Invited Speakers - A Closer Look

Mohammed Abouzaid
Stanford University
Photo Credit: Stanford University

AMS Invited Address

One Hundred Years of Morse Theory

Wednesday, January 8, 2025, 2:10 p.m.-3:15 p.m.

Marston Morse addressed this society on December 28 1923, and subsequently on December 30 1924, on the subject that would later be known as Morse theory. This lecture will begin by re-casting Morse's results in modern terms, using the formalism that Witten developed in the 1980's, in terms of the existence of a chain complex, built from the critical points of a function and the gradient flow lines connecting them, and whose homology computes ordinary homology. I will then describe modern developments, initiated by Floer, for describing more delicate information about a manifold, using gradient flow trajectories, than are encoded in ordinary homology. Finally, I will address some of the motivations for the recent progress, arising in the areas of symplectic topology and Hamiltonian dynamics..

Jarod Alper
University of Washington

Current Events Bulletin Lecture III

Embracing AI and Formalization: Experimenting with Tomorrow's Mathematical Tools

Friday, January 10, 4:00 p.m.- 5:00 p.m.

You've heard the buzz: AI and formalization will revolutionize mathematics. Computers will soon surpass humans in solving olympiad-style problems. Humans will transition from proving research-level theorems on their own to guiding computers to prove their theorems. As our standards of proofs change, journals will begin requiring formalized proofs in LEAN to accompany our informal exposition.

And maybe you even believe the hype. Other than pulling out your hair fearing the day when computers take your job, what can you do? If you are like most career mathematicians, you are already overwhelmed with too many academic responsibilities reserving any precious spare work hours for research. How are you going to find the time to learn LEAN or machine learning techniques to get involved?

While I don't have the answers, I can share how I have embraced the potential of AI and formalization in mathematics. Over the last few years, I've been experimenting with LEAN with undergraduate students as part of the eXperimental Lean Lab (XLL) at the University of Washington. By summarizing the lessons we've learned leading XLL projects and by offering suggestions for how you too might get involved, I hope to inspire you to take an active role in guiding the transformation of our field.

The lecture by Jarod Alper is supported by the Bose, Datta, Mukhopadhyay and Sarkar Fund, to bring appreciation for mathematics to a broader audience.

Maria-Florina Balcan
Carnegie Mellon University

ASL Invited Address

Machine Learning Theory: New Challenges and Connections

Friday, January 10, 9:00 a.m.-10:00 a.m.

In recent years, machine learning has been applied to increasingly complex settings rendering classic learning theoretic approaches insufficient for reasoning about their performance. In this talk I will discuss new frontiers of learning theory and its interplay with other fields for analyzing learning of more complex objects (algorithms rather than simple classifiers) and learning in challenging environments with strategic agents.

At the interface between machine learning and algorithm design, I will survey recent work on learning algorithms for solving problems that are hard in classic frameworks. The classic theory of computing framework considers hand-designed algorithms and focuses on worst-case guarantees. Since such hand-designed algorithms have weak worst-case guarantees for many problems, in practice machine learning components are often incorporated in algorithm design. In this talk, I will describe recent work in our group that provides theoretical foundations for such learning augmented algorithms. I will describe both specific case studies (from data science to operations research to computational biology) and general principles applicable broadly to a variety of combinatorial algorithmic problems. I will then show how we can loop back and use these tools to learn machine learning algorithms themselves!

I will also survey work that employs learning theory lenses to relax assumptions traditionally made in game theory, including learning utilities of agents from data and leveraging contextual feature information widely used in machine learning but often ignored in game theory.

Anton Bernshteyn
UCLA

ASL Invited Address

Some Recent Progress in Descriptive Combinatorics

Friday, January 10, 10:00 a.m.-11:00 a.m.

A common theme throughout mathematics is the search for “constructive” solutions as opposed to mere existence results. For problems on $\mathbb{R}$ and other well-behaved spaces, this idea is conveniently captured by the concept of a Borel construction. For example, one can seek Borel solutions to such combinatorial problems as graph coloring, perfect matching, etc. The area studying these questions is called descriptive combinatorics. Unfortunately, many classical facts in graph theory—for example, Brooks’ theorem—turn out to be inherently “ non-constructive” in this sense. That being said, recent years have seen the emergence of geometric tools that make it possible to solve many combinatorial problems “constructively.” In this talk I will describe these tools, outline the general techniques for using them, and give a number of applications. Some of the results in this talk come from separate joint works with Abhishek Dhawan, Felix Weilacher, and Jing Yu.

Dmitriy Bilyk
University of Minnesota

AIM Alexanderson Award Lecture

Energy minimization problems in analysis and discrete geometry

Thursday, January 9, 10:50 a.m.-11:55 a.m.

We discuss problems of minimizing discrete and continuous pairwise interaction energies, i.e. expressions of the form $$ \sum_{i\neq j} K(x_i, x_j)\,\,\, \mathrm{ and } \int_\Omega \int_\Omega K(x,y) d\mu (x) d \mu(y),$$ where $x_1,\dots, x_N $ are points in a given domain $\Omega$ and $\mu$ is a probability measure on $\Omega$. Minimizing such energies can be interpreted as finding the optimal distribution of $N$ particles or of the continuous unit charge under the interaction defined by the kernel $K$. Such problems naturally arise in numerous areas of mathematics: discrete and metric geometry, analysis, potential theory, signal processing, geometric measure theory, discrepancy theory, mathematical physics etc. In many classical settings, the minimizing distribution is in some sense uniform, however, a peculiar effect can be observed for some energies: minimizers are discrete or are supported on small sets, i.e. optimal distributions tend to cluster. We shall discuss a variety of problems and energies (sums of distances, Riesz energies, $p$-frame energies) with a particular focus on this mysterious clustering phenomenon.

The talk, which is based on joint work with A. Glazyrin, R. Matzke, J. Park, and O. Vlasiuk, will be accessible to a wide mathematical audience.

Tai-Danae Bradley
SandboxAQ
Photo credit: Jon Meadows

NAM Claytor-Woodard Lecture

Structure in Language: A Category Theoretical Perspective

Thursday, January 9, 2025: 2:10 p.m.-3:15 p.m.

Mad Libs is a fun language game, played by filling in the blanks of a story with different words: nouns, verbs, adjectives, and so on. Different word choices result in different, sometimes amusing, stories that are all related by a common template. In some ways, category theory can be thought of as the Mad Libs of mathematics. Originating in the 1940s, category theory provides a framework or “template” that unites many ideas, constructions, and themes across the mathematical landscape. A category theoretical viewpoint can also extend to disciplines outside of mathematics, providing new ways to think about new (and old) problems. Mathematical structure in natural language is one example, as some ideas behind today’s large language models seem to be pleasantly compatible with existing tools in category theory. In this talk, I’ll share more about these tantalizing connections, starting with a friendly introduction to this modern branch of mathematics.

Eugenia Cheng
School of the Art Institute of Chicago

JPBM Communications Award Lecture

Saturday, January 11, 10:50 a.m.-11:55 a.m.

Semyon Dyatlov
MIT

AMS Invited Address

Uncertainty Principles in Quantum Chaos

Saturday, January 11, 2025, 3:20 p.m.-4:25 p.m.

In quantum mechanics, the pure states of a particle are the eigenfunctions of the associated quantum Hamiltonian. A commonly studied setting is that of compact Riemannian manifolds, where the pure quantum states are the eigenfunctions of the Laplace-Beltrami operator. The behavior of eigenfunctions in the high energy limit (i.e. with the eigenvalue going to infinity) is known to be connected to the dynamics of the geodesic flow on the manifold.

A fundamental problem in quantum chaos is to understand localization of eigenfunctions when the geodesic flow has chaotic behavior. This can be done using semiclassical measures, which are limiting objects on the cotangent bundle of the manifold capturing concentration of a high energy sequence of eigenfunctions in the position/frequency space. The Quantum Ergodicity theorem, going back to the 1970s--80s, states that most eigenfunctions equidistribute, that is, converge to the volume measure. The Quantum Unique Ergodicity (QUE) conjecture claims that the entire sequence of eigenfunctions equidistributes, that is the only semiclassical measure is the volume measure.

In this talk I will discuss two partial results towards the QUE conjecture. Each of these says that every semiclassical measure is delocalized: one is the statement that the measure has full support, and the other one is a lower bound on the entropy of the measure. I will highlight the role played in the proofs by uncertainty principles such as the Fractal Uncertainty Principle and the Entropic Uncertainty Principle.

Based on joint works with Jean Bourgain, Long Jin, St\'ephane Nonnenmacher, Jayadev Athreya, Nicholas Miller, and Alex Cohen.

Joan Ferrini-Mundy
University of Maine

TPSE Invited Address

Learning, Teaching, and Doing Mathematics in the Era of AI: New Challenges and Opportunities

Friday, January 10, 2025, 8:30 a.m.-9:35 a.m.

In a university, the learning, teaching, and doing of mathematics go on every day. And as the world changes and the AI era is here, students, faculty, and researchers have new opportunities and challenges while continuing to advance mathematics and mathematics learning. How can department chairs understand those opportunities and challenges in real time in order to make good decisions about the resources and tools their faculty and students need? Drawing on the perspectives of leading mathematicians and mathematics educators who are deeply engaged in these issues, I will share some thoughts on how the mathematical sciences community can shape the policies and investments in AI in higher education.

Elena Fuchs
UC Davis

Current Events Bulletin, Lecture I

Apollonian packings: the rise and fall of the local to global conjecture

Friday, January 10, 2:00 p.m.-3:00 p.m.

Nearly 20 years ago, a paper of Graham-Lagarias-Mallows-Wilks-Yan on the number theory of Apollonian circle packings sparked an interest in the number theory community, which was just developing tools to handle arithmetic problems involving so-called thin groups. At the time, these packings were the only naturally occurring example of such an arithmetic problem, and naturally number theorists sprang upon the opportunity to discover all their rich properties, thinness notwithstanding. In 2010, building upon conjectures of Graham-Lagarias-Mallows-Wilks-Yan, the speaker together with her co-author Katherine Sanden gave evidence to what they called the Local to Global Conjecture for Apollonian circle packings, stating that in any integral packing, any large enough integer that satisfied certain congruence conditions modulo 24 must appear as a curvature in the packing. For 13 years, most everyone believed this conjecture to be true. In this talk, we will explore the history of this conjecture, and its fascinating downfall after Haag-Kertzer-Rickards-Stange proved that, in fact, infinitely many integral Apollonian packings fail to abide by the local to global principle, and come with extra obstructions from quadratic and quartic reciprocity.

Wilfrid Gangbo
UCLA

CRM-PIMS-AARMS Invited Address

Viscosity Solutions In Non-Commutative Variables

Friday, January 10, 2025, 9:40 a.m.-10:45 a.m.

Motivated by parallels between mean eld games and random matrix theory, we develop stochastic optimal control problems and viscosity solutions to Hamilton-Jacobi equa- tions in the setting of non-commutative variables. Rather than real vectors, the inputs to the equation are tuples of self-adjoint operators from a tracial von Neumann algebra. The individual noise from mean eld games is replaced by a free semicircular Brownian motion, which describes the large-n limit of Brownian motion on the space of self-adjoint matrices. We introduce a classi- cal common noise from mean eld games into the non-commutative setting as well, allowing the problems to combine both classical and non-commutative randomness. Under certain convexity assumptions, we show that the value of the optimal control problems in the non-commutative setting describes the large-n limit of control problems on tuples of self-adjoint matrices. (This talk is based on works in collaboration with D. Jekel, K. Nam and A. Palmer).

Bonnie Ghosh Dastidar
American Statistical Association, Rand Corporation

ASA Invited Address

Informing Policy and Countering Misinformation

Thursday, January 9, 2025: 3:20 p.m.-4:25 p.m.

A primary purpose of statistics is to promote the discovery, understanding, and communication of facts and uncertainty through data and modeling. We live in an era of massive, and ever expanding, data with a simultaneous increase in pervasive misinformation. Statistical thinking is essential to generate the reliable information on which a healthy democracy depends. Statisticians and data scientists confront the challenges of algorithmic inequality and misuse of statistics in a world of emerging artificial intelligence technologies. Statistics play a critical role in the development and implementation of evidence-based policy, serving as the foundation for informed decision-making across various sectors. Statisticians provide policymakers with the evidence needed to evaluate the effectiveness of existing programs and predict the outcomes of proposed interventions. The strategic application of statistical methods ensures that policy decisions are grounded in robust evidence, leading to better societal outcomes.

Anne Greenbaum
University of Washington

ILAS Invited Address

Are Iterative Linear System Solvers Backward Stable?

Thursday, January 9, 2025, 9:40 a.m.-10:45 a.m.

This is a question that I often discuss with colleagues but we seldom come to a definitive conclusion. One reason is that “backward stability” is not precisely defined – it is defined in a number of different ways throughout the literature. Roughly, a backward stable algorithm for solving $Ax=b$, when implemented on a machine with unit roundoff $u$, produces a result $\hat{x}$ that satisfies $(A+\Delta A)\hat{x}=b+\Delta b$, where $\|\Delta A\|\leq \epsilon \|A\|$ and $\|\Delta b\|\leq \epsilon\| b\|$, where $\epsilon=p(n)u+O(u^2)$ and $p(n)$ is a polynomial in the problem size $n$. However, one must be careful to specify the domain of allowable matrices $A$, which may depend on $u$, as well as the details of the implementation. Additionally, for some iterative methods, there is a dependence on $k$, the number of steps for which the algorithm is run or, perhaps, a bound on the number of steps needed for a corresponding problem assuming exact arithmetic; thus we may need to replace $p(n)$ by $p(n, k)$.

In this talk, I will survey the literature on the attainable accuracy of iterative methods such as simple iteration (Jacobi, Gauss-Seidel, SOR, iterative refinement), steepest descent, and the conjugate gradient and Lanczos algorithms.

Neena Gupta
Indian Statistical Institute

AWM-AMS Noether Lecture

The Abhyankar-Sathaye Conjecture for Linear Hyperplanes

Thursday, January 9, 2025, 3:20 p.m.-4:25 p.m.

In a famous survey paper published in 1996, H. Kraft compiled a list of eight challenging problems on affine $n$-spaces, including the Zariski Cancellation Problem (ZCP), the Embedding Problem and the Linearisation Problem.

The ZCP, which asks whether the affine $n$-space is cancellative, has an affirmation solution for $n=1$ and $n=2$ and the speaker had shown that in positive characteristic, the affine $n$-space is not cancellative when $n>2$.

In characteristic zero, the Embedding Problem has an affirmative solution when $n=2$; it is the famous Epimorphism Theorem of Abhyankar-Moh and Suzuki. The Abhyankar-Sathaye Conjecture, a special case of the Embedding Problem, asserts that any embedding of the affine $n-1$-space in affine $n$-space is rectifiable for any integer $n \ge 3$. The problem is open in general. When $n=3$, any linear plane was shown to be a coordinate by A. Sathaye (in characteristic zero) and P. Russell (in general).

In recent decades some of the central problems on affine spaces crucially involved the settling of questions of the type:

(i) whether a specified linear polynomial $H$ in $k[X_1, \dots, X_n]$ is a hyperplane and

(ii) whether linear hyperplanes of a certain form are coordinates.

Problem (i) for certain specified linear polynomials defined by M. Koras and P. Russell was crucial for the Linearization Problem. Again, central to the speaker's researches around the ZCP was the settling of Problems (i) and (ii) for a generalized version of certain linear polynomials defined by T. Asanuma. Note that Problem (ii) is a special case of the Abhyankar-Sathaye Conjecture.

In this talk we shall present certain new families of linear hyperplanes where the Abhyankar-Sathaye Conjecture holds. They arose from joint works with Parnashree Ghosh and Ananya Pal.

Pamela E. Harris
University of Wisconsin-Milwaukee

AMS Lecture on Education

The Myth of the Symmetric Difference: Mathematics and Mathematics Education

Saturday, January 11, 2025, 9:40 a.m.-10:45 a.m.

At various stages of our academic careers, we have likely prioritized between our research and our teaching, dedicating more time and energy to one over the other. For example, an undergraduate student might prioritize their time as a tutor, both because they enjoy helping their peers and because tutoring can be a lucrative part time job. While a PhD student or early career faculty member might prioritize their research, as they need to prove those final results to complete their thesis or a mathematical research article in time for graduation or promotion. We understand that such a prioritization does not make what we prioritize intrinsically better. However, the cultures of our institutions assign value to our research and education work, often creating a total order on the worth of each activity. Sadly, prioritizing work in a way contrary to institutional values\hierarchy, may lead to our own value being questioned. In this talk, I want to discuss how certain prioritizations of my professional work has impacted my life and the life of those around me, and how staying true to my own values above those of any institution has helped me find my place in mathematics at the intersection of research and education.

Eric Hsu
San Francisco State University

MAA Lecture on Teaching & Learning

Precalculus and calculus: Why do we teach it and who is allowed to learn it?

Thursday, January 9, 2025, 10:50 a.m.-11:55 a.m.

We analyze the historical development of precalculus and calculus as a changing academic subject; the effectiveness of different institutional placement strategies for deciding who is allowed to learn it; how the usual measuring stick of course passing rates conceals important information; and the challenges of promoting real curricular change.

Svetlana Jitomirskaya
University of California Berkeley

AMS Colloquium Lectures: Lecture I

Quantum mechanics meets arithmetics. The ten martini problem.

Wednesday, January 8, 2025, 1:00 p.m. - 2:05 p.m.

The spectral theory of discrete quasiperiodic Schrödinger operators is a field with origins in and strong ongoing ties to physics. It features a fascinating competition between randomness (ergodicity) and order (periodicity), which is often resolved on a deep arithmetic level. This leads to an especially rich collection of phenomena, many of which we are only beginning to understand.

In particular, the Hofstadter butterfly, a plot of the band spectra of almost Mathieu operators at rational frequencies, has become a pictorial symbol of the field and has gained renewed prominence through the experimental study of moire materials. It is visually clear from this plot that for all irrational frequencies the spectrum must be a Cantor set, a statement that has been dubbed the ten martini problem. We will present the fascinating history of the topic through the lens of this problem.

The recent solution of its robust version, as well as a range of other robust results, were based on the duality approach to Avila's global theory of analytic quasiperiodic SL(2,C) cocycles, concept of dual Lyapunov exponents, and hidden partial hyperbolicity. This approach and related concepts were developed as a consequence of multiplicative generalization of the classical Jensen's formula and will be presented in the second lecture.

Svetlana Jitomirskaya
University of California Berkeley

AMS Colloquium Lectures: Lecture II

Quantitative global theory, dual Lyapunov exponents, and robust spectral results.

Thursday, January 9, 2025, 1:00 p.m. - 2:05 p.m.

Svetlana Jitomirskaya
University of California Berkeley

AMS Colloquium Lectures: Lecture III

Small denominators without KAM. Robust arithmetic Spectral transitions.

Friday, January 10, 2025, 1:00 p.m. - 2:05 p.m.

Small denominator problems appear in various areas of analysis, PDE, and dynamical systems, including spectral theory of quasiperiodic Schrödinger operators, non-linear Schrödinger equations, and non-linear wave equations. These problems have traditionally been approached by KAM-type constructions. We will discuss the new methods, originally developed for the spectral theory of quasiperiodic Schrödinger operators, that are both considerably simpler and lead to results unattainable through KAM techniques. For quasiperiodic operators, these methods have enabled precise treatment of various types of resonances and their combinations, leading to .proofs of sharp (arithmetic) spectral transitions and universality of various spectral features.

Rajesh Kasturirangan
Socratus Foundation

SIGMAA on the Philosophy of Mathematics Guest Speaker

Friday, January 10, 5:45 p.m. - 6:45 p.m. Room 609

Mathematicians, like other white collar professionals, are watching developments in AI with a mixture of horror and excitement. We don't know if an AI will discover a five page proof of FLT, but this potential phase transition in the history of mathematics also gives us the room to re-imagine some foundational practices. In particular, I argue that as design needs to be explicitly incorporated into mathematics as it starts becoming more of an engineering discipline aided by automation tools. Once we recognize the value of design in a future AI enabled mathematical world, we will acknowledge the value design brings to mathematics today, for design is how mathematics can incorporate cognition and philosophy in everyday practice.

This talk will illustrate these theoretical arguments with concrete examples of 'design thinking' applied to mathematical practice.

Kristin Lauter
Meta

AMS Erdős Lecture for Students

AI for Crypto

Wednesday, January 8, 2025, 10:50 a.m.-11:55 a.m.

AI is taking off and we could say we are living in “the AI Era”. Progress in AI today is based on mathematics and statistics under the covers of machine learning models. This talk will explain at a high level how these techniques work, and some important applications. In particular, I will explain recent work on AI4Crypto, where we train AI models to attack Post Quantum Cryptography (PQC) schemes based on lattices. Understanding the concrete security of these standardized PQC schemes is important for the future of e-commerce and internet security. So instead of saying that we are living in a “Post-Quantum” era, we should say that we are living in a “Post-AI” era!

Yann LeCun
Meta
Photo Credit: Kimberly Wang

AMS Josiah Willard Gibbs Lecture

Thursday, January 9, 2025, 5:00 p.m.-6:00 p.m.

Steven Lee
Department of Education Office of Science

SIAM Invited Address

Friday, January 10, 2025, 11:00 a.m.-12:05 p.m.

Lester Mackey
Stanford University

AMS John von Neumann Lecture

Stein’s Method, Learning, and Inference

Wednesday, January 8, 2025, 9:40 a.m.-10:45 a.m.

Stein’s method is a powerful tool from probability theory for bounding the distance between probability distributions. In this talk, I’ll describe how this tool designed to prove central limit theorems can be adapted to assess and improve the quality of practical inference procedures. Along the way, I’ll highlight applications to Markov chain Monte Carlo sampler selection, goodness-of-fit testing, generative modeling, de novo sampling, post-selection inference, distribution compression, bias correction, and nonconvex optimization, and I’ll close with opportunities for future work..

Victor Moll
Tulane University

MAA-SIAM-AMS Hrabowski-Gates-Tapia-McBay Lecture

Integral tales: Some Unexpected Connections

Friday, January 10, 2025, 9:00 a.m.-10:00 a.m.

During the process of learning Calculus one observes that there is a well-defined list of rules to compute derivatives: product, quotient and chain rules are among the first taught in every class. On the other hand, when one tries to compute integrals, the student is left with a feeling that now there is simply a collection of tricks. There is no clear reason of why one can integrate $e^{x}$ is a simple manner, but the integral of $e^{x^{2}}$ is more complicated. One learns these tricks from the instructor, by talking to older classmates or by searching for them online. At the end, there seems to be no systematic way of doing this.

It is remarkable that, in the search of producing closed-forms of definite integrals, one finds many interesting connections with apparently disjoint parts of Mathematics. Examples will include $(1)$ properties of a collection of positive integers appearing in the evaluation of a rational functions, $(2)$ a planar dynamical system connected with a variation of the arithmetic-geometric mean and $(3)$ a list of definite integrals involving the gamma function.

The lecture will be suitable for undergraduates and it will include stories about how the speaker got involved in such projects.

Emma Murphy
University of Toronto

Spectra Lavender Lecture

Symplectic topology of Stein manifolds

Thursday, January 9, 2025, 2:10 p.m.-3:15 p.m.

A Stein manifold is any complex manifold which admits a proper holomorphic embedding into C^N, for some large N. Stein manifolds are more ``flexible'' or topological when compared to compact complex manifolds -- one instance of this being the classical theorems A and B of Cartan. A culmination of this approach is Cieliebak-Eliashberg's work, which characterizes the deformation classes of Stein manifolds purely in terms of another geometry: symplectic geometry. This symplectic-first perspective on Stein manifolds allows us to import a number of powerful tools, such as Weinstein handlebody theory, contact h-principle dichotomies, and the rich algebra of wrapped Fukaya categories. This talk is intended for general audiences, and won't assume any background in symplectic/complex geometry.

Omayra Ortega

Sonoma State University

NAM Cox-Talbot Lecture

Friday, January 10, 7:45 p.m.-8:45 p.m.

Sarah Peluse
Stanford University

Current Events Bulletin Lecture IV

Finding Arithmetic Progressions in Dense Sets of Integers

Friday, January 10, 5:00 p.m.-6:00 p.m.

Determining how large a subset of the first $N$ integers can be before it is forced to contain $k$ elements forming an arithmetic progression is one of the central problems in additive combinatorics. Around 25 years ago, Gowers proved the first reasonable upper bounds in this problem when $k\geq 4$ and created the field of higher order Fourier analysis. In this talk, I will report on some very recent progress in higher order Fourier analysis and how it has led to the first ever quantitative improvement on Gowers's upper bounds when $k\geq 5$, along with other applications.

Daniel Pomerleano
University of Massachusetts, Boston

Current Events Bulletin Lecture II

Some Recent Progress in Descriptive Combinatorics

Friday, January 10, 3:00 a.m.-4:00 p.m.

In his retiring presidential address to the AMS in 1971, Oscar Zariski proposed various notions of equivalence for hypersurface singularities. He introduced the concept of "equisingularity" as follows:

The relation of equivalence which we are trying to spell out and which we shall designate by the term "equisingularity" should formalize our vague and not very intuitive idea of singularities of the same type, of the same degree of complexity. One thing is clear: it must be an equivalence relation which is much weaker than an analytical isomorphism.

Among the definitions he proposed was "topological equisingularity." The multiplicity of a holomorphic germ $f$ is defined as the degree of the lowest non-zero term in its power series expansion. Zariski asked whether topological equisingularity implies equality of multiplicity. This problem remains open and is known as the Zariski Multiplicity Conjecture.

Very recently, Javier Fernández de Bobadilla and Tomasz Pełka proved a parameterized variant of the Zariski Multiplicity Conjecture for isolated hypersurface singularities. Remarkably, their solution employs tools from symplectic geometry. Specifically, they build on earlier results by Mark McLean, which characterize the multiplicity of an isolated hypersurface singularity in terms of Floer cohomology groups associated with the symplectic monodromy of the symplectic Milnor fiber of $f$.

This talk will provide a gentle introduction to these symplectic concepts and an overview of the proof of these results. If time permits, we will also explore how other invariants of isolated singularities might conjecturally appear in these Floer cohomology groups.

Theodore A. Slaman
University of California, Berkeley

ASL Invited Address

Extending Borel's Conjecture from Measure to Dimension

Saturday, January 11, 1:00 p.m.-2:00 p.m.

Borel (1919) defined a set of real numbers $A$ to have strong measure zero if for every sequence of positive numbers $(\epsilon_i: i \in \omega)$ there is an open cover of $A$, $(U_i: i \in \omega)$, such that for each $i$, the diameter of $U_i$ is less than $\epsilon_i.$ Besicovitch (1956) showed that $A$ has strong measure zero if and only if $A$ has strong dimension zero, which means that for every gauge function $f$, $A$ is null for its associated measure $H^f$. We say that a subset of $A$ of $R^n$ has strong dimension $f$ if and only if $H^f(A)>0$ and for every gauge function $g$ of higher order $H^g(A)=0.$ Here, $g$ has higher order than $f$ when $\lim_{t \to 0^+} g(t)/f(t)=0.$

Borel conjectured that a set of strong measure zero must be countable. This conjecture naturally extends to the assertion that a set has strong dimension $f$ if and only if it is $\sigma$-finite for $H^f$. Sierpinski (1928) used the continuum hypothesis to give a counterexample to Borel's conjecture and Besicovitch (1963) did the same for its generalization. Laver (1976) showed that Borel's conjecture is relatively consistent with ZFC, the conventional axioms of set theory including the axiom of choice. We will discuss the proof that its generalization to strong dimension is also relatively consistent with ZFC.

Daniel Turetsky
Victoria University of Wellington

ASL Invited Address

True Stages for Computability and Effective Descriptive Set Theory

Saturday, January 11, 9:00 a.m.-10:00 a.m.

Priority arguments are hard. The true stages machinery was conceived as a technique for organizing complex priority constructions in computability theory, much like Ash's metatheorem. With a little modification, however, it can prove remarkably useful in descriptive set theory. Using this machinery, we can obtain nice proofs of results of Wadge, Hausdorff and Kuratowski, and Louveau, sometimes strengthening the result in the process. I will give the ideas behind the machinery and some examples of how it applies to both computability theory and descriptive set theory.

Ravi Vakil
Stanford University
Photo credit: Rod Searcey

MAA-AMS-SIAM Gerald and Judith Porter Public Lecture

The Mathematics of Doodling

Saturday, January 11, 2025, 2:15 p.m.-3:20 p.m.

Doodling is a creative and fundamentally human activity, resulting in doodles with intricate and often hidden implicit structure. We will treat doodles as an example for how mathematics is done — by starting with some doodles, we will ask ourselves some natural questions and see where they take us. They will lead us to some unexpected places, and to some sophisticated mathematics.

Kirsten Wickelgren
Duke University

AMS Invited Address

Arithmetic Aspects of Enumerative Geometry

Friday, January 10, 2025, 2:10 p.m.-3:15 p.m.

Enumerative geometry answers questions such as "How many lines meet four lines in space?" and "how many conics pass through 8 general points of the plane?" Fixed answers to such questions (here, 2 and 12) are obtained by working over an algebraically closed field like the complex numbers. Some of the solutions may be real, or integral, or defined over Q[i], but the fixed count does not see the difference. Homotopy theory on the other hand, studies continuous deformations of maps. In its modern form, it provides a framework to study shape in more algebraic and analytic contexts. This talk will introduce some interactions of homotopy theory with the arithmetic of solutions to enumerative problems in geometry. The study of such interactions was initiated in joint work with Jesse Kass and independently by Marc Hoyois and Marc Levine.

Rebecca Willett
University of Chicago

AAAS-AMS Invited Address

Mathematics in Scientific Machine Learning

Friday, January 10, 2025, 10:50-11:55 a.m.

Artificial intelligence (AI) and machine learning (ML) are poised to revolutionize the pace and nature of scientific discovery. The widespread adoption of AI in the sciences has the potential to integrate scientific inquiry with modes of hypothesis generation, data analysis, experimental design, and simulation, transforming our capacity to address scientific problems that currently seem insurmountable. The mathematical foundations of AI and ML are crucial for high-quality, reproducible, AI-enabled scientific research. However, blindly applying AI and ML poses significant risks, such as the rapid acceleration of the “reproducibility crisis” in science. In this talk, I will discuss fundamental machine learning challenges and opportunities that are particularly relevant to scientific discovery, such as emulators, generative models, and inverse problems. These problems underscore the importance of incorporating mathematical and physical models as well as numerical algorithms into ML frameworks, highlighting exciting directions for future work.

Daniela Witten
University of Washington

PME J. Sutherland Frame Lecture

Selective Inference for Real-World Problems

Thursday, January 9, 2025, 9:40 a.m.-10:45 a.m.

Statistics courses suppose an idealized version of the scientific process, in which a researcher is faced with a specific scientific question, which they then formulate as a statistical null hypothesis. The researcher then (i) collects data to test this null hypothesis, (ii) either rejects or fails to reject the null hypothesis, and then (iii) publishes the result (or not).

However, real life often proceeds quite differently. Data are often collected at massive scale before a specific scientific question is ever asked; in fact, often the goal of data collection is to explore the data in order to identify interesting scientific questions, and then to answer those questions based on the same data. Unfortunately, if the null hypothesis tested is a function of the data (rather than having been specified in advance), then the hypothesis tests, confidence intervals, and other measures of uncertainty taught in our statistics courses will not be valid.

Selective inference is a relatively new subfield of statistics aimed at bridging the aforementioned gap between the idealized version of the scientific process, and the actual way that science is conducted. In a nutshell, the philosophy behind selective inference is as follows: to test a null hypothesis generated from the data, we must account for the fact that the data led us to generate this particular null hypothesis. We will discuss a number of approaches for selective inference, ranging from the simple (sample splitting) to the more complex (conditional selective inference).

Alexander Wright
University of Michigan

AMS Maryam Mirzakhani Lecture

Curve graphs and totally geodesic subvarieties of moduli spaces of Riemann surfaces

Friday, January 10, 2025, 3:20 p.m.-4:25 p.m.

Given a surface, the associated curve graph has vertices corresponding to certain isotopy classes of curves on the surface, and edges for disjoint curves. Starting with work of Masur and Minsky in the late 1990s, curve graphs became a central tool for understanding objects in low dimensional topology and geometry. Since then, their influence has reached far beyond what might have been anticipated. Part of the talk will be an expository account of this remarkable story.

Much more recently, non-trivial examples of totally geodesic subvarieties of moduli spaces have been discovered, in work of McMullen-Mukamel-Wright and Eskin-McMullen-Mukamel-Wright. Part of the talk will be an expository account of this story and its connections to dynamics.

The talk will conclude with new joint work with Francisco Arana-Herrera showing that the geometry of totally geodesic subvarieties can be understood using curve graphs, and that this is closely intertwined with the remarkably rigid structure of these varieties witnessed by the boundary in the Deligne-Mumford compactification.

Jinhe Ye
University of Oxford

ASL Invited Address

Lang-Weil Estimate in Finite Difference Fields

Saturday, January 11, 10:00 a.m.-11:00 a.m.

A difference field is a field equipped with a given automorphism and a difference variety is the natural analogue of an algebraic varieties in this setting. Complex numbers with complex conjugation or finite fields with the Frobenius automorphism are natural examples of difference fields.

For finite fields and varieties over them, the celebrated Lang-Weil estimate gives a universal estimate of number of rational points of varieties over finite fields in terms of several notions of the complexities of the given variety. In this talk, we will discuss an analogue to Lang-Weil estimate for difference varieties in finite difference fields. The proof uses pseudofinite difference fields, where the automorphism is the nonstandard Frobenius. This is joint work with Martin Hils, Ehud Hrushovski and Tingxiang Zou.

Andy Zucker
University of Waterloo

ASL Invited Address

Topological Dynamics and Continuous Logic

Friday, January 10, 2:00 p.m.-3:00 p.m.

Motivated by the recent construction of the author of a notion of ultracoproduct of flows of topological groups and by recent work of Ben Yaacov and Goldbring which offers a formalization of unitary representations of locally compact groups in the language of continuous logic, we present some of the key ideas needed for such a formalization for flows of topological groups. A crucial addition is the need to add several sorts onto the algebra of continuous functions of the flow which serve as “external declarations” of moduli of $G$-continuity, obtaining a richer structure that we call a graded $G$-flow. Working with graded $G$-flows, the theory of ultracoproducts and weak containment dramatically simplifies, while also shedding light on why things are so much more difficult in the “ungraded” setting.

I. Ben Yaacov and I.Goldbring, Unitary representations of locally compact groups as metric structures, Notre Dame J. Formal Logic 64(2) (2023), 159--172.

A. Zucker, Ultracoproducts and weak containment for flows of topological groups, https://arxiv.org/abs/2401.08000..

Top