-
ing, San Francisco, Jan. 2024.
Synergistic activities: Organizer of Special Sessions at
AMS Meetings, Jan. 2002, Jan. 2014, Jan. 2017; Co-
founder and Leadership Team Member, Philadelphia Area
Math Teachers’ Circle, 2011–2020; Rubin Scholar Mentor
for students from underrepresented groups in STEM, 2011–
2024; Lester R. Ford Awards Committee for MAA, 2012–
2016.
Additional experience/qualications you bring to the
position: Chair of the Department of Mathematics and
Statistics, 2016–2020, 2021–2024; member of AWM, MAA,
and Sigma Xi.
Candidate statement: I would be honored to serve on
the AMS Nominating Committee. The AMS plays a crucial
role in advocating, promoting, and supporting our disci-
pline. Through its organization of meetings, it promotes
the dissemination of knowledge and the creation of com-
munity. Through its publications, it advances our schol-
arship. And through its advocacy, it continually advances
mathematical research and education. Since all of these
activities are vitally important, it is crucial that the AMS of-
cers include representation from a diverse group of peo-
ple who can use their passion for mathematics to further
the goals of the society.
Although my research is in pure mathematics, the six
years I have spent as Chair of a department of mathematics
and statistics, which includes a strong applied math com-
ponent, has given me a wide view of the pathways people
take in their mathematical careers. I have also seen how
important it is for the future of our discipline to have rep-
resentation along all axes of our society. As a member of
the Nominating Committee, I would work to nd a diverse
slate of candidates that can bring their energy and vision
to the leadership of the AMS.
Lily Signe Khadjavi
Professor of Mathematics, Loy-
ola Marymount University
PhD: U. C. Berkeley, 1999.
AMS ofces and commit-
tees: Committee on Equity, Di-
versity, and Inclusion, 2022–
2025; Council, 2022–2025; Hu-
man Rights of Mathematicians,
2023–2026.
Selected publications or
other forms of scholarship: 1.
Driving while black in the City
of Angels, Chance 19 (2006), no. 2, 43–46, MR2247023;
2. Edited with G. Karaali, Mathematics for social justice: Re-
sources for the college classroom, Classroom Resource Materi-
als, vol. 60, MAA Press, Providence, RI, 2019, vii+277 pp.,
MR3967051; 3. with R. Bryant, R. Buckmire, and D. Lind,
The origins of Spectra, an organization for LGBT mathe-
maticians, Notices Amer. Math. Soc. 66 (2019), no. 6, 875–
882, MR3929579; 4. with R. Malek-Madani and T. Moore,
Navigating an Uncharted Path: The Life and Legacy of Dr.
Gladys B. West, Notices Amer. Math. Soc. 68 (2021), no.
3, 357–364, MR4218169; 5. with T. Moore and K. Weems,
The Innite Possibilities Conference: Creating Moments
of Belonging, Notices Amer. Math. Soc. 71 (2024), no. 3,
349–355, excerpted from Count Me In: Community and Be-
longing in Mathematics, edited by D. Haunsberger and D.
Dumbaugh, Classroom Resource Materials, vol. 68, MAA
Press, Providence, RI, 2022, 145–155, ISBN: 978-1-4704-
6566-7.
Selected addresses or public presentations: Invited
speaker, AMS Special Session on Number Theory and
Cryptography, JMM, Seattle, 2016; Plenary address on
“Women and Mathematics: Inspiration, Obstacles, and
Opportunities,” Celebrating the Mathematical Legacy of
Professor Maryam Mirzakhani, UCLA, 2017; Plenary ad-
dress on “Policing and the Issue of Racial Proling in Los
Angeles,” Latinx in the Mathematical Sciences Conference,
IPAM, 2018; Plenary address, Math for All conference, vir-
tual/New Orleans, February 2022; Invited speaker, “Em-
powering students through authentic engagement,” Edu-
cating at the Intersection of Data Science and Social Justice,
ICERM, Providence, 2023.
Synergistic activities: NSF award 1135426, 2011–2012;
Co-chair of the Innite Possibilities Conference support-
ing BIPOC women in mathematics, 2012, 2015, 2018,
and Board Member of Building Diversity in Science; Co-
PI, NSF award 1464089, 2015–2016; Co-PI Board Mem-
ber, Harvard Gender Sexuality Caucus, 2016–2018; Prin-
cipal Investigator, NSF award 1642548, 2016–2021; Co-
organizer, “The Mathematics and Mathematicians Behind
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1103
Election Candidate Biographies
Hidden Figures,” JMM, 2017; Principal Investigator, NSF
award 2015440, for AWM Travel/Mentoring grants, 2020–
2024; Member, SLMath Broadening Participation Advisory
Committee (formerly MSRI HRAC), 2020–2023; Board
member, Spectra, 2022; Steering Committee Member, In-
nite Possibilities Conference, 2025.
Additional experience/qualications you bring to the
position: Appointee of the Attorney General of Califor-
nia to the Racial Identity and Proling Advisory Board,
Stop Data Analysis Subcommittee, State and Local Poli-
cies and Accountability Subcommittee, 2020–2024; Chair,
Loyola Marymount University Mathematics Department,
2021–2024; Representative, AAAS Human Rights Coali-
tion, 2022–2025; Mary and Ale Gray Award for Social
Justice, inaugural awardee, AWM, 2023.
Candidate statement: I am deeply honored to be con-
sidered for the Nominating Committee. Serving on the
AMS Council has given me a practical perspective into the
many functions of the Society, from supporting the inter-
nal health of the mathematical community to serving as a
public face of the mathematical sciences. I believe the AMS
is strengthened when an array of perspectives and experi-
ences is represented, as we promote mathematical research
and address the needs of the community. My service on ad-
visory boards, overseeing NSF grants with the AWM, and
organizing conferences aimed at broadening participation
in mathematics have all given me the wonderful opportu-
nity to interact with researchers across a broad range of
mathematical elds and at many different types of institu-
tions. If serving on the Nominating Committee, I would
be excited to collaborate with committee members to de-
velop a diverse slate of candidates. In this way, AMS can
strengthen its commitment to supporting meetings, publi-
cations, and other scientic programming; engaging in ad-
vocacy for mathematics; addressing systemic issues around
inclusion; and responding to sudden challenges, such as
the vulnerable position of students and professionals in
the face of travel bans.
Kasso A. Okoudjou
Professor, Tufts University
PhD: Georgia Institute of
Technology, 2003.
AMS ofces and commit-
tees: Member at Large, Coun-
cil, 2019–2022; Committee on
Science Policy, 2019–2022; Co-
Chair, Task Force on Under-
standing and Documenting the
Historical Role of the AMS in
Racial Discrimination, 2020–
2021; Executive Committee,
2020–2024; Student Mathematics Library Editorial Com-
mittee, 2024–2028.
Selected publications or other forms of scholarship:
1. with R. S. Strichartz, Weak uncertainty principles on
fractals, J. Fourier Anal. Appl. 11 (2005), no. 3, 315–
331, MR2167172; 2. with A. Benyi, K. Grochenig, and
L. Rogers, Unimodular Fourier multipliers on modula-
tion spaces, J. Funct. Anal. 246 (2007), no. 2, 366–
384, MR2321047; 3. with M. Ehler, Minimization of the
probabilistic p-frame potential, J. Statist. Plann. Inference
142 (2012), no. 3, 645–659, MR2853573; 4. Extension
and restriction principles for the HRT conjecture, J. Fourier
Anal. Appl. 25 (2019), no. 4, 1874–1901, MR3977139; 5.
with T. R. Inniss, W. J. Lewis, I. Mitrea, A. Salerno, F. Su,
and D. Thurston, Towards a fully inclusive mathematics
profession—one year later, Notices Amer. Math. Soc.69
(2022), no. 7, 1214–1219, MR4454145.
Selected addresses or public presentations: Keynote
Speaker, International Conference on Technology, En-
gineering & Mathematics (TEM’18), Kenitra, Morocco,
March 2018; Invited Speaker, 24th Conference for
African American Researchers in the Mathematical Sci-
ences (CAARMS 24), Institute for Advanced Study, Prince-
ton, NJ, July 2018; Keynote Speaker, Undergraduate Re-
search Conference, Georgia Institute of Technology, At-
lanta, GA, April 2022; Lecture at the 2022 Workshop on
Operator Theory with an Eye on Linear Systems and Hy-
percomplex Analysis, Chapman University, Orange, CA,
May 2022; 8th International Conference on Computa-
tional Harmonic Analysis, Ingolstadt, Germany, Septem-
ber 2022; Spring 2023 Christie Lecturer, Bowdoin College,
April 2023.
Synergistic activities: IMU Volunteer Lecturer Program,
IMSP, Benin, January 2014; Member of MIT’s Department
of Mathematics’ Diversity and Community Building Com-
mittee, 2018–2020; Co-Chair, Task Force on Understand-
ing and Documenting the Historical Role of the AMS in
Racial Discrimination, 2020–2021; Co-Organizer of the
(virtual) Gene Golub SIAM Summer School, AIMS South
Africa, 2021; Member of Tufts University’s AS&E Diversity
Fund Committee, 2021–2022.
Candidate statement: I am honored to run for elec-
tion as a member of the Nominating Committee. Aligned
with its mission statement, the AMS is dedicated to ad-
vancing the interests of mathematical research and scholar-
ship, contributing to the national and international com-
munity through publications, meetings, advocacy, and var-
ious programs. In pursuit of this mission, the AMS is ac-
tively fostering inclusivity by promoting the involvement
of all mathematicians in its initiatives. Moreover, achiev-
ing these goals necessitates integrating diverse perspectives
into the AMS governance structure. Therefore, if elected, I
1104 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
Election Candidate Biographies
commit to collaborating with the Nominating Committee
to inspire greater engagement from mathematicians and
enhance their participation in the AMS and its governance.
Gigliola Staffilani
Abby Rockefeller Mauze Profes-
sor, MIT
PhD: University of Chicago.
AMS ofces and commit-
tees: Invited Address Com-
mittee For National Meetings,
2009–2012 (Chair, 2011–2012);
Graduate Studies in Mathemat-
ics, 2010–2024 (Chair, 2020–
2024); AMS-MAA Joint Lecture
Committee, 2018–2019; Bul-
letin Chief Editor Search Com-
mittee, 2018–2020; Committee on the Profession, 2018–
2021; Council, 2018–2023 (Member at Large, 2018–2021,
Executive Committee Representative, 2019–2023); Chair,
Bôcher Memorial Prize Selection Committee, 2019–2020;
Executive Committee of the Council, 2019–2023; Long
Range Planning, 2020–2022; Editor, Communications of
the American Mathematical Society, 2020–2025; Nominat-
ing Committee of the ECBT, 2021–2022; Journal of the
AMS Associate Editor, 2022–2026; Committee on Com-
mittees, 2023–2025; Invited Address Committee For Na-
tional Meeting, 2023–2026 (Chair, 2023–2024).
Selected publications or other forms of scholarship:
1. with J. Colliander, M. Keel, H. Takaoka, and T.
Tao, Global well-posedness and scattering for the energy-
critical nonlinear Schrödinger equation in ℝ3,Ann. of
Math. (2) 167 (2008), no. 3, 767–865, MR2415387; 2.
with J. Colliander, M. Keel, H. Takaoka, and T. Tao, Trans-
fer of energy to high frequencies in the cubic defocusing
nonlinear Schrödinger equation, Invent. Math.181 (2010),
no. 1, 39–113, MR2651381; 3. with A. R. Nahmod, T.
Oh, and L. Rey-Bellet, Invariant weighted Wiener measures
and almost sure global well-posedness for the periodic de-
rivative NLS, J. Eur. Math. Soc. (JEMS) 14 (2012), no.
4, 1275–1330, MR2928851; 4. with D. Mendelson, A. R.
Nahmod, N. Pavlovi´c, and M. Rosenzweig, Poisson com-
muting energies for a system of innitely many bosons,
Adv. Math.406 (2022), Paper No. 108525, 148 pp.,
MR4441152; 5. with M. A. Garrido, R. Grande, and K.
M. Kurianski, Large deviations principle for the cubic NLS
equation, Comm. Pure Appl. Math. 76 (2023), no. 12,
4087–4136, MR4655361.
Selected addresses or public presentations: Hendrik
Lecture Series, MAA, Colorado, 2018; Simons Foundation
Public Lecture, NYC, 2021;Floer Lectures, University of
Bochum, Germany, 2023; Inaugural Noether Lecture Se-
ries, IAS, Princeton, 2023; Alice Roth Lecture, ETH, Zurich,
2023.
Synergistic activities: Co-designer of the online single
variable calculus class offered on MITx, 2015–2016; Or-
ganizes the Women in Math group at MIT (co-designed
the web page https://math.mit.edu/wim/); Chairs the
committee on Diversity and Community Building in
the math department at MIT (co-designed the web page
https://math.mit.edu/diversity/).
Candidate statement: I am honored to have been nom-
inated to run as a member of the Nominating Committee
of the American Mathematical Society. Over the years I
have served in several committees of the AMS, and I have
appreciated the thoughtfulness, dedication, and willing-
ness to serve the community that the AMS ofcers have
demonstrated. The AMS is a wonderful society that is help-
ing the mathematical community in a variety of manners,
and the ofcers are some of the most important parts of its
organization. If elected I will be proud to continue to give
back to the society by recruiting the most capable and ded-
icated future ofcers. I am sure members of the AMS have
realized that leading the society has proved to be a chal-
lenging experience. It is important to have ofcers who
will maintain the highest standards of our discipline, who
will value diversity of backgrounds, and who will work not
just to benet our mathematical community, but also to
make the value of analytic thinking and evidence-based ar-
guments more popular.
Jared Wunsch
Professor of Mathematics,
Northwestern University
PhD: Harvard, 1998.
AMS ofces and commit-
tees: Central Section Program
Committee, 2011–2013; Prize
Oversight Committee, 2019–
2026 (Chair, 2019–2024);
Mathematical Surveys and
Monographs Editorial Commit-
tee, 2023–2027; Invited Ad-
dress Committee for National
Meetings, 2024–2027; Collected Works Editorial Commit-
tee, 2024–2028.
Selected publications or other forms of scholarship:
1. with R. Melrose, Propagation of singularities for the
wave equation on conic manifolds, Invent. Math. 156
(2004), no. 2, 235–299, MR2052609; 2. with A. Hassell,
The Schrödinger propagator for scattering metrics, Ann. of
Math. (2) 162 (2005), no. 1, 487–523, MR2178967;
3. with M. Zworski, Resolvent estimates for normally
hyperbolic trapped sets, Ann. Henri Poincar´e 12 (2011),
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1105
Election Candidate Biographies
no. 7, 1349–1385, MR2846671; 4. with D. Baskin and
A. Vasy, Asymptotics of radiation elds in asymptotically
Minkowski space, Amer. J. Math.137 (2015), no. 5, 1293–
1364, MR3405869; 5. with D. Lafontaine and E. A. Spence,
For most frequencies, strong trapping has a weak effect in
frequency-domain scattering, Comm. Pure Appl. Math. 74
(2021), no. 10, 2025–2063, MR4303013.
Selected addresses or public presentations: MSRI
Evans Lecture, MSRI, 2008; CMI Summer School on Evolu-
tion Equations, 2008; AMS Fall Central Sectional meeting,
2010; Journ´ees EDP, Biarritz, France, 2012; S´eminaire Lau-
rent Schwartz, IHES France, 2016.
Synergistic activities: Organized summer schools in
Zürich (2008) and at Northwestern (2019, 2024); Mentor
for Causeway Postbaccalaureate program, 2021–2022.
Candidate statement: I have been an AMS member
for my whole professional life, and have been grateful for
the Society’s efforts to further our work as mathematicians
and to share that work with the larger world. I have had
some direct involvement in these efforts through service
on AMS committees, most notably as Chair of the Prize
Oversight Committee from 2019–2024. My committee
service and my time as Department Chair at Northwest-
ern (2012–2015) have given me experience in coping with
distinct—and sometimes competing—demands on an or-
ganization with nite resources.
My own research is mainly in pure analysis and PDE,
with a avor of mathematical physics. An additional re-
cent research direction has been in the more applied direc-
tion of numerical analysis. I consequently have a broad
view of what is interesting in mathematics and where it
can be found. In 2024 I served on the Invited Address
Committee, and especially enjoyed the process of select-
ing some dishes from the rich banquet table of current
mathematical research. The Nominating Committee has
a related mission, and I would be excited for the chance
to cast a wide net in seeking excellent candidates for AMS
leadership positions.
The Nominating Committee, like the AMS as a whole,
faces challenges of balancing the Society’s role as the ad-
vocate for mathematics research and its missions in educa-
tion, outreach, and the broadening of participation in the
profession. These missions work in synergy, and I aim to
advance them all.
Editorial Boards Committee
Ivan Corwin
Professor, Columbia University
PhD: New York University,
2011.
AMS ofces and commit-
tees: Centennial Fellowship
Selection Committee, 2017–
2019 (Chair, 2018–2019); Asso-
ciate Editor for Bulletin Articles,
2018–2028.
Selected publications or
other forms of scholarship: 1.
with G. Amir and J. Quastel,
Probability distribution of the free energy of the contin-
uum directed random polymer in 1+1 dimensions, Comm.
Pure Appl. Math.64 (2011), no. 4, 466–537, MR2796514;
2. The Kardar-Parisi-Zhang equation and universality class,
Random Matrices Theory Appl.1(2012), no. 1, 1130001,
MR2930377; 3. with A. Borodin, Macdonald processes,
Probab. Theory Related Fields 158 (2014), no. 1-2, 225–400,
MR3152785; 4. with A. Hammond, Brownian Gibbs prop-
erty for Airy line ensembles, Invent. Math.195 (2014), no.
2, 441–508, MR3152753; 5. with N. O’Connell, T. Sep-
päläinen, and N. Zygouras, Tropical combinatorics and
Whittaker functions, Duke Math. J. 163 (2014), no. 3, 513–
563, MR3165422.
Selected addresses or public presentations: ICM
Seoul, Invited Lecture, 2014; Mahler, Lipschitz, Chern-
Simon, Pinsky Lectures.
Synergistic activities: Scientic board member for
ICERM (2020–2023) and SLMath (2021–present); Lead or-
ganizer of GROW 2024 and 2025 conferences.
Additional experience/qualications you bring to the
position: Editorial board member at 10 journals in gen-
eral mathematics as well as probability and mathematical
physics.
Candidate statement: I am happy to contribute to the
ongoing mission of ensuring excellent editorial oversight
of AMS journals and book series.
1106 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
Election Candidate Biographies
Irene Fonseca
Kavˇci´c-Moura University Pro-
fessor of Mathematics, Carnegie
Mellon University
PhD: University of Minneso-
ta, 1985.
AMS ofces and commit-
tees: Representative to AMS-
IMS-SIAM Evaluation Panel for
the NSF Mathematical Sciences
Postdoctoral Research Fellow-
ships, 2006–2009; AMS Nomi-
nating Committee, 2009–2011;
AMS Short Course Subcommittee, 2016–2019; Mathemat-
ics Research Communities Advisory Board, AMS, 2017–
2020; AMS Fellows Selection Committee, 2017–2020;
AMS Prize Oversight Committee, 2019–2025; CAMS, Com-
munications of the AMS, Senior Editor, 2020–2025; AMS
Bôcher Memorial Prize Selection Committee, 2021–2024;
AMS Editorial Boards Committee, 2022–2025; AMS-SIAM
Committee to Select the Winner of the 2024 Birkhoff Prize,
Chair, 2023–2026; AMS Vice President, 2024–2027.
Selected publications or other forms of scholarship:
1. with G. Bouchitt´e and L. Mascarenhas, A global method
for relaxation, Arch. Rational Mech. Anal. 145 (1998),
51–98, MR1656477; 2. with S. Müller, 𝒜-quasiconvexity,
lower semicontinuity, and Young measures, SIAM J. Math.
Anal. 30 (1999), 1355–1390, MR1718306; 3. with G.
Dal Maso and G. Leoni, Asymptotic analysis of second or-
der nonlocal Cahn-Hilliard-type functionals, Trans. Amer.
Math. Soc. 370 (2018), 2785–2823, MR3748585; 4.
with N. Fusco, M. Morini, and G. Leoni, A model for
dislocations in epitaxially strained elastic lms, J. Math.
Pures Appl. 111 (2018), 126–160, MR3760751; 5. with R.
Choksi, J. Lin, and R. Venkatraman, Anisotropic surface
tensions for phase transitions in periodic media, Calc. Var.
Partial Differential Equations 61 (2022), no. 107, 41 pp.,
MR4404852; 6. with N. Fusco, G. Leoni, and M. Morini,
Global and local energy minimizers for a nanowire growth
model, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 40 (2023),
919–957.
Selected addresses or public presentations: AWM-
SIAM Sonia Kovalevsky Lecturer, SIAM Annual Meeting,
Boston, 2006;Invited Lecture, ICM 2022, July 6–14, 2022.
Synergistic activities: Mentoring of junior faculty
(pre-tenure) at the Department of Mathematical Sciences,
CMU, 2016–2023; AWM (Association for Women in Math-
ematics) Scientic Advisory Committee (Chair, 2021).
Additional experience/qualications you bring to the
position: Director of the Center for Nonlinear Analysis;
Past President of SIAM, 2013–2014; SIAM SIAG APDE
Chair, 2019–2020; SIAM SIAG MS Chair, 2023–2024; PI
of large network grants; Co-PI in international mobility
programs, including with SISSA (Trieste); Serves on 20 Ed-
itorial Boards, including Advances in Calculus of Variations,
Archive for Rational Mechanics and Analysis,Communications
of the AMS (CAMS), ESAIM: COCV (SMAI), Journal of Non-
linear Science,Mathematical Models and Methods in Applied
Sciences (M3AS), and SIAM Journal on Mathematical Anal-
ysis; Member of the ICM 2026 Local Organizing Commit-
tee, and Chair of several advisory and scientic boards of
research centers and institutes, international prize commit-
tees, and review and evaluation panels of multiple univer-
sities in the US and abroad.
Candidate statement: I am honored and grateful to ac-
cept the nomination to join the EBC. It is a responsibility
that I take seriously, and I welcome this opportunity to
continue upholding the standards of excellence set forth
by the AMS and to contribute to the advancement of our
eld through editorial guidance.
Christopher Hacon
McMinn Presidential Endowed
Chair, Distinguished Professor,
University of Utah
PhD: UCLA, 1998.
AMS ofces and commit-
tees: Fellows Program Selec-
tion Committee, 2013–2016;
Cole Prize Selection Commit-
tee, 2015; Western Section Pro-
gram Committee, 2015–2016
(Chair, 2016).
Selected publications or
other forms of scholarship: 1. with C. Birkar, P. Cascini,
and J. McKernan, Existence of minimal models for vari-
eties of log general type, J. Amer. Math. Soc.23 (2010), no.
2, 405–468, MR2601039; 2. with J. McKernan, Existence
of minimal models for varieties of log general type II, J.
Amer. Math. Soc.23 (2010), no. 2, 469–490, MR2601040;
3. with J. McKernan and C. Xu, ACC for log canonical
thresholds, Ann. of Math.(2) 180 (2014), no. 2, 523–571,
MR3224718; 4. with C. Xu, On the three-dimensional
minimal model program in positive characteristic, J. Amer.
Math. Soc.28 (2015), no. 3, 711–744, MR3327534; 5.
with J. McKernan and C. Xu, On the birational automor-
phisms of varieties of general type, Ann. of Math. (2) 177
(2013), no. 3, 1077–1111, MR3034294.
Selected addresses or public presentations: AMS Fall
Sectional Meeting, University of California, Riverside, Ple-
nary Speaker, 2009; Algebraic and Complex Geometry
Session, ICM, Invited Lecture, 2010; British Mathematical
Colloquium, Edinburgh, Plenary Speaker, 2010; European
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1107
Election Candidate Biographies
Congress of Mathematics, Krakow, Plenary Speaker, 2012;
JMM, Baltimore, Plenary Speaker, 2014.
Additional experience/qualications you bring to the
position: Clay Research Award, 2007; Frank Nelson Cole
Prize in Algebra, 2009; Editor, Journal of Algebraic Geom-
etry, since 2009; Associate Editor, Journal of the American
Mathematical Society, 2009–2017; Antonio Feltrinelli Prize
in Mathematics, Mechanics, and Applications, 2011; MSRI
Science Advisory Committee, 2012–2018 (Co-chair, 2015–
2018); Fellow of the AMS, 2013; Associate Editor, An-
nals of Mathematics, 2013–2020; Associate Editor, Bollettino
dell’Unione Matematica Italiana, since 2013; ICM commit-
tee for the selection of sectional speakers in Algebraic and
Complex Geometry, 2014 and 2022; Selection Commit-
tee, Alfred P. Sloan Research Fellowships in Mathematics,
2015–2021; Associate Editor, Cambridge Journal of Math-
ematics, since 2016; Associate Editor, Journal of Pure and
Applied Algebra, since 2016; E. H. Moore Research Article
Prize, 2016; Member of the American Academy of Arts and
Sciences, 2017; Breakthrough Prize, 2018; Member of the
National Academy of Sciences, 2018; Selection Committee,
Breakthrough Prize in Mathematics, 2019–present (Chair,
since 2021); Fellow of the Royal Society, 2019.
Candidate statement: I am honored to be nominated
for election to the AMS Editorial Boards Committee.
The books and journals published by the AMS are an
important resource of the highest quality for mathemati-
cians worldwide. I believe that if elected to the AMS Edi-
torial Boards Committee, my experience as an author and
editor will allow me to contribute to the ongoing success
of these publications.
Michael J. Larsen
Distinguished Professor, Math-
ematics, Indiana University,
Bloomington
PhD: Princeton University,
1988.
AMS ofces and commit-
tees: Transactions and Memoirs
Editorial Committee, 2000–
2001; Committee on Publica-
tions, 2014–2017; AMS Coun-
cil, Member at Large, 2014–
2017; Transactions and Memoirs
Editorial Committee, 2015–2019; Frank Nelson Cole Prize
Selection Committee, 2016–2017; Journal of the AMS, Asso-
ciate Editor, 2016–2021; Journal of the AMS Editorial Com-
mittee, 2021–2025.
Selected publications or other forms of scholarship:
1. with M. H. Freedman, A. Kitaev, and Z. Wang, Topolog-
ical quantum computation. Mathematical challenges of
the 21st century (Los Angeles, CA, 2000), Bull. Amer. Math.
Soc. (N.S.) 40 (2003), no. 1, 31–38, MR194313; 2. with R.
Pink, Finite subgroups of algebraic groups, J. Amer. Math.
Soc.24 (2011), no. 4, 1105–1158, MR2813339; 3. with A.
Shalev and P. H. Tiep, Probabilistic Waring problems for
nite simple groups, Ann. of Math. (2) 190 (2019), no.
2, 561–608, MR3997129; 4. with V. A. Lunts, Irrational-
ity of motivic zeta functions, Duke Math. J. 169 (2020),
no. 1, 1–30, MR4047547; 5. with L. Hesselholt and A.
Lindenstrauss, On the K-theory of division algebras over
local elds, Invent. Math.219 (2020), no. 1, 281–329,
MR4050106.
Selected addresses or public presentations: Algebraic
Geometry: Seattle (plenary lectures), 2005; Texas A & M
University Frontier Lecture Series, 2008; Binghamton Uni-
versity Dean’s Speaker Series in Geometry/Topology, 2011;
Cornell University Chelluri Lecture, 2022; International
Congress of Mathematicians (online), 2022.
Synergistic activities: Founded the Bloomington Math
Circle; Wrote Putnam Exam problems, originally as a
member of the Putnam Committee (MAA), and recently
as an ”additional contributor.”
Candidate statement: AMS publications have meant
a lot to me over my professional life. My shelves are
crammed with them, from the beaten-up old volumes of
Automorphic Forms, Representations, and L-functions, to re-
cent acquisitions, like the handsome collected works of
Tate. My hope is that in the face of rapid change, the AMS
can remain a relevant and viable alternative to high-priced
commercial publishers.
I have only been involved with AMS books as a reader,
but on the journal side, I have served as an AMS editor
since 2015 (rst for the Transactions and the Memoirs and
now for JAMS). In that capacity, I have come to appreci-
ate the enormous amount of labor that referees put into
making the journal system work. If I am elected to the
Editorial Boards Committee, I will try to ensure that our
editors are worthy of that effort: fair, broadly knowledge-
able, mathematically open minded, and as transparent as
possible given the constraints of the anonymous refereeing
process.
1108 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
105 Notices of the AmericAN mAthemAticAl society Volume 66, Number 1
Special Section
2019 Election
FROM THE AMS SECRETARY
Ballots
AMS members will receive email with instructions for vot-
ing online by August 20, or a paper ballot by September 20.
If you do not receive this information by that date, please
contact the AMS (preferably before October 1) to request
a ballot. Send email to ballot@ams.org or call the AMS
at 800-321-4267 (within the US or Canada) or 401-455-
4000 (worldwide). The deadline for receipt of ballots is
November 2, 2018.
Write-in Votes
It is suggested that names for write-in votes be accompa-
nied by the institution or web address of the individual for
whom the vote is cast.
Replacement Ballots
A member who has not received a ballot by September 20,
2018, or who has received a ballot but has accidentally
spoiled it, may write to ballot@ams.org or Secretary of
the AMS, 201 Charles Street, Providence, RI 02904-2213,
USA, asking for a second ballot. The request should in-
clude the individual’s member code and the address to
which the replacement ballot should be sent. Immediately
upon receipt of the request a second ballot, indistin-
guishable from the original, will be sent by first class or
airmail. However, the deadline for receipt of ballots will
not be extended.
Biographies of Candidates
The next several pages contain biographical information
about all candidates. All candidates were given the oppor-
tunity to provide a statement of not more than 200 words
to appear at the end of their biographical information.
Photos were supplied by the candidates.
Description of Offices
The vice president and the members at large of the
Council serve for three years on the Council. That body
determines all scientific policy of the Society, creates and
oversees numerous committees, appoints the treasurers
and members of the Secretariat, makes nominations of
candidates for future elections, and determines the chief
editors of several key editorial boards. Typically, each of
these new members of the Council will also serve on one
of the Society’s five policy committees. Current and past
members of the Council may be found here: www.ams
.org/comm-all.html#COUNCIL.
Vice President
(one to be elected)
Sara Billey
Abigail Thompson
Board of Trustees
(one to be elected)
Matthew Ando
Rick Miranda
Member at Large
of the Council
(five to be elected)
Dan Freed
Fernando Q. Gouvêa
Christopher D. Hacon
Daniel Krashen
Susan Loepp
Lenhard Ng
Kasso A. Okoudjou
Maria Cristina Pereyra
Hal Schenck
Melanie Matchett Wood
Nominating Committee
(three to be elected)
Sami H. Assaf
Ricardo Cortez
Rebecca Garcia
Yongbin Ruan
David Savitt
Deane Yang
Editorial Boards Committee
(two to be elected)
Ian Agol
David Marker
James McKernan
Terence Tao
List of Candidates–2019 Election
Update in
master page
MP: a-ballot-opener
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2025 Election
Call for Suggestions
YOUR SUGGESTIONS ARE WANTED BY:
the Nominating Committee, for the following contested seats in the 2025 AMS elections:
vice president, trustee, and ve members at large of the Council.
Deadline for suggestions: November 1, 2024
the president, for the following contested seats in the 2025 AMS elections:
three members of the Nominating Committee and two members of the Editorial Boards
Committee.
Deadline for suggestions: January 31, 2025
the Editorial Boards Committee, for appointments to various editorial boards of
AMS publications.
Deadline for suggestions: Can be submitted any time
Send your suggestions for any of the above to:
Boris Hasselblatt, Secretary
American Mathematical Society
201 Charles Street
Providence, RI 02904-2213, USA
secretary@ams.org
or submit them online at
www.ams.org/committee-nominate
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1109
105 Notices of the AmericAN mAthemAticAl society Volume 66, Number 1
Special Section
2019 Election
FROM THE AMS SECRETARY
Ballots
AMS members will receive email with instructions for vot-
ing online by August 20, or a paper ballot by September 20.
If you do not receive this information by that date, please
contact the AMS (preferably before October 1) to request
a ballot. Send email to ballot@ams.org or call the AMS
at 800-321-4267 (within the US or Canada) or 401-455-
4000 (worldwide). The deadline for receipt of ballots is
November 2, 2018.
Write-in Votes
It is suggested that names for write-in votes be accompa-
nied by the institution or web address of the individual for
whom the vote is cast.
Replacement Ballots
A member who has not received a ballot by September 20,
2018, or who has received a ballot but has accidentally
spoiled it, may write to ballot@ams.org or Secretary of
the AMS, 201 Charles Street, Providence, RI 02904-2213,
USA, asking for a second ballot. The request should in-
clude the individual’s member code and the address to
which the replacement ballot should be sent. Immediately
upon receipt of the request a second ballot, indistin-
guishable from the original, will be sent by first class or
airmail. However, the deadline for receipt of ballots will
not be extended.
Biographies of Candidates
The next several pages contain biographical information
about all candidates. All candidates were given the oppor-
tunity to provide a statement of not more than 200 words
to appear at the end of their biographical information.
Photos were supplied by the candidates.
Description of Offices
The vice president and the members at large of the
Council serve for three years on the Council. That body
determines all scientific policy of the Society, creates and
oversees numerous committees, appoints the treasurers
and members of the Secretariat, makes nominations of
candidates for future elections, and determines the chief
editors of several key editorial boards. Typically, each of
these new members of the Council will also serve on one
of the Society’s five policy committees. Current and past
members of the Council may be found here: www.ams
.org/comm-all.html#COUNCIL.
Vice President
(one to be elected)
Sara Billey
Abigail Thompson
Board of Trustees
(one to be elected)
Matthew Ando
Rick Miranda
Member at Large
of the Council
(five to be elected)
Dan Freed
Fernando Q. Gouvêa
Christopher D. Hacon
Daniel Krashen
Susan Loepp
Lenhard Ng
Kasso A. Okoudjou
Maria Cristina Pereyra
Hal Schenck
Melanie Matchett Wood
Nominating Committee
(three to be elected)
Sami H. Assaf
Ricardo Cortez
Rebecca Garcia
Yongbin Ruan
David Savitt
Deane Yang
Editorial Boards Committee
(two to be elected)
Ian Agol
David Marker
James McKernan
Terence Tao
List of Candidates–2019 Election
Update in
master page
MP: a-ballot-opener
PS: NEWS SUBHEAD
CS: BCS: I
PS: TEXT NO INDENT
— SINGLE LINE SPACE —
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INDENT
1 PICA
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2025 Election
Nominations by Petition
Vice President or Member at Large
One position of vice president and member of the Council
ex ofcio for a term of three years is to be lled in the elec-
tion of 2025. The Council intends to nominate at least two
candidates, among whom may be candidates nominated
by petition as described in the rules and procedures below.
Five positions of member at large of the Council for a
term of three years are to be lled in the same election.
The Council intends to nominate at least ten candidates,
among whom may be candidates nominated by petition in
the manner described in the rules and procedures below.
Petitions are presented to the Council, which, according
to Section 2 of Article VII of the bylaws, makes the nomi-
nations.
Prior to presentation to the Council, petitions in sup-
port of a candidate for the position of vice president or
of member at large of the Council must have at least fty
valid signatures and must conform to several rules and pro-
cedures, which are described below. Petitioners can facil-
itate the procedure by accompanying the petitions with a
signed statement from the candidate giving consent.
Editorial Boards Committee
Two places on the Editorial Boards Committee will be
lled by election. There will be four continuing members
of the Editorial Boards Committee.
The president will name at least four candidates for
these two places, among whom may be candidates nomi-
nated by petition in the manner described in the rules and
procedures.
The candidate’s assent and petitions bearing at least 100
valid signatures are required for a name to be placed on
the ballot. In addition, several other rules and procedures,
described below, should be followed.
Nominating Committee
Three places on the Nominating Committee will be lled
by election. There will be six continuing members of the
Nominating Committee.
The president will name at least six candidates for these
three places, among whom may be candidates nominated
by petition in the manner described in the rules and pro-
cedures.
The candidate’s assent and petitions bearing at least 100
valid signatures are required for a name to be placed on
the ballot. In addition, several other rules and procedures,
described below, should be followed.
Rules and Procedures
Use separate copies of the form for each candidate for vice
president, member at large, or member of the Nominating
or Editorial Boards Committees.
1. To be considered, petitions must be addressed to Sec-
retary, American Mathematical Society, 201 Charles
Street, Providence, RI 02904-2213, USA, and must ar-
rive by 24 February 2025.
2. The name of the candidate must be given as it appears
in the American Mathematical Society’s membership
records and must be accompanied by the member
code. If the member code is not known by the can-
didate, it may be obtained by the candidate contact-
ing the AMS headquarters in Providence (amsmem@ams
.org).
3. The petition for a single candidate may consist of sev-
eral sheets each bearing the statement of the petition,
including the name of the position, and signatures.
The name of the candidate must be exactly the same
on all sheets.
4. On the next page is a sample form for petitions. Peti-
tioners may make and use photocopies or reasonable
facsimiles.
5. A signature is valid when it is clearly that of the mem-
ber whose name and address is given in the left-hand
column.
6. When a petition meeting these various requirements
appears, the secretary will ask the candidate to indicate
willingness to be included on the ballot.
1110 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
Election Nominations by Petition
Nominations by Petition
____________________________________________________________________________
The undersigned members of the American Mathematical Society propose the name of
______________________________________________________________ as a candidate for the position of (check one):
□Vice President (term beginning 02/01/2026)
□Member at Large of the Council (term beginning 02/01/2026)
□Member of the Nominating Committee (term beginning 01/01/2026)
□Member of the Editorial Boards Committee (term beginning 02/01/2026)
of the American Mathematical Society.
Return petitions by February 24, 2025 to:
Secretary, AMS, 201 Charles Street, Providence, RI 02904-2213, USA
Name, address, and AMS member code,
if available (printed or typed)
Signature
Signature
Signature
Signature
Signature
Signature
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1111
NEWS
AMS Updates
Early-Bird Registration
Open for December Joint
International Meeting
Register now for the joint meeting of the New Zealand,
Australian, and American mathematical societies, taking
place December 9–13, 2024, in Auckland, New Zealand.
Early-bird registration is available until October 25, 2024.
Plenary speakers at the joint meeting of the NZMS,
AustMS, and AMS include Persi Diaconis (Stanford Uni-
versity), Rachael Ka’ai-Mahuta (Auckland University of
Technology), Svitlana Mayboroda (University of Min-
nesota, ETH Zurich), Michael Miller (Victoria University
of Wellington), and James Saunderson (Monash Univer-
sity).
General plenary speakers at the joint meeting in-
clude Lara Alcock (Loughborough University), Richard
Kenyon (Yale University), Eamonn O’Brien (University of
Auckland), Priya Subramanian (University of Auckland),
Katharine Turner (Australian National University), and
Geordie Williamson (University of Sydney).
Learn more and register at https://ms-meet-2024
.blogs.auckland.ac.nz/registration/.
—AMS Communications
Last Call for AMS
Fellowship Applications
September 30, 2024, is the deadline to apply for the fol-
lowing AMS fellowships for 2025–2026:
•Stefan Bergman Fellowship ($25,000), an early-
career research fellowship for mathematicians
who specialize in the areas of real analysis, com-
plex analysis, or partial differential equations;
•Joan and Joseph Birman Fellowship for Women
Scholars ($50,000), a mid-career research fellow-
DOI: https://doi.org/10.1090/noti3014
ship for women mathematicians, specially de-
signed to t the unique needs of women;
•AMS Centennial Research Fellowship ($50,000),
a research fellowship for mathematicians cur-
rently holding a tenured, tenure-track, postdoc-
toral, or comparable (at the discretion of the se-
lection committee) position at an institution in
North America. Recipients of the fellowship shall
have held their doctoral degree for at least three
years and not more than twelve years at the incep-
tion of the award;
•AMS Claytor-Gilmer Fellowship ($50,000), a
mid-career research fellowship to further excel-
lence in mathematics research and to help gen-
erate wider and sustained participation by Black
mathematicians.
More information is available at https://www.ams
.org/programs/ams-fellowships/ams-fellowships.
—AMS Communications
Teens Win Menger Awards
at ISEF
The American Mathematical Society (AMS) presented the
Karl Menger Awards at the 2024 Regeneron International
Science and Engineering Fair (Regeneron ISEF), held in Los
Angeles on May 17, 2024. The winners were high-school
students who earned the right to compete at the Regeneron
ISEF by winning a top prize at a local, regional, state, or
national science fair. All winners received a one-year AMS
membership and a booklet on Karl Menger.
Quang Tran of Patrick F. Taylor Science and Technol-
ogy Academy, Harvey, LA, received the rst-place prize of
$2,000 for Divisors.
Second awards ($1,000): Anna Oliva, Carnegie
Vanguard High School, Houston, TX, Symmetry, Fixed
Points and Quantum Billiards and Emma Rueter, Leibniz-
Gymnasium Berlin, Berlin, Germany, Integration of Se-
quences
1112 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
Third awards ($500): Arda Ozcelebi, Izmir Ozel Ege
Lisesi, Izmir, Turkey, p-Euler-phi Partitions and Their Prop-
erties; Yoonsang Lee, Korea Science Academy of KAIST,
Seoul, South Korea, A Study on Arc Index of Theta Curves;
Anay Aggarwal and Manu Isaacs, Jesuit High School, Port-
land, OR, Fast Modular Exponentiation with Factored Modu-
lus; Helena Welch, Los Alamos High School, Los Alamos,
NM, Modeling an Ancient Musical Instrument
Certicates of Honorable Mention: Yunjia Quan, Char-
lotte Country Day School, Charlotte, NC, Enhancing
Ethereum’s Security With LUMEN; Austin Luo, Morgan-
town High School,Morgantown, WV, Injective Chromatic
Index of Packet Radio Networks; Ayush Jain, Shri Ram
School - Aravali Campus, Gurgaon, Haryana, India, De-
tecting Causality Using Symplectic Quandles; Joseph Vulakh,
Paul Laurence Dunbar High School, Lexington, KY, Twisted
Homogeneous Racks; Arav Chand, Half Hollow Hills High
School West, Dix Hills, NY, Proofs of Fibonacci Analogues
of Two Theorems; Songtianze Huang, Hangzhou Foreign
Languages School, Hangzhou, Zhejiang, China, Group of
seventh chord transformations; Sarah Lu, Centro Residen-
cial de Oportunidades Educativas de Mayaguez, Mayaguez,
Puerto Rico, Enhancing Federated Learning Using Math and
Coding
AMS participation in the Regeneron ISEF is supported
in part by funds from the Karl Menger Fund, which was
established by the family of the late Karl Menger. Awards
at the fair are given to pre-college students in mathematics
as well as to mathematically oriented projects in computer
science, physics, and engineering.
—AMS Communications
Deaths of AMS Members
Jacob P. Murre, of the Netherlands, died on April 9, 2023.
Born on September 18, 1929, he was a member of the So-
ciety for 68 years.
William W. Adams, of Silver Spring, Maryland, died on
February 15, 2024. Born on July 23, 1937, he was a mem-
ber of the Society for 63 years.
George L. Csordas, of Honolulu, Hawaii, died on April
23, 2024. Born on September 20, 1941, he was a member
of the Society for 57 years.
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1113
NEWS
Mathematics People
Sarnak Wins 2024 Shaw Prize
Peter Sarnak of the Institute for Advanced Study (IAS) and
Princeton University was awarded the 2024 Shaw Prize in
Mathematical Sciences for his development of the arith-
metic theory of thin groups and the afne sieve by bringing
together number theory, analysis, combinatorics, dynam-
ics, geometry, and spectral theory.
Sarnak is the Gopal Prasad Professor of Mathematics
at IAS and Eugene Higgins Professor of Mathematics at
Princeton, where he also has served as department chair.
Born in Johannesburg, South Africa, Sarnak received his
PhD in Mathematics from Stanford University in 1980. He
has taught at Stanford and the Courant Institute of Math-
ematical Sciences, New York University. Sarnak is a mem-
ber of the US National Academy of Sciences and a Fellow
of both the American Mathematical Society (AMS) and the
Royal Society of London.
The Shaw Prize Foundation awards three annual prizes,
in astronomy, life science and medicine, and mathemat-
ical sciences, each bearing an award of US$1.2 million.
The presentation ceremony is scheduled for November 12,
2024, in Hong Kong.
—AMS Communications
Ambrosio Receives 2024
Nemmers Prize
Luigi Ambrosio, Scuola Normale Superiore, Pisa, Italy, re-
ceived the Frederic Esser Nemmers Prize in Mathematics.
Ambrosio was honored for his “deep and numerous con-
tributions to calculus of variations and geometric mea-
sure theory, and broad and far-reaching inuence on these
elds,” as announced by Northwestern University.
Ambrosio, a professor of mathematical analysis, will re-
ceive US$300,000 and will present lectures, participate in
seminars, and engage with faculty and students in other
scholarly activities.
DOI: https://doi.org/10.1090/noti3015
“Together with his PhD advisor, Ennio De Giorgi, Am-
brosio founded the theory of free discontinuity problems,
a class of problems in the calculus of variations that in-
volves the combination of volume and surface energies. In
this class, it is possible to frame problems coming from im-
age segmentation and fracture mechanics,” according to a
press release.
“In the second part of his career, Ambrosio moved to
the theory of currents in geometric measure theory, intro-
ducing a far-reaching extension of the Federer-Fleming the-
ory to metric spaces and to the theory of ows associated
to non-smooth vector elds. His present research interests
include optimal transport and analysis in metric measure
spaces.”
Ambrosio was a plenary speaker at the 2018 Interna-
tional Congress of Mathematics (ICM) and previously gave
an invited section lecture at ICM 2002. He was awarded
the Caccioppoli Prize (1999), Fermat Prize (2003), Balzan
Prize (2019), and Riemann Prize (2022).
Northwestern’s biennial Nemmers Prizes recognize top
scholars in earth sciences, economics, and mathematics
for their lasting contributions to new knowledge, outstand-
ing achievements, and the development of signicant new
modes of analysis.
—AMS Communications
Daubechies Named to
Royal Society
Ingrid C. Daubechies of Duke University has been elected
a Foreign Member of the Royal Society, the United King-
dom’s national academy of sciences.
In 2024, more than 90 researchers from around the
world were elected to the Fellowship of the Royal Society
as fellows and foreign members. “This new cohort has al-
ready made signicant contributions to our understand-
ing of the world around us and continues to push the
boundaries of possibility in academic research and indus-
try,” said Sir Adrian Smith, president of the Royal Society,
in a press release.
1114 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 71, NUMBER 8
Mathematics People
NEWS
At Duke, Daubechies is James B. Duke Professor, De-
partment of Mathematics and Department of Electrical
and Computer Engineering. She received her PhD in 1980
from Vrije Universiteit Brussel.
—AMS Communications
Nelson, Newton, Thorne
Named Clay Research
Awardees for 2024
The Clay Mathematics Institute announced the 2024 Clay
Research Awards, which will be presented at the Clay Re-
search Conference in Oxford, UK, on October 2, 2024.
A joint award was made to James Newton (University of
Oxford) and Jack Thorne (University of Cambridge) “in
recognition of their remarkable proof of the existence of
the symmetric power functorial lift for Hilbert modular
forms.... The proof marks a milestone in work on the
Langlands program.”
Paul Nelson (Aarhus University) received a Clay Re-
search Award in recognition of his contributions to the
analytic theory of automorphic forms. “His work has re-
sulted in the rst convexity breaking bounds for a large
class of L-functions on the critical line (including all the
standard ones of GL(n)),” according to a press release.
“This marks a signicant advance in a eld initiated one
hundred years ago by Hermann Weyl in the context of the
Riemann Zeta function.”
—Clay Mathematics Institute
Kupers Awarded 2024
Aisenstadt Prize
The 2024 André Aisenstadt Prize in Mathematics was
awarded to Alexander Kupers, University of Toronto Scar-
borough (UTSC). Created in 1991 by the Centre de
recherches mathématiques (CRM), the Aisenstadt Prize
recognizes outstanding research results in pure or applied
mathematics by a young Canadian mathematician. It in-
cludes a scholarship and a medal.
Born in the Netherlands, Kupers received his PhD in
2016 from Stanford University. Following a postdoctoral
position at the University of Copenhagen and a Benjamin
Peirce Fellowship at Harvard University, he joined UTSC’s
department of computer and mathematical sciences in
2020. Kupers’ work is “motivated by the modern version
of a classical problem: the classication of smooth mani-
folds,” according to a press release.
—Centre de recherches math´ematiques
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1115
Submission deadlines and eligibility info:
www.ams.org/student-travel
AMS Graduate Student
Apply for support for doctoral student
travel to the Joint Mathematics Meetings
• Connect with fellow researchers
• Advance your career
• Present your research
• Explore new mathematical areas
• Expand your knowledge of
professional and educational matters
Classied Advertising
Employment Opportunities
NEW JERSEY
Program in Applied and Computational Mathematics
Princeton University
Postdoctoral Research Associate
The Program in Applied and Computational Mathematics
invites applications for Postdoctoral Research Associate or
more senior positions, to join in research efforts of interest
to its faculty. Domains of interest include nonlinear par-
tial differential equations, computational uid dynamics,
material science, dynamical systems, numerical analysis,
stochastic analysis, graph theory and applications, mathe-
matical biology, nancial mathematics, mathematical ap-
proaches to signal analysis, information theory, structural
biology and image processing.
The term of appointment is based on rank. Positions
at the postdoctoral rank are for one year with the pos-
sibility of renewal pending satisfactory performance and
continued funding; those hired at more senior ranks may
have multi-year appointments. For details on specic fac-
ulty members and their research interests, please go to
https://www.pacm.princeton.edu/sites/default
/files/2023-10/Faculty%20Interests%2023-24
.pdf.
The Notices Classied Advertising section is devoted to listings of current employment opportunities. The publisher reserves the right to reject any listing not in
keeping with the Society’s standards. Acceptance shall not be construed as approval of the accuracy or the legality of any information therein. Advertisers are
neither screened nor recommended by the publisher. The publisher is not responsible for agreements or transactions executed in part or in full based on classied
advertisements.
The 2024 rate is $3.65 per word. Advertisements will be set with a minimum one-line headline, consisting of the institution name above body copy, unless additional
headline copy is specied by the advertiser. Headlines will be centered in boldface at no extra charge. Ads will appear in the language in which they are submitted.
There are no member discounts for classied ads. Dictation over the telephone will not be accepted for classied ads.
Upcoming deadlines for classied advertising are as follows: November 2024—August 23, 2024; December 2024—September 27, 2024.
US laws prohibit discrimination in employment on the basis of color, age, sex, race, religion, or national origin. Advertisements from institutions outside the US
cannot be published unless they are accompanied by a statement that the institution does not discriminate on these grounds whether or not it is subject to US laws.
Submission: Send email to classads@ams.org.
.
DOI: https://doi.org/10.1090/noti3017
Applicants must submit a cover letter, CV, bibli-
ography/publications list, statement of research and
three letters of recommendation online at https://www
.mathjobs.org/jobs. PhD is required. This position is
subject to the University background check policy. The
work location for this position is in-person on campus at
Princeton University.
Princeton University is an Equal Opportunity/
Afrmative Action Employer: https://rrr.princeton
.edu/2023/equal-opportunity-policy/equal
-opportunity-policy and all qualied applicants will re-
ceive consideration for employment without regard to age,
race, color, religion, sex, sexual orientation, gender iden-
tity or expression, national origin, disability status, pro-
tected veteran status, or any other characteristic protected
by law.
4
SEPTEMBER 2024 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1117
NEW BOOKS
New Books Offered by the AMS
1118 Notices of the AmericAN mAthemAticAl society Volume 71, Number 8
General Interest
The Mathematical
Playground
People and Problems from 31
Years of Math Horizons
Alissa S. Crans, Loyola Mary-
mount University, Los Angeles, CA,
and Glen T. Whitney, Prison
Math Project, Phoenix, AZ, Editors
Welcome to The Mathematical
Playground, a book celebrating
more than thirty years of the
problems column in the MAA
undergraduate magazine, Math Horizons. Anecdotes, in-
terviews, and historical sketches accompany the puzzles,
conveying the vibrancy of the “Playground” community.
The lively prose and humor used throughout the book
reveal the enthusiasm and playfulness that have become
the column’s hallmark.
Each chapter features a theme that helps illustrate
community: from the Opening Acts—chronicling how in-
teresting questions snowball into original research—to the
Posers and Solvers themselves. These stories add an engaging
dimension beyond the ample mathematical challenge. A
particular highlight is a chapter introducing the seven ed-
itors who have produced “The Playground”, revealing the
perspectives of the individuals behind the column.
The Mathematical Playground has plenty to offer both
novice and experienced solvers. The lighthearted, conversa-
tional style, together with copious hints, a problem-solving
primer, and a detailed glossary, welcomes newcomers,
regardless of their background, to the puzzle-solving
world. The more seasoned solver will find over twenty new
problems plus open-ended challenges and suggestions for
further investigation. Whether you’re a long-time Math
Horizons reader, or encountering “The Playground” for the
first time, you are invited into this celebration of the rich
culture of recreational mathematics. Just remember the
most important rule … Have fun!
Discrete Mathematics
and Combinatorics
Exploring
Discrete Geometry
Thomas Q. Sibley, St. John’s
University, Collegeville, MN
Together with its clear mathe-
matical exposition, the prob-
lems in this book take the reader
from an introduction to discrete
geometry all the way to its fron-
tiers. Investigations start with
easily drawn figures, such as
dividing a polygon into triangles
or finding the minimum number of “guards” for a polygon
(“art gallery” problem). These early explorations build in-
tuition and set the stage. Variations on the initial problems
stretch this intuition in new directions. These variations
on problems together with growing intuition and under-
standing illustrate the theme of this book: “When you have
answered the question, it is time to question the answer.”
Numerous drawings, informal explanations, and careful
reasoning build on high school algebra and geometry.
This item will also be of interest to those working in geometry
and topology.
Anneli Lax New Mathematical Library, Volume 56
August 2024, 156 pages, Softcover, ISBN: 978-1-4704-
7807-0, LC 2024015227, 2020 Mathematics Subject Clas-
sification: 52–01, List US$69, AMS Individual member
US$51.75, AMS Institutional member US$55.20, MAA
members US$51.75, Order code NML/56
bookstore.ams.org/nml-56
NEW BOOKS
september 2024 Notices of the AmericAN mAthemAticAl society 1119
Geometry and Topology
Trees of Hyperbolic Spaces
Michael Kapovich, University of
California, Davis, CA, and Pra-
nab Sardar, Indian Institute of
Science Education and Research,
Mohali, India
This book offers an alternative
proof of the Bestvina–Feighn
combination theorem for trees
of hyperbolic spaces and de-
scribes uniform quasigeodesics
in such spaces. As one of the
applications of their description of uniform quasigeodesics,
the authors prove the existence of Cannon–Thurston maps
for inclusion maps of total spaces of subtrees of hyperbolic
spaces and of relatively hyperbolic spaces. They also ana-
lyze the structure of Cannon–Thurston laminations in this
setting. Furthermore, some group-theoretic applications
of these results are discussed. This book also contains
background material on coarse geometry and geometric
group theory.
Mathematical Surveys and Monographs, Volume 282
August 2024, 278 pages, Softcover, ISBN: 978-1-4704-7425-
6, LC 2024009568, 2020 Mathematics Subject Classifica-
tion: 20F67, 51F30, List US$135, AMS members US$108,
MAA members US$121.50, Order code SURV/282
bookstore.ams.org/surv-282
Frontiers in Geometry and
Topology
Paul M. N. Feehan, Rutgers, The
State University of New Jersey,
Piscataway, NJ, Lenhard L. Ng,
Duke University, Durham, NC,
and Peter S. Ozsváth, Princeton
University, NJ, Editors
This volume contains the pro-
ceedings of the summer school
and research conference “Fron-
tiers in Geometry and Topol-
ogy”, celebrating the sixtieth birthday of Tomasz Mrowka,
which was held from August 1–12, 2022, at the Abdus
Salam International Centre for Theoretical Physics (ICTP).
The summer school featured ten lecturers and the re-
search conference featured twenty-three speakers covering a
range of topics. A common thread, reflecting Mrowka’s own
Problem Books, Volume 38
August 2024, 458 pages, Softcover, ISBN: 978-1-4704-
7752-3, LC 2024014160, 2020 Mathematics Subject Clas-
sification: 00A07, 00A08, 00A27, 97D50, 01A07, 01A80,
List US$39, AMS Individual member US$29.25, AMS
Institutional member US$31.20, MAA members US$29.25,
Order code PRB/38
bookstore.ams.org/prb-38
Teaching Mathematics
Through Cross-Curricular
Projects
Elizabeth A. Donovan, Murray
State University, KY, Lucas A.
Hoots, Morehead State University,
KY, and Lesley W. Wiglesworth,
Centre College, Danville, KY,
Editors
This book offers engaging
cross-curricular modules to sup-
plement a variety of pure mathe-
matics courses. Developed and tested by college instructors,
each activity or project can be integrated into an instructor’s
existing class to illuminate the relationship between pure
mathematics and other subjects. Every chapter was carefully
designed to promote active learning strategies.
The editors have diligently curated a volume of twen-
ty-six independent modules that cover topics from fields as
diverse as cultural studies, the arts, civic engagement, STEM
topics, and sports and games. An easy-to-use reference table
makes it straightforward to find the right project for your
class. Each module contains a detailed description of a
cross-curricular activity, as well as a list of the recommended
prerequisites for the participating students. The reader will
also find suggestions for extensions to the provided activi-
ties, as well as advice and reflections from instructors who
field-tested the modules.
Teaching Mathematics Through Cross-Curricular Projects is
aimed at anyone wishing to demonstrate the utility of pure
mathematics across a wide selection of real-world scenarios
and academic disciplines. Even the most experienced in-
structor will find something new and surprising to enhance
their pure mathematics courses.
This item will also be of interest to those working in math ed-
ucation.
Classroom Resource Materials, Volume 72
August 2024, 351 pages, Softcover, ISBN: 978-1-4704-7466-
9, LC 2024006104, 2020 Mathematics Subject Classifica-
tion: 00–XX, 97–XX, List US$65, AMS Individual member
US$48.75, AMS Institutional member US$52, MAA mem-
bers US$48.75, Order code CLRM/72
bookstore.ams.org/clrm-72
Mathematical
Surveys
and
Monographs
Volume 282
Trees of
Hyperbolic
Spaces
Michael Kapovich
Pranab Sardar
Volume 109
Proceedings of Symposia in
Proceedings of Symposia in
ATHEMATICS
ATHEMATICS
P
URE
URE
M
Frontiers in
Geometry
and Topology
Paul M. N. Feehan
Lenhard L. Ng
Peter S. Ozsváth
Editors
NEW BOOKS
1120 Notices of the AmericAN mAthemAticAl society Volume 71, Number 8
This item will also be of interest to those working in probability
and statistics.
Titles in this series are co-published with the Courant Institute of
Mathematical Sciences at New York University.
Courant Lecture Notes, Volume 32
September 2024, approximately 205 pages, Softcover, ISBN:
978-1-4704-5618-4, LC 2024013482, 2020 Mathematics
Subject Classification: 60–XX, 81–XX, 82–XX, List US$55,
AMS Individual member US$40.88, AMS Institutional
member US$43.60, MAA members US$49.05, Order code
CLN/32
bookstore.ams.org/cln-32
New in Contemporary
Mathematics
Algebra and
Algebraic Geometry
A Glimpse into Geometric
Representation Theory
Mahir Bilen Can, Tulane Univer-
sity, New Orleans, LA, and Jörg
Feldvoss, University of South Al-
abama, Mobile, AL, Editors
This volume contains the pro-
ceedings of the AMS Special
Session on Combinatorial and
Geometric Representation The-
ory, held virtually on November
20–21, 2021. The articles offer
an engaging look into recent advancements in geometric
representation theory.
Despite diverse subject matters, a common thread unit-
ing the articles of this volume is the power of geometric
methods. The authors explore the following five contem-
porary topics in geometric representation theory: equivar-
iant motivic Chern classes; equivariant Hirzebruch classes
and equivariant Chern-Schwartz-MacPherson classes of
Schubert cells; locally semialgebraic spaces, Nash mani-
folds, and their superspace counterparts; support varieties
of Lie superalgebras; wreath Macdonald polynomials; and
equivariant extensions and solutions of the Deligne-Simp-
son problem.
Each article provides a well-structured overview of its
topic, highlighting the emerging theories developed by the
authors and their colleagues.
work, was the rich interplay among the fields of analysis,
geometry, and topology.
Articles in this volume cover topics including knot the-
ory; the topology of three and four-dimensional manifolds;
instanton, monopole, and Heegaard Floer homologies;
Khovanov homology; and pseudoholomorphic curve
theory.
Proceedings of Symposia in Pure Mathematics, Volume
109
August 2024, 284 pages, Softcover, ISBN: 978-1-4704-
7087-6, LC 2024004662, 2020 Mathematics Subject Clas-
sification: 53D45, 57K10, 57K18, 57K33, 57K41, 57R58,
List US$139, AMS members US$111.20, MAA members
US$125.10, Order code PSPUM/109
bookstore.ams.org/pspum-109
Mathematical Physics
Lattice Models and
Conformal Field Theory
Franck Gabriel, Université Lyon
1, France, Clément Hongler,
EPFL, Lausanne, Switzerland, and
Francesco Spadaro, Zürich, Swit-
zerland
This book introduces the math-
ematical ideas connecting Statis-
tical Mechanics and Conformal
Field Theory (CFT). Building
advanced structures on top of
more elementary ones, the authors map out a well-posed
road from simple lattice models to CFTs.
Structured in two parts, the book begins by exploring
several two-dimensional lattice models, their phase transi-
tions, and their conjectural connection with CFT. Through
these lattice models and their local fields, the fundamental
ideas and results of two-dimensional CFTs emerge, with a
special emphasis on the Unitary Minimal Models of CFT.
Delving into the delicate ideas that lead to the classification
of these CFTs, the authors discuss the assumptions on the
lattice models whose scaling limits are described by CFTs.
This produces a probabilistic rather than an axiomatic or
algebraic definition of CFTs.
Suitable for graduate students and researchers in mathe-
matics and physics, Lattice Models and Conformal Field Theory
introduces the ideas at the core of Statistical Field Theory.
Assuming only undergraduate probability and complex
analysis, the authors carefully motivate every argument and
assumption made. Concrete examples and exercises allow
readers to check their progress throughout.
32
FRANCK GABRIEL
FRANCK GABRIEL
CL
C L
É
É
MENT HONGLER
MENT HONGLER
FRANCESCO SPADARO
FRANCESCO SPADARO
Lattice Models and
Conformal Field Theory
A Glimpse
into Geometric
Representation Theory
Mahir Bilen Can
Jörg Feldvoss
Editors
ONTEMPORARY
ATHEMATICS
C
M
804
NEW BOOKS
september 2024 Notices of the AmericAN mAthemAticAl society 1121
Contemporary Mathematics, Volume 804
September 2024, 203 pages, Softcover, ISBN: 978-1-4704-
7090-6, 2020 Mathematics Subject Classification: 14C17,
14M15, 14P20, 58A50, 17B56, 20G10, 05E05, 33D52,
14D24, 20G25, List US$135, AMS members US$108, MAA
members US$121.50, Order code CONM/804
bookstore.ams.org/conm-804
New in Memoirs
of the AMS
Algebra and Algebraic
Geometry
Reconstructing Orbit Closures from their
Boundaries
Paul Apisa, University of Wisconsin–Madison, Wisconsin, and
Alex Wright, University of Michigan, Ann Arbor, Michigan
Memoirs of the American Mathematical Society, Volume
298, Number 1487
August 2024, 141 pages, Softcover, ISBN: 978-1-4704-6911-
5, 2020 Mathematics Subject Classification: 32G15, 37D40,
14H15, List US$85, AMS members US$68, MAA members
US$76.50, Order code MEMO/298/1487
bookstore.ams.org/memo-298-1487
Modular Representation Theory and
Commutative Banach Algebras
David J. Benson, University of Aberdeen, Scotland, United
Kingdom
This item will also be of interest to those working in analysis.
Memoirs of the American Mathematical Society, Volume
298, Number 1488
August 2024, 118 pages, Softcover, ISBN: 978-1-4704-7029-
6, 2020 Mathematics Subject Classification: 20C20; 46J99,
16T05, List US$85, AMS members US$68, MAA members
US$76.50, Order code MEMO/298/1488
bookstore.ams.org/memo-298-1488
Reflexive Modules on Normal Gorenstein Stein
Surfaces, Their Deformations and Moduli
Javier Fernández de Bobadilla, Basque Foundation for Sci-
ence, Bilbao, Basque Country, Spain, and Basque Center for Ap-
plied Mathematics, Bilbao, Basque Country, Spain, and Agustín
Romano-Velázquez, Universidad Nacional Autónoma de
México, Cuernavaca, Morelos, México
Memoirs of the American Mathematical Society, Volume
298, Number 1493
August 2024, 94 pages, Softcover, ISBN: 978-1-4704-7053-
1, 2020 Mathematics Subject Classification: 13C14, 13H10,
14E16, 32S25, List US$85, AMS members US$68, MAA
members US$76.50, Order code MEMO/298/1493
bookstore.ams.org/memo-298-1493
Homotopy in Exact Categories
Jack Kelly, Lincoln College, Oxford University, United Kingdom
Memoirs of the American Mathematical Society, Volume
298, Number 1490
August 2024, 160 pages, Softcover, ISBN: 978-1-4704-
7041-8, 2020 Mathematics Subject Classification: 18G35,
18N40, 18N70; 12J05, 18G80, 18M05, List US$85, AMS
members US$68, MAA members US$76.50, Order code
MEMO/298/1490
bookstore.ams.org/memo-298-1490
Analysis
Multi-scale Sparse Domination
David Beltran, Universitat de Valencia, Burjassot, Spain, Joris
Roos, University of Massachusetts Lowell, Massachusetts, and
Andreas Seeger, University of Wisconsin–Madison, Wisconsin
Memoirs of the American Mathematical Society, Volume
298, Number 1491
August 2024, 104 pages, Softcover, ISBN: 978-1-4704-7042-
5, 2020 Mathematics Subject Classification: 42B15, 42B20,
42B25, List US$85, AMS members US$68, MAA members
US$76.50, Order code MEMO/298/1491
bookstore.ams.org/memo-298-1491
Differential Equations
Asymptotic Completeness for a Scalar
Quasilinear Wave Equation Satisfying the
Weak Null Condition
Dongxiao Yu, University of California, Berkeley, CA
Memoirs of the American Mathematical Society, Volume
298, Number 1492
August 2024, 136 pages, Softcover, ISBN: 978-1-4704-7048-
7, 2020 Mathematics Subject Classification: 35L70, List
US$85, AMS members US$68, MAA members US$76.50,
Order code MEMO/298/1492
bookstore.ams.org/memo-298-1492
NEW BOOKS
1122 Notices of the AmericAN mAthemAticAl society Volume 71, Number 8
also reflect on the gender publication gap in mathematics
and focus on one of the central pillars of zbMATH Open:
the community of reviewers.
A publication of the European Mathematical Society (EMS). Distributed
within the Americas by the American Mathematical Society.
June 2024, 110 pages, Softcover, ISBN: 978-3-98547-073-0,
2020 Mathematics Subject Classification: 01A74; 01–06,
01A60, 01A61, 01A65, List US$35, AMS members US$28,
Order code EMSZBMATH
bookstore.ams.org/emszbmath
Geometry and Topology
Configuration Spaces of
Manifolds with Boundary
Ricardo Campos, Institut de
Mathématiques de Toulouse,
Universiteé de Toulouse, CNRS,
France, Najib Idrissi, Université
de Paris, France, Pascal Lam-
brechts, Université catholique de
Louvain, Louvain-la- Neuve, Bel-
gium, and Thomas Willwacher,
ETH Zúrich, Ramistrasse, Swit-
zerland
The authors study ordered configuration spaces of compact
manifolds with boundary. They show that for a large class
of such manifolds, the real homotopy type of the configura-
tion spaces only depends on the real homotopy type of the
pair consisting of the manifold and its boundary. Moreover,
they describe explicit real models of these configuration
spaces using three different approaches. They do this by
adapting previous constructions for configuration spaces
of closed manifolds which relied on Kontsevich’s proof of
the formality of the little disks operads.
The authors also prove that our models are compatible
with the richer structure of configuration spaces, respec-
tively a module over the Swiss-Cheese operad, a module
over the associative algebra of configurations in a collar
around the boundary of the manifold, and a module over
the little disks operad.
This item will also be of interest to those working in algebra and
algebraic geometry.
A publication of the Société Mathématique de France, Marseilles (SMF),
distributed by the AMS in the U.S., Canada, and Mexico. Orders from
other countries should be sent to the SMF. Members of the SMF receive
a 30% discount from list.
Number Theory
On p -Adic L-Functions for Hilbert Modular Forms
John Bergdall, Bryn Mawr College, PA, and David Hansen,
Max Planck Institute for Mathematics, Bonn, Germany
Memoirs of the American Mathematical Society, Volume
298, Number 1489
August 2024, 125 pages, Softcover, ISBN: 978-1-4704-
7031-9, 2020 Mathematics Subject Classification: 11F67,
11F85; 11F41, 11F03, 11F80, 11F33, List US$85, AMS
members US$68, MAA members US$76.50, Order code
MEMO/298/1489
bookstore.ams.org/memo-298-1489
New AMS-Distributed
Publications
General Interest
90 Years of zbMATH
Klaus Hulek, Leibniz Universi-
tät Hannover, Germany, Octavio
Paniagua Taboada, FIZ Karl-
sruhe, Germany, and Olaf Te-
schke, FIZ Karlsruhe, Germany,
Editors
zbMATH Open, the world’s most
comprehensive and longest-run-
ning abstracting and reviewing
service in pure and applied
mathematics, was founded by
Otto Neugebauer in 1931. It celebrated its 90th anniversary
by becoming an open access database. In December 2019,
the Joint Science Conference (Gemeinsame Wissenschafts-
konferenz) agreed that the Federal and State Governments
of Germany would support FIZ Karlsruhe in transforming
zbMATH into an open platform. In the future, zbMATH
Open will link mathematical services and platforms so as
to provide considerably more content for further research
and collaborative work in mathematics and related fields.
This book explains how zbMATH Open has reacted to
a rapidly changing digital era. Topics covered include: the
linkage of zbMATH Open with different community plat-
forms and digital maths libraries, the use of zbMATH Open
as a bibliographical tool, API solutions, current advance-
ments in author profiles, the indexing of mathematical
software packages (swMATH), and issues concerning math-
ematical formula search in zbMATH Open. The authors
NEW BOOKS
september 2024 Notices of the AmericAN mAthemAticAl society 1123
Astérisque, Number 449
June 2024, 482 pages, Softcover, ISBN: 978-2-85629-
990-6, 2020 Mathematics Subject Classification: 55R80,
18M75, 18M70, 55P62, 55P48, List US$63, AMS members
US$50.40, Order code AST/449
bookstore.ams.org/ast-449
Number Theory
Les Suites Spectrales de
Hodge-Tate
Ahmed Abbes, Laboratoire Al-
exander Grothendieck, CNRS,
IHES, Université Paris-Saclay,
Bures-sur-Yevette, France, and Mi-
chel Gros, Université de Rennes,
CNRS, France
This book presents two import-
ant results in p -adic Hodge the-
ory following the approach ini-
tiated by Faltings, namely (i) his
main p -adic comparison theorem, and (ii) the Hodge-Tate
spectral sequence. The authors establish for each of these
results two versions: an absolute one and a relative one.
While the absolute statements can reasonably be consid-
ered as well as understood, particularly after their extension
to rigid varieties by Scholze, Faltings’ initial approach for
the relative variants has remained much less studied.
Although the authors follow the same strategy as that
used by Faltings to establish his main p -adic comparison
theorem, part of their proofs is based on new results.
The relative Hodge-Tate spectral sequence is new in this
approach.
This item will also be of interest to those working in algebra and
algebraic geometry.
A publication of the Société Mathématique de France, Marseilles (SMF),
distributed by the AMS in the U.S., Canada, and Mexico. Orders from
other countries should be sent to the SMF. Members of the SMF receive
a 30% discount from list.
Astérisque, Number 448
May 2024, 482 pages, Softcover, ISBN: 978-2-85629-988-3,
2020 Mathematics Subject Classification: 11G25, 11F80,
14F05, 14F20, 14F30, 14F35, 14G20, List US$116, AMS
members US$92.80, Order code AST/448
bookstore.ams.org/ast-448
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September 2024 NoticeS of the AmericAN mAthemAticAl Society 1125
Meetings & Conferences of the AMS
September Table of Contents
Meetings in this Issue
2024
September 14–15 San Antonio, Texas p. 1127
October 5–6 Savannah, Georgia p. 1
129
October 19–20 Albany, New York p. 1
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October 26–27 Riverside, California p. 1
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December 9–13 Auckland, New Zealand p. 1
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2025
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March 8–9 Clemson, South Carolina p. 1
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April 5–6 Hartford, Connecticut p. 1
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May 3–4 San Luis Obispo, CA p. 1135
October 18–19 St. Louis, Missouri p. 1
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December 6–7 Denver, Colorado p. 1
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April 18–19 Fargo, North Dakota p. 1
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The Meetings and Conferences section of the Notices gives
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Important Information About AMS Meetings: Potential
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Associate Secretaries of the AMS
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MEETINGS & CONFERENCES
Meetings & Conferences
of the AMS
IMPORTANT information regarding meetings programs: AMS Sectional Meeting programs do not appear in the print
version of the Notices. However, comprehensive and continually updated meeting and program information with links
to the abstract for each talk can be found on the AMS website. See https://www.ams.org/meetings.
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New: Sectional Meetings Require Registration to Submit Abstracts. In an effort to spread the cost of the sectional
meetings more equitably among all who attend and hence help keep registration fees low, starting with the 2020 fall
sectional meetings, you must be registered for a sectional meeting in order to submit an abstract for that meeting.
You will be prompted to register on the Abstracts Submission Page. In the event that your abstract is not accepted or
you have to cancel your participation in the program due to unforeseen circumstances, your registration fee will be
reimbursed.
September 2024 NoticeS of the AmericAN mAthemAticAl Society 1127
San Antonio, Texas
University of Texas, San Antonio
September 14–15, 2024
Saturday – Sunday
Meeting #1198
Central Section
Associate Secretary for the AMS: Betsy Stovall
Program first available on AMS website: To be announced
Issue of Abstracts: Volume 45, Issue 4
Deadlines
For organizers: Expired
For abstracts: Expired
The scientific information listed below may be dated. For the latest information, see https://www.ams.org/amsmtgs
/sectional.html.
Invited Addresses
James A M Alvarez, The University of Texas at Arlington, Leveraging Research on Mathematics Teaching and Learning to
Reimagine Pathways to Mathematics.
Jason R Schweinsberg, University of California San Diego, Using coalescent theory to analyze genetic data from growing
tumors.
Anne Shiu, Texas A&M University, Dynamics of Biochemical Reaction Networks.
Special Sessions
If you are volunteering to speak in a Special Session, you should send your abstract as early as possible via the abstract submission
form found at https://www.ams.org/cgi-bin/abstracts/abstract.pl.
Additive Number Theory and Modular Forms I, Debanjana Kundu, University of Texas - Rio Grande Valley, and Brandt
Kronholm, University of Texas Rio Grande Valley.
MEETINGS & CONFERENCES
1128 NoticeS of the AmericAN mAthemAticAl Society Volume 71, Number 8
Advances in Coding Theory and Cryptography I, Henry Chimal-Dzul, University of Notre Dame, and Jingbo Liu, Texas
A&M University-San Antonio.
Advances in Differential Equations: Theory, Methods, and Applications I, Faranak Rabiei, Texas A & M University Kingsville,
Aden Omar Ahmed, Texas A&M University-Kingsville, and Dongwook Kim, Texas A & M University Kingsville.
Advances in Mathematical and Numerical Analysis of Partial Differential Equations for Application-Oriented Computations I,
Bruce A Wade, University of Louisiana at Lafayette, Qin Sheng, Baylor University, Abdul Q.M. Khaliq, Middle Tennes-
see State University, JaEun Ku, Oklahoma State University, and Xiang-Sheng Wang and Yangwen Zhang, University of
Louisiana at Lafayette.
Applications of Algebraic Geometry I, Frank Sottile, Texas A&M University, Alperen Ergur, University of Texas at San
Antonio, and Anne Shiu, Texas A&M University.
Applications of model theory in analysis, topology and set theory I, Eduardo Dueñez and Jose N Iovino, The University of
Texas at San Antonio.
Applications of Probability in Biology I, Jason R Schweinsberg, University of California San Diego.
A Showcase of Algebraic Geometry at Undergraduate Institutions I, David Swinarski, Fordham University, Julie Rana,
Lawrence University, and Han-Bom Moon, Fordham University.
Commutative algebra and connections to combinatorics I, Michael Robert DiPasquale, Louiza Fouli, and Arvind Kumar,
New Mexico State University.
Differential Geometry, Alvaro Pampano, Texas Tech University, Bogdan D. Suceava, California State University Fullerton,
and Magdalena Daniela Toda, Texas Tech University and NSF.
Dynamical systems: Statistical properties, spectral theory, and fractal geometry I, Mrinal Kanti Roychowdhury, School of
Mathematical and Statistical Sciences, University of Texas Rio Grande Valley, and William R Ott, University of Houston.
Enumerative Combinatorics I, Brian K. Miceli, Trinity University, and Lara Pudwell, Valparaiso University.
Geometric Group Theory and Low-Dimensional Topology I, George Domat and Khanh Le, Rice University, Jing Tao, Uni-
versity of Oklahoma, and Christopher Jay Leininger, Rice University.
Graph Theory I, Youngho Yoo and Chun-Hung Liu, Texas A&M University.
Harmonic Analysis, Geometric Measure Theory and PDE I, Dorina I. Mitrea and Marius Mitrea, Baylor University.
Homological and combinatorial methods in noncommutative algebra I, Amrei Oswald and Be”eri Greenfeld, University of
Washington.
Homological Commutative Algebra I, Luigi Ferraro, University of Texas Rio Grande Valley, and Alexis Hardesty, Texas
Woman’s University.
Inquiry Oriented Learning in the Mathematics Classroom I, Carolyn Luna, University of Texas At San Antonio, and Jennifer
Austin, University of Texas at Austin.
L-functions and Automorphic Forms I, Lea Beneish, University of North Texas, and Melissa Emory, Oklahoma State
University.
Link invariants and surfaces in 4-manifolds I, Michael Willis and Sherry Gong, Texas A&M University.
Machine Learning, Data Science and Related Fields I, Hansapani Rodrigo, The University of Texas Rio Grande Valley, and
Lakshmi Roychowdhury and Mrinal Kanti Roychowdhury, University of Texas Rio Grande Valley.
Mathematical Modeling at the Interface of Ecology, Epidemiology, and Human Behavior I, Tamer Oraby, University of Texas
- Rio Grande Valley, Lale Asik, University of the Incarnate Word, Ummugul Bulut, ubulut@uiwtx.edu, and Md Rafiul
Islam, University of the Incarnate Word.
Mathematical Physics and Numerical Methods I, Vu Hoang and Jose Morales, University of Texas at San Antonio.
Mathematics of Infectious Disease Emergence, Spread, and Control I, Zhuolin Qu, University of Texas at San Antonio, and
Michael Andrew Robert, Virginia Tech.
Mathematics: The gateway to Social Justice I, Juan B. Gutiérrez, University of Texas at San Antonio, James Broda, Wash-
ington and Lee University, Funda Gultepe, University of Toledo, Ron Buckmire, Occidental College, Matthew Salomone,
Bridgewater State University, Joseph Edward Hibdon, Northeastern Illinois University, and Terrance Pendleton, Drake
University.
Methods & Applications of Data-driven Manufacturing I, Kristen Lee Hallas, The University of Texas Rio Grande Valley,
and Benjamin Peters and Jianzhi Li, University of Texas Rio Grande Valley.
Modeling and analysis in biological and epidemiological systems I, Michael Lindstrom, The University of Texas Rio Grande
Valley, and Erwin Suazo and Zhaosheng Feng, University of Texas Rio Grande Valley.
Non-Archimedean, Algebraic, Tropical Geometry and applications I, Jackson S. Morrow, University of North Texas, and
Farbod Shokrieh, University of Washington.
MEETINGS & CONFERENCES
September 2024 NoticeS of the AmericAN mAthemAticAl Society 1129
Noncommutative Geometry and Analysis I, Zhizhang Xie, Guoliang Yu, Bo Zhu, and Simone Cecchini, Texas A&M
University.
Operator algebras, quantum information and computation I, Jose A Morales Escalante, University of Texas at San Antonio,
and Marius Junge, University of Illinois, Urbana and Champaign.
Periodicity in Quantum Systems I, Long Li, Rice University, Wencai Liu, Texas A&M University, and Tal Malinovitch,
Rice University.
Quasi-periodic and Disordered Systems I, Alberto Takase, Rice University, Omar Hurtado, University of California, Irvine,
and Matthew H Faust, Texas A&M University.
Recent developments on local and nonlocal PDEs I, Fernando Charro, Wayne State University, and Thialita Nascimento,
Iowa State University.
Recent studies in topics related to ion channel problems I, Mingji Zhang, New Mexico Institute of Mining and Technology,
and Saulo Orizaga, New Mexico Tech.
Recent trends in differential equations applied to biological processes I, Rachidi B. Salako, University of Nevada, Las Vegas,
and Markjoe O. Uba and Maria Amarakristi Onyido, Northern Illinois University.
Research in Post-Secondary Teaching and Learning of Mathematics I, James A M Alvarez, The University of Texas at Arling-
ton, and Paul Christian Dawkins, Texas State University.
Spectral Theory of Schrödinger Operators and Related Topics I, Christoph Fischbacher, Fritz Gesztesy, and Jon Harrison,
Baylor University.
The many scales of mathematical analysis of fluid I, Xin Liu, Texas A&M University, Quyuan Lin, Clemson University,
and Cheng Yu, University of Florida.
Theoretical and Numerical Aspects of Nonlinear Dispersive Wave Equations I, Baofeng Feng, University of Texas Rio Grande
Valley, and Geng Chen and Yannan Shen, University of Kansas.
Topics in Convexity I, Zokhrab Mustafaev, University of Houston-Clear Lake.
Contributed Paper Sessions
AMS Contributed Paper Session, Betsy Stovall, University of Wisconsin-Madison.
Savannah, Georgia
Georgia Southern University
October 5–6, 2024
Saturday – Sunday
Meeting #1199
Southeastern Section
Associate Secretary for the AMS: Brian D. Boe
Program first available on AMS website: To be announced
Issue of Abstracts: Volume 45, Issue 4
Deadlines
For organizers: Expired
For abstracts: August 13, 2024
The scientific information listed below may be dated. For the latest information, see https://www.ams.org/amsmtgs
/sectional.html.
Invited Addresses
Peter Bubenik, University of Florida, Topological Data Analysis: from geometry, algebra and combinatorics to analysis, learn-
ing and applications.
Akos Magyar, University of Georgia, To Be Announced.
Sarah Peluse, Princeton/IAS, Arithmetic Patterns in Dense Sets.
Special Sessions
If you are volunteering to speak in a Special Session, you should send your abstract as early as possible via the abstract submission
form found at https://www.ams.org/cgi-bin/abstracts/abstract.pl.
Advanced Topics in Graph Theory and Combinatorics. (Code: SS 4A), Songling Shan, Auburn University, and Zi-Xia Song,
University of Central Florida.
Advances in applied algebraic geometry (Code: SS 15A), Kisun Lee and Michael Byrd, Clemson University.
MEETINGS & CONFERENCES
1130 NoticeS of the AmericAN mAthemAticAl Society Volume 71, Number 8
Advances in the theory of integrable partial differential equations (Code: SS 27A), Barbara Prinari, University at Buffalo,
and Zechuan Zhang, SUNY Buffalo.
Algebraic, combinatorial and geometric aspects of representation theory. (Code: SS 19A), Cornelius Pillen, University of
South Alabama, Aparna Upadhyay, University at Buffalo, SUNY, and Arik Wilbert, University of South Alabama.
Applicable Analysis of Multi-physics Partial Differential Equations Systems (Code: SS 17A), George Avalos, University of
Nebraska-Lincoln, and Justin Thomas Webster, University of Maryland, Baltimore County.
Biological Systems Modeling and Analysis: recent progress and current challenges (Code: SS 16A), Dawit Denu, Georgia
Southern University.
Commutative Algebra (Code: SS 1A), Saeed Nasseh, Tricia Muldoon Brown, and Alina C. Iacob, Georgia Southern
University.
Control, PDEs and Inverse Problems. (Code: SS 29A), Tien Khai Nguyen, North Carolina State University, Loc Hoang
Nguyen, UNC Charlotte, and Thuy T. Le, North Carolina State University.
Convexity, Probability, and Asymptotic Geometric Analysis (Code: SS 12A), Galyna Livshyts, Georgia Institute of Technology,
Steven Hoehner, Longwood University, and Stephanie Mui, Georgia Institute of Technology.
Deterministic and Stochastic PDEs: Theoretical and Numerical Analyses (Code: SS 8A), Pelin Guven Geredeli, Clemson
University, and Xiang Wan, Loyola University Chicago.
Dynamical Systems and Control Systems with Applications (Code: SS 13A), Yan Wu, Georgia Southern University, and
Liancheng Wang, Kennesaw State University.
Ergodic theory and discrete analysis. (Code: SS 23A), Neil Lyall and Tomasz Szarek, University of Georgia.
Exploring the Geometry for Teachers (GeT) Course (Code: SS 24A), Tuyin An, Georgia Southern University, and Erin
Krupa, North Carolina State University.
Extremal and structural graph theory. (Code: SS 9A), Ruth Luo, University of South Carolina, and Zhiyu Wang, Georgia
Institute of Technology.
Extremal Problems of Approximation Theory and Harmonic Analysis (Code: SS 35A), Yuliya Babenko, Kennesaw State
University, and Scott Kersey, Georgia Southern University.
Fluids, Waves, and Free Boundaries. (Code: SS 2A), David M. Ambrose, Drexel University, and Michael Siegel, New
Jersey Institute of Technology.
Game Theories in Network Security (Code: SS 32A), Zheni Utic, Georgia Southern University.
Geometric Maximal Operators and Related Topics. (Code: SS 3A), Paul Hagelstein, Baylor University, and Alex Stokolos,
Georgia Southern University.
Harmonic analysis, fractals, and related topics in memory of Ka-Sing Lau and Robert Strichartz (Code: SS 14A), Sze-Man
Ngai, Georgia Southern University, and Alexander Teplyaev, University of Connecticut.
Interactions, Discrepancies, Approximations: From Energy Optimization to Dynamics (Code: SS 26A), Ryan W Matzke, Van-
derbilt University, and Ihsan Topaloglu, Virginia Commonwealth University.
Modules over Commutative Rings (Code: SS 10A), Laura Ghezzi, New York City College of Technology and The Graduate
Center-Cuny, and Joseph P Brennan, University of Central Florida.
Noncommutative Algebras, Quantum Groups, and Related Topics (Code: SS 31A), Garrett Johnson, North Carolina Cen-
tral University, Xin Tang, Math & Computer Science, Fayetteville State University, and Xingting Wang, Louisiana State
University.
Nonlinear Dispersive Equations (Code: SS 34A), Iryna Petrenko, Florida International University, Justin Holmer, Brown
University, and Svetlana Roudenko, Florida International University.
Number theory and additive combinatorics (Code: SS 28A), Sarah Peluse, Princeton/IAS, and Giorgis Petridis, University
of Georgia.
Partitions and q-series (Code: SS 25A), Andrew V. Sills, Georgia Southern University, and Robert Schneider, University
of Georgia.
Poisson geometry, Diffeology and Singular Spaces. (Code: SS 21A), Yi Lin, Georgia Southern University, Jordan Watts,
Central Michigan University, and Francois Ziegler, Georgia Southern University.
Recent Advances in Contact and Symplectic Topology (Code: SS 30A), Nur Saglam, Georgia Tech, and Eduardo Fernández,
University of Georgia.
Recent advances in Molecular based Computational and Mathematical Bioscience (Code: SS 18A), Shan Zhao, University of
Alabama, and Zhan Chen, Georgia Southern University.
Recent Advances in Theory and Practice of Data Science (Code: SS 33A), Divine Wanduku and Ionut Iacob, Georgia
Southern University.
MEETINGS & CONFERENCES
September 2024 NoticeS of the AmericAN mAthemAticAl Society 1131
Recent Advances of PDEs in Modern Mathematical Physics: Theory and Applications (Code: SS 7A), Yuanzhen Shao, The
University of Alabama, and Yi Hu and Shijun Zheng, Georgia Southern University.
Recent developments in applications of complex analysis. (Code: SS 22A), Ashley Ran Zhang, Vanderbilt University, and
Burak Hatinoglu, UC Santa Cruz.
Recent Progress in Numerical Methods for PDEs (Code: SS 11A), Xuejian Li and Leo Rebholz, Clemson University.
Topics in commutative algebra and algebraic geometry (Code: SS 6A), Prashanth Sridhar and Michael Brown, Auburn
University.
Topological Data Analysis, Theory and Applications (Code: SS 5A), Peter Bubenik and Kevin P. Knudson, University of
Florida.
Trees in many contexts. (Code: SS 20A), Hua Wang, Department of Mathematical Sciences, Georgia Southern University,
and Heather Smith Blake, Davidson College.
Albany, New York
University at Albany
October 19–20, 2024
Saturday – Sunday
Meeting #1200
Eastern Section
Associate Secretary for the AMS: Steven H. Weintraub
Program first available on AMS website: To be announced
Issue of Abstracts: Volume 45, Issue 4
Deadlines
For organizers: Expired
For abstracts: August 27, 2024
The scientific information listed below may be dated. For the latest information, see https://www.ams.org/amsmtgs
/sectional.html.
Invited Addresses
Jennifer Balakrishnan, Boston University, Title to be announced.
Jose Perea, Northeastern University, Title to be announced.
Richard Rimanyi, UNC, Title to be Announced.
Special Sessions
If you are volunteering to speak in a Special Session, you should send your abstract as early as possible via the abstract submission
form found at https://www.ams.org/cgi-bin/abstracts/abstract.pl.
Agent-Based and Mean-Field Modeling for Complex Social Systems (Code: SS 1A), Daniel Brendan Cooney, University
of Illinois Urbana-Champaign, Jeungeun Park, SUNY at New Paltz, and Rebecca Hardenbrook, Dartmouth College.
Applied and Computational Topology (Code: SS 2A), Barbara Giunti and Håvard Bakke Bjerkevik, University at Albany,
SUNY, and Justin Michael Curry, University at Albany SUNY.
Data-Driven Modeling and Analysis of Complex Dynamical Systems (Code: SS 3A), Felix X.-F Ye, SUNY Albany, and Weiqi
Chu, UMass Amherst.
Ergodic Theory - In Memory of Nathaniel Friedman (1938 - 2020) (Code: SS 4A), Karin B. Reinhold, University at Albany,
SUNY, Cesar E. Silva, Williams College, and Terry Adams, University at Albany.
Explicit Methods in Arithmetic Geometry (Code: SS 5A), Manami Roy, Lafayette College, and Alexander J Barrios, Uni-
versity of St. Thomas.
Generalized Schubert Calculus and Recent Progress (Code: SS 6A), Changlong Zhong, SUNY Albany, and Richard Rimanyi,
UNC.
Geometric Group Theory (Code: SS 7A), Matt Zaremsky, University at Albany, Emily Stark, Wesleyan University, and
Daniel Studenmund, Binghamton University.
Harmonic Analysis, Theory of Function Spaces and Their Applications (Code: SS 8A), Liding Yao, Chian Yeong Chuah, and
Jan Lang, The Ohio State University.
Holomorphic Function Spaces and Operators on Them (Code: SS 9A), Kehe Zhu, University at Albany, SUNY, and Zhijian
Wu, University of Nevada, Las Vegas.
MEETINGS & CONFERENCES
1132 NoticeS of the AmericAN mAthemAticAl Society Volume 71, Number 8
Homotopy Theory and Algebraic K-Theory (Code: SS 10A), Marco Varisco, University at Albany, State University of New
York, and Brenda Johnson, Union College.
Interactions Between Lie Theory and Combinatorics of Symmetric Functions (Code: SS 11A), Hadi Salmasian, University of
Ottawa, and Siddhartha Sahi, Rutgers University, New Brunswick NJ.
Invariants of Knots, Links, and Low-dimensional Manifolds (Code: SS 12A), Adam M. Lowrance, Vassar College, Patricia
Cahn, Smith College, Moshe Cohen, State University of New York At New Paltz, and Caitlin Leverson, Bard College.
Mathematics and the Arts in Memory of Nat Friedman (Code: SS 13A), David A Reimann, Albion College, Ergun Akleman,
Texas A&M University, and Alex Feingold, Binghamton University State University of New York.
Matroids, Quivers, 𝔽1-geometry, and Connections with Algebra (Code: SS 14A), Jaiung Jun, SUNY New Paltz, Chris Ep-
polito, The University of the South, and Alexander Sistko, Manhattan College.
Multivariable Operator Theory (Code: SS 15A), Rongwei Yang, Hyun-Kyoung Kwon, Alea L Wittig, and Kate Howell,
University at Albany.
Nonlocal Analysis and Geometric Measure Theory (Code: SS 16A), Cornelia Mihaila, Saint Michael’s College, and Brian
Seguin, Loyola University Chicago.
Nonsmooth Analysis and Geometry (Code: SS 17A), Matthew Badger, University of Connecticut, Ryan Alvarado, Amherst
College, and Lisa Naples, Fairfield University, Fairfield CT USA.
Permutation Patterns (Code: SS 18A), Megan A. Martinez, Ithaca College, and Rebecca Nicole Smith, SUNY Brockport.
Probabilistic and Analytic Aspects in Convexity (Code: SS 19A), Michael Roysdon, Case Western Reserve University, Sergii
Myroshnychenko, University of the Fraser Valley, Kateryna Tatarko, University of Waterloo, Yiming Zhao, Syracuse Uni-
versity, and Elisabeth M Werner, Case Western Reserve University.
Quantum Mathematics for Computation (Code: SS 20A), Hanmeng (Harmony) Zhan, Worcester Polytechnic Institute,
and Christino Tamon, Clarkson University.
Random Processes and Probability (Code: SS 21A), Martin V. Hildebrand, University at Albany, SUNY.
Recent Advances in Harmonic Analysis (Code: SS 22A), Joshua Brough Isralowitz, University At Albany, SUNY, and
David Cruz-Uribe, University of Alabama.
Recent Advances in Vertex Operator Algebras (Code: SS 23A), Antun Milas, SUNY at Albany, and Shashank Kanade,
University of Denver.
Recent Developments in Automorphic Forms and Representation Theory (Code: SS 24A), Moshe Adrian, Queens College,
City University of New York, and Anantharam Raghuram, Fordham University.
Recent Developments in Graph Theory (Code: SS 25A), Nathan Kahl and John T. Saccoman, Seton Hall University, and
Kerry E Ojakian, Bronx Community (CUNY).
Recent Developments in Physics Informed Machine Learning for Inverse Problems (Code: SS 26A), Taufiquar Khan and Sudeb
Majee, University of North Carolina at Charlotte.
Regularity of Nonlinear Equations and Free Boundary Problems (Code: SS 27A), Maria Soria-Carro and Iñigo Urtiaga
Erneta, Rutgers University, and Daniel Restrepo, Johns Hopkins University.
Singularities in Commutative Algebra (Code: SS 28A), Josh Pollitz and Claudia Miller, Syracuse University, and Jason
Howell, University at Albany - State University of New York.
Symmetric Functions and Applications (Code: SS 29A), Olya Mandelshtam, University of Waterloo, and Rosa C. Orel-
lana, Dartmouth College.
Topics in Recreational Math and Finite Geometry (Code: SS 30A), Lauren L Rose, Bard College, Kelly Isham, Colgate
University, and Elizabeth McMahon and Gary Gordon, Lafayette College.
Contributed Paper Sessions
AMS Contributed Paper Session (Code: CP 1A), Steven H Weintraub, Lehigh University.
MEETINGS & CONFERENCES
September 2024 NoticeS of the AmericAN mAthemAticAl Society 1133
Riverside, California
University of California, Riverside
October 26–27, 2024
Saturday – Sunday
Meeting #1201
Western Section
Associate Secretary for the AMS: Michelle Ann Manes
Program first available on AMS website: To be announced
Issue of Abstracts: Volume 45, Issue 4
Deadlines
For organizers: Expired
For abstracts: September 3, 2024
The scientific information listed below may be dated. For the latest information, see https://www.ams.org/amsmtgs
/sectional.html.
Invited Addresses
Matthew D. Blair, University of New Mexico, Title to be announced.
Hannah K. Larson, UC Berkeley, Title to be announced.
Tianyi Zheng, University of California San Diego, Title to be announced.
Special Sessions
If you are volunteering to speak in a Special Session, you should send your abstract as early as possible via the abstract submission
form found at https://www.ams.org/cgi-bin/abstracts/abstract.pl.
Advances in Extremal Combinatorics (Code: SS 27A), Emily Heath and Shira Zerbib, Iowa State University.
Advances in Understanding of Student Thinking in Lower Division Mathematics Courses (Code: SS 26A), Sara Lapan, Uni-
versity of California, Riverside, Jeffrey S Meyer, California State University, San Bernardino, and Rasha Issa, University
of California, Riverside.
Applied Partial Differential Equations and Inverse Problems (Code: SS 11A), Amir Moradifam, University of California at
Riverside, and Yat Tin Chow, University of California, Riverside.
Calculating Probabilities using Matrix Methods with Applications to Markovian, Gaussian or Queueing Models (Code: SS 22A),
Alan Krinik, California State Polytechnic University, and Randall J. Swift, California State Polytechnic University, Pomona.
Conformal Geometry, Einstein Metrics, and General Relativity (Code: SS 16A), Andrew K. Waldron and Jaroslaw Kopinski,
University of California, Davis.
Dynamical Systems (Code: SS 5A), Agnieszka Zelerowicz and Zhenghe Zhang, UC Riverside.
Dynamics of Solutions to Wave Equations (Code: SS 21A), Michael McNulty and Willie Wong, Michigan State University,
and Po-Ning Chen, UC Riverside.
Finite groups, their representations, and related structures (Code: SS 6A), Nariel Monteiro, University of California Santa
Cruz, Robert Boltje, University of California, Santa Cruz, and Mandi A. Schaeffer Fry, University of Denver.
Gender Equity in the Mathematical Sciences (GEMS) of Combinatorics (Code: SS 2A), Aleyah Dawkins, George Mason
University, Andrés Vindas Meléndez, University of Kentucky, and Katie Waddle, University of Michigan.
Geometric and Categorical Representation Theory (Code: SS 14A), Carl Mautner, UC Riverside, and Tom Gannon, Uni-
versity of California - Los Angeles.
Geometry and Topology of Contact and Symplectic Manifolds (Code: SS 9A), Bahar Acu, Pitzer College, Wenyuan Li, Uni-
versity of Southern California, and Hyunki Min, UCLA.
Geometry, topology and dynamics of character varieties (Code: SS 24A), Filippo Mazzoli, University of Virginia, and Brian
Collier, Unviersity of California, Riverside.
Graphical Calculus in Representation Theory and Low-Dimensional Topology (Code: SS 19A), Emily McGovern, North Car-
olina State University, and Agustina Czenky, University of Oregon.
Harmonic Analysis and Applications (Code: SS 7A), Rodolfo H. Torres, University of California, Riverside, and Arpad
Benyi, Western Washington University.
Harmonic Analysis, Partial Differential Equations, and Spectral Theory associated with Invited Address by Matthew Blair (Code:
SS 4A), Matthew D. Blair, University of New Mexico, and Xiaoqi Huang, Louisiana State University.
Logic in SoCal (Code: SS 10A), Meng-Che Ho, California State University, Northridge, Scott Cramer, California State
University, San Bernardino, Sheila Miller Edwards, Arizona State University, and Name Trang, University of North Texas.
MEETINGS & CONFERENCES
1134 NoticeS of the AmericAN mAthemAticAl Society Volume 71, Number 8
Moduli associated with Invited Address by Hannah Kerner Larson (Code: SS 3A), Hannah K. Larson, UC Berkeley, Jesse
Kass, USC, and Patricio Gallardo, UC Riverside.
Non-commutative Algebras in Representation Theory and Topology (Code: SS 20A), Peter Samuelson and Pallav Goyal,
University of California, Riverside, and Boris Tsvelikhovskiy, UC Riverside.
Non-commutative birational geometry, cluster structures and canonical bases (Code: SS 8A), Jacob Greenstein, University of
California Riverside, Vladimir Retakh, Rutgers University, and Arkady Berenstein, University of Oregon Eugene.
Probability and Mathematical Physics (Code: SS 25A), David E Weisbart and Rahul D. Rajkumar, University of California
Riverside.
Random matrices, related structures, and applications (Code: SS 18A), John Peca-Medlin, University of Arizona, and Yizhe
Zhu, University of California Irvine.
Random walks on groups and dynamics of group actions associated with Invited Address by Tianyi Zheng (Code: SS 29A),
Omer Tamuz, California Institute of Technology, Gil Goffer, University of California at San Diego, and Tianyi Zheng,
University of California San Diego.
Recent Advances in Modeling and Simulation of Complex Fluids (Code: SS 12A), Yiwei Wang and Weitao Chen, University
of California, Riverside, and Siting Liu, University of California, Los Angeles.
Several Complex Variables: New developments and trends (Code: SS 23A), Ziming Shi, University of California - Irvine,
John N Treuer, Texas A&M University, and Bun Wong, University of California, Riverside.
Structural Features in Mathematical Physics (Code: SS 17A), Adam M. Yassine, Pomona College, and Andrea Stine,
University of California, Riverside.
Surfaces, 3-manifolds and hyperbolic geometry (Code: SS 13A), Julien Paupert, Thi Hanh VO, and Puttipong Pongtana-
paisan, Arizona State University.
Topics in Algebraic Geometry (Code: SS 15A), Javier Gonzalez Anaya, Harvey Mudd College, Courtney George, University
of California, Riverside, and Jose Gonzalez, University of California at Riverside.
Topics on Geometric Analysis (Code: SS 1A), Xiaolong Li, Wichita State University, Lihan Wang, California State Uni-
versity, Long Beach, and Qi S Zhang, UC Riverside.
Topological and Geometric Methods in Combinatorics (Code: SS 28A), Zoe Wellner, Carnegie Mellon University, and Zilin
Jiang, Arizona State University.
Auckland, New Zealand
December 9–13, 2024
Monday – Friday
Associate Secretary for the AMS: Steven H. Weintraub
Program first available on AMS website: To be announced
Issue of Abstracts: To be announced
Deadlines
For organizers: Expired
For abstracts: September 30, 2024
Seattle, Washington
Seattle Convention Center and the Sheraton Grand Seattle
January 8–11, 2025
Wednesday – Saturday
Meeting #1203
Associate Secretary for the AMS: Brian D. Boe
Program first available on AMS website: To be announced
Issue of Abstracts: Volume 46, Issue 1
Deadlines
For organizers: Expired
For abstracts: September 10, 2024
MEETINGS & CONFERENCES
September 2024 NoticeS of the AmericAN mAthemAticAl Society 1135
Clemson, South Carolina
Clemson University
March 8–9, 2025
Saturday – Sunday
Meeting #1204
Southeastern Section
Associate Secretary for the AMS: Brian D. Boe, University
of Georgia
Program first available on AMS website: To be announced
Issue of Abstracts: Volume 46, Issue 2
Deadlines
For organizers: August 13, 2024
For abstracts: January 14, 2025
Lawrence, Kansas
University of Kansas
March 29–30, 2025
Saturday – Sunday
Meeting #1205
Central Section
Associate Secretary for the AMS: Betsy Stovall, University
of Wisconsin-Madison
Program first available on AMS website: To be announced
Issue of Abstracts: Volume 46, Issue 2
Deadlines
For organizers: August 27, 2024
For abstracts: February 4, 2025
Hartford, Connecticut
Hosted by University of Connecticut; taking place at the Connecticut Convention Center and Hartford
Marriott Downtown
April 5–6, 2025
Saturday – Sunday
Meeting #1206
Eastern Section
Associate Secretary for the AMS: Steven H. Weintraub
Program first available on AMS website: To be announced
Issue of Abstracts: To be announced
Deadlines
For organizers: September 17, 2024
For abstracts: February 11, 2025
San Luis Obispo, California
California Polytechnic State University, San Luis Obispo
May 3–4, 2025
Saturday – Sunday
Meeting #1207
Western Section
Associate Secretary for the AMS: Michelle Ann Manes
Program first available on AMS website: Not applicable
Issue of Abstracts: Volume 46, Issue 3
Deadlines
For organizers: October 1, 2024
For abstracts: March 5, 2025
MEETINGS & CONFERENCES
1136 NoticeS of the AmericAN mAthemAticAl Society Volume 71, Number 8
St. Louis, Missouri
Saint Louis University
October 18–19, 2025
Saturday – Sunday
Central Section
Associate Secretary for the AMS: Betsy Stovall, University
of Wisconsin-Madison
Program first available on AMS website: To be announced
Issue of Abstracts: To be announced
Deadlines
For organizers: March 18, 2025
For abstracts: August 26, 2025
Denver, Colorado
University of Denver
December 6–7, 2025
Saturday – Sunday
Western Section
Associate Secretary for the AMS: Michelle Ann Manes
Program first available on AMS website: Not applicable
Issue of Abstracts: Volume 46, Issue 4
Deadlines
For organizers: May 6, 2025
For abstracts: October 14, 2025
Washington, District of Columbia
Walter E. Washington Convention Center and Marriott Marquis Washington DC
January 4–7, 2026
Sunday – Wednesday
Associate Secretary for the AMS: Betsy Stovall
Program first available on AMS website: To be announced
Issue of Abstracts: To be announced
Deadlines
For organizers: To be announced
For abstracts: To be announced
Boise, Idaho
Boise State University
March 7–8, 2026
Saturday – Sunday
Western Section
Associate Secretary for the AMS: Michelle Ann Manes, AIM
Program first available on AMS website: To be announced
Issue of Abstracts: To be announced
Deadlines
For organizers: To be announced
For abstracts: To be announced
Fargo, North Dakota
North Dakota State University
April 18–19, 2026
Saturday – Sunday
Central Section
Associate Secretary for the AMS: Betsy Stovall, University
of Wisconsin-Madison
Program first available on AMS website: To be announced
Issue of Abstracts: To be announced
Deadlines
For organizers: To be announced
For abstracts: To be announced
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of your brain by providing you with a math challenge for every day of
the year. The solution is always the date, but the fun lies in fi guring out
how to arrive at the answer, and possibly discovering more than one
method of arriving there.
2024; 14 pages; ISBN: 978-1-4704-7800-1; List US$20; AMS members US$16;
MAA members US$18; Order code MBK/151
IT’S HERE!
Your Daily Epsilon of Math
Wall Calendar 2025
Visit bookstore.ams.org/mbk-151
