# Departmental Colloquium 2021-2022

*The schedule for last year, 2020-2021, can be found here.*

## Spring 2022

__ April 14 (Thursday), 4pm__ -

*In person, JWB 335*

**Speaker:**Jeff Calder, University of Minnesota

**Title:**

*Boundary estimation and Hamilton-Jacobi equations on point clouds*

**Abstract:**This talk will discuss recent work on estimating the boundary of a domain from iid samples, and on solving Hamilton-Jacobi equations on graphs. We'll present a method for boundary estimation that is scalable to large datasets in high dimensional settings, and provide statistical guarantees that the method identifies all samples within a desired distance of the boundary. We'll show that the method can be used to solve PDEs on point clouds with Dirichlet boundary conditions, which has applications to problems like data depth and machine learning. We will also present some work on robust approximations of graph distances via the solution of particular Hamilton-Jacobi equations on graphs, and discuss applications to data depth and semi-supervised learning. This is joint work with Dejan Slepcev, Sangmin Park, and Mahmood Ettehad.

__ April 7 (Thursday), 4pm__ -

*In person, JWB 335*

**Speaker:**Elizabeth Munch, Michigan State University

**Title:**

*Interleaving Distances for Reeb Graphs*

**Abstract:**Reeb graphs and other related graphical signatures have extensive use in applications, but only recently has there been intense interest in finding metrics for these objects. The idea is that graphical signatures such as Reeb graphs, merge trees, and contour trees encode data in both a space and a real valued function, and we want to build metrics that are sensitive to this information. In this talk, we will focus on a particular metric for comparing Reeb graphs known as the interleaving distance which is a categorical reformulation of the eponymous metric from persistence modules arising in Topological Data Analysis. These ideas come from viewing the data of a Reeb graph as stored in a sheaf, allowing both for more combinatorial views of the distance, as well as generalizations to other categorical frameworks which fit this model.

__ March 31 (Thursday), 4pm__ -

*In person, JWB 335*

**Speaker:**Jayadev Athreya, University of Washington

**Title:**

*Variance estimates for geometric counting problems*

**Abstract:**We'll discuss three examples (lattices, translation surfaces, hyperbolic surfaces) of computing the variance of natural counting problems for random geometric structures. All of them can be viewed as generalizations of the space of flat structures on the two-dimensional torus. We'll carefully discuss this base example, and the three different generalizations. This talk should be accessible to those with some familiarity with real and complex analysis at a first-year graduate level. In the first example, we'll discuss lattices, and some joint work with G. Margulis; in the second, translation surfaces, and joint work with Y. Cheung and H. Masur, and in the third, hyperbolic surfaces, and joint work with F. Arana-Herrera.

__ March 24 (Thursday), 4pm__ -

*In person, JWB 335*AWM Speaker Series

**Speaker:**Mary Silber, University of Chicago

**Title:**

*Vegetation Pattern Formation in Drylands*

**Abstract:**A beautiful example of spontaneous pattern formation occurs in certain dryland environments around the globe. Stripes of vegetation alternate with stripes of bare soil, with striking regularity and on a scale readily monitored via satellites. Though the vegetation is a showstopping spectacle, water, which is the limiting resource for these ecosystems, is the unseen player behind the scenes. Water concentrates into the vegetated zones, essentially reinforcing vegetation patterning, via positive feedbacks, and its dynamics play out on the short timescales of the rare storms. In contrast, the vegetation may change very little over decades. Mathematicians, physicists and theoretical ecologists have had a blast creating and analyzing models of the slow vegetation dynamics. They have suggested, based on model simulations, monitoring changes to pattern characteristics as early warning signs of ecosystem collapse, e.g., under climate change. In this talk I will tell you a little bit about the empirical data, the models, the analysis and numerical simulations. I will also advocate for a shift in focus to the water resource that is entering the system, and the need to observe and model it better on its timescale. Our work suggests some alternative perspectives on these fascinating patterns, and new questions about how they might respond to changes in precipitation characteristics, such as seasonality, storm strength and storm frequency, that will likely occur as a consequence of climate change.

__ March 17 (Thursday), 4pm__ -

*In person, JWB 335*

**Speaker:**Nicolas Garcia Trillos, University of Wisconsin Madison

**Title:**

*On multiclass adversarial training, perimeter minimization, and multimarginal optimal transport problems*

**Abstract:**Adversarial training is a framework widely used by machine learning practitioners to enforce robustness of learning models. Despite the development of several computational strategies for adversarial training and some theoretical development in the broader distributionally robust optimization literature, there are still several theoretical questions about adversarial training that remain relatively unexplored. One such question is to understand, in more precise mathematical terms, the type of regularization enforced by adversarial training in modern settings like non-parametric classification as well as classification with deep neural networks. In this talk, I will present a series of connections between adversarial training and several problems in the calculus of variations, geometric measure theory, and multimarginal optimal transport. These connections reveal a rich geometric structure of adversarial problems and conceptually all aim at answering the question: what is the regularization effect induced by adversarial training? In concrete terms, I will discuss an equivalence between a family of adversarial training problems for non-parametric classification and a family of regularized risk minimization problems where the regularizer is a nonlocal perimeter functional. I will also present a result with interesting computational implications: to solve certain adversarial training problems for classification, it is enough to solve a suitable multimarginal optimal transport problem where the number of marginals is equal to the number of classes in the original classification problem.

This talk is based on joint works with Ryan Murray, Camilo García Trillos, Leon Bungert, Jakwang Kim, Matt Jacobs, and Meyer Scetbon.

__ February 24 (Thursday), 4pm__ -

*In person, JWB 335*

**Speaker:**Robin Pemantle, University of Pennsylvania

**Title:**

*ACSV - an adventure in combinatorics through many foreign lands, such as analysis, algebraic geometry and topology*

**Abstract:**Analytic Combinatorics in Several Variables (ACSV) seeks to estimate coefficients of Laurent series F(x,y,z) where the series has some nice form. Motivation comes from many corners of combinatorics and discrete probability theory, including random walks, lattice paths, quantum walks, random tilings, and many other "exactly solvable" models, i.e., models for which closed form generating functions exist.

The general solution to the coefficient estimation problem requires a combination of techniques from harmonic analysis, computer algebra / algebraic geometry and algebraic topology, specifically stratified Morse theory and singularity theory. The title of this talk borrows from our 2019 AMS Notices article, in advance of the recent AMS-sponsored Math Research Community on ACSV.

In this talk for a general math audience, I will amplify on the motivation for the problem of coefficient extraction, show pictures of diverse phenomena, provide a map of the areas of math intersected by the problem, and give an idea of the main constructions, arguments and results.

__ January 25 (Tuesday), 11am__ -

*Zoom*

**Speaker:**Jody Reimer, University of Leeds

**Title:**

*Mathematics for a changing Arctic – from polar bears to algae*

**Abstract:**Sea ice is one of the largest and most dynamic biomes on Earth. This extreme, ephemeral environment supports organisms ranging from ice-adapted algae to charismatic polar bears. I will highlight several ways in which mathematics plays a vital role in understanding this ecosystem and anticipating future changes. For example, polar bear behavior can be understood through the lens of optimal control theory (e.g., stochastic dynamic programming). The responses of seal populations to environmental change can be modeled using matrix population models. Finally, ice algae provide the foundation for this ecosystem, and our understanding of regional algal blooms is improved by using methods from uncertainty quantification. I will discuss how my work in these areas has revealed underexplored mathematical connections between seemingly disparate mathematical approaches and propose future research questions at the interface between applied mathematics and ecological modeling.

__ January 20 (Thursday), 4:00pm__ -

*In person*

**Speaker:**Rebecca Bellovin, University of Glasgow

**Title:**

*p-adic variation of Galois representations and modular forms*

**Abstract:**Galois representations and modular forms are important objects of study in modern algebraic number theory. To study the relationship between them, it is often fruitful to study congruences between them. I will give an introduction to this theory, and I will conclude by discussing some recent results and applications.

__ January 13 (Thursday), 4:00pm__ -

*In person*

**Speaker:**Alice Nadeau, Cornell University

**Title:**

*Mathematical Challenges in Modeling Exoplanet Climates*

**Abstract:**In recent years over 4500 planets have been discovered outside of our solar system. These planets range from small rocky planets that harbor liquid water like Earth to gas giants like Jupiter as hot as 7000 degrees Fahrenheit. However, only limited data can be collected for each planet and dynamical models are needed to understand what these distant worlds might be like. Luckily, mathematicians have been modeling Earth's climate for at least 200 years and it is possible to adapt these Earth models to learn about our distant neighbors. Surprisingly, adaptations that we make to Earth models to understand exoplanet climates can help resolve long-standing debates about the Earth models themselves. In this talk I'll focus on the basic tools and intuition needed to model exoplanet climate and highlight my recent work on the likelihood of exoplanets with partial ice cover. The results of this work have the ancillary benefit of addressing the issue of the small ice cap instability for Earth. I’ll conclude with prospects for exoplanet modeling in the advent of the next generation of space telescopes.

__ January 13 (Thursday), 11:00am__ -

*In person*

**Speaker:**Wenyu Pan, University of Chicago

**Title:**

*Exponential mixing of flows for geometrically finite hyperbolic manifolds with cusps*

**Abstract:**Let \mathbb{H}^n be the hyperbolic n-space and \Gamma be a geometrically finite discrete subgroup in Isom_{+}(\mathbb{H}^n) with parabolic elements. We investigate whether the geodesic flow (resp. the frame flow) over the unit tangent bundle T^1(\Gamma\backslash \mathbb{H}^n) (resp. the frame bundle F(\Gamma\backslash \mathbb{H}^n)) mixes exponentially. This result has many applications, including spectral theory, prime geodesic theorems, orbit counting, equidistribution, etc. I will start with a survey of the past results, methods, and related problems on this topic. Along the way, I will present the joint work with Jialun Li, Pratyush Sarkar.

__ January 11 (Tuesday), 4:00pm__ -

*In person*

**Speaker:**Geordie Williamson, University of Sydney

**Title:**

*Towards the combinatorial invariance conjecture*

**Abstract:**The combinatorial invariance conjecture is a fascinating conjecture in Representation Theory. Basically it says that certain important polynomials are determined in a very non-trivial way by a directed graph. About 2 years ago, I began working on this problem with DeepMind, an AI lab based in London. (DeepMind is famous for their AlphaGo program which was the first program to defeat the best human Go players.) Our goal was to try to discover whether modern Machine Learning techniques are helpful in approaching problems in mathematics. I will outline how the Machine Learning models work, how successful they were and (by far the trickiest part) how we went about extracting new mathematics from these models. The result is a formula which sheds considerable light on the combinatorial invariance conjecture. This is joint work with the DeepMind team: Charles Blundell, Lars Buesing, Alex Davies and Petar Veličković.

__ January 10 (Monday), 4:00pm__ -

*In person*

**Speaker:**Junshan Lin, Auburn University

**Title:**

*Electromagnetic Wave in Novel Materials and Devices: Mathematical Analysis and Numerical Computation*

**Abstract:**The advance in fabrication technology allows for the manipulation of electromagnetic waves at various scales and with high efficiency by novel materials and devices. The significant applications of these materials and devices in physics and engineering have driven the need for mathematical studies to guide their experimental designs. In particular, rigorous mathematical theories need to be established for new types of wave-matter interactions and efficient computational methods need to be developed for the modeling of wave phenomena in complex media. Furthermore, the mathematical problems arising from the design of materials/devices and their realistic applications in many cases are inverse problems or optimization problems.

In this talk, I will exemplify how mathematical research contributes to these aspects by two research topics. The first topic is resonant wave scattering in nano-hole structures. I will present quantitative mathematical theories for various resonant scattering phenomena and fast numerical methods for their computational modeling. I will also introduce the mathematical framework for the application of resonances in biosensing and imaging. The second topic is on topological photonic materials. I will present mathematical theory to quantify the Dirac point and the interface mode induced from the topological index for one-dimensional photonic structures with both time-reversal and inversion symmetry. The studies for the two-dimensional photonic structures and related inverse design problems will also be discussed.

__ January 6 (Thursday), 4:00pm__ -

*In person*

**Speaker:**Jingni Xiao, Rutgers University

**Title:**

*Scattering, Nonscattering, and Inverse Scattering*

**Abstract:**Scattering studies the response of a known medium when probed with waves. In inverse scattering one seeks for information of an unknown medium from the exterior measurement of the scattered waves.

One of the interesting questions in both scattering and inverse scattering is whether the scattering response could be zero in the exterior when a medium is probed by certain waves. I plan to describe some results concerning this question in two different cases. The first is when the medium has corner(s) in its shape, for which we show that corners "almost" always scatter, with some exceptions. We also apply this result in inverse scattering. The other is when a medium has smooth boundary, in which case we prove the finiteness of nonscattering wavenumbers. This question is also related to invisibility, Schiffer's conjecture (and Pompeiu's problem), as well as free boundary problems.

__ January 6 (Thursday), 2:30pm__ -

*In person*

**Speaker:**Spencer Leslie, Duke University

**Title:**

*Periods, L-values, and stabilization*

**Abstract:**The study of period integrals of automorphic forms originates in deep questions about cohomology of locally symmetric spaces. A particularly powerful tool for studying periods is a relative trace formula, which often allows one to relate these integrals to other arithmetic objects like L-functions. In this talk, I review some of this story, discuss the modern approach to relating period integrals to L-functions, and introduce a new case of interest: unitary Friedberg-Jacquet periods. These periods are conjecturally related to central values of certain L-functions and are thus connected to deep conjectures on the cohomology of the associated locally symmetric spaces.

To prove these conjectural relationships, a promising approach is to use a relative trace formula. However, new problems (known as instability) arise in this setting that must be overcome if one is to prove this relation. I will discuss my work on a theory of endoscopy and stable relative trace formulae to overcome these problems. This gives a refinement of the relative trace formula amenable to proving this conjecture.

__ January 3 (Monday), 4:00pm__ -

*In person*

**Speaker:**Michael Lindsey, Courant Institute New York University

**Title:**

*Tools for multimodal sampling*

**Abstract:**The task of sampling from a probability distribution with known density arises almost ubiquitously in the mathematical sciences, from Bayesian inference to computational chemistry. The most generic and widely-used method for this task is Markov chain Monte Carlo (MCMC), though this method typically suffers from extremely long autocorrelation times when the target density has many modes that are separated by regions of low probability. We present several new methods for sampling that can be viewed as addressing this common problem, drawing on techniques from MCMC, graphical models, and tensor networks.

## Fall 2021 - past

__ September 23 (Thursday), 4:00pm__ -

*Virtual*

**Speaker:**David Constantine, Wesleyan University

**Title:**

*Geodesic flows on locally CAT(-1) spaces*

**Abstract:**Geodesic flows on compact, negatively curved Riemannian manifolds famously have lots of extremely nice dynamical properties. To what extent do those properties hold for geodesic flows on metric spaces that are negatively curved? In this talk I'll discuss how we can consider geodesic flows on general metric spaces, and then discuss some results on the geodesic flow of a compact, locally CAT(-1) space. It turns out that the CAT(-1) condition is sufficient for us to recover many nice properties. This is joint work with Jean-Francois Lafont and Daniel Thompson.

__ September 30 (Thursday), 4:00pm__ -

*In person*

**Speaker:**James Murphy, Tufts University

**Title:**

*Geometric and Statistical Approaches to Shallow and Deep Clustering*

**Abstract:**We propose approaches to unsupervised clustering based on data-dependent distances and dictionary learning. By considering metrics derived from data-driven graphs, robustness to noise and ambient dimensionality is achieved. Connections to geometric analysis, stochastic processes, and deep learning are emphasized. The proposed algorithms enjoy theoretical performance guarantees on flexible data models and in some cases guarantees ensuring quasilinear scaling in the number of data points. Applications to image processing and bioinformatics will be shown, demonstrating state-of-the-art empirical performance. Extensions to active learning, generative modeling, and computational geometry will be discussed.

__ October 7 (Thursday), 4:00pm__ -

*Virtual*

**Speaker:**Jennifer Balakrishnan, Boston University

**Title:**

*Rational points on curves and quadratic Chabauty*

**Abstract:**Let C be a smooth projective curve defined over the rational numbers with genus at least 2. It was conjectured by Mordell in 1922 and proved by Faltings in 1983 that C has finitely many rational points. However, Faltings' proof does not give an algorithm for finding these points, and in practice, given a curve, provably finding its set of rational points can be quite difficult.

In the case when the Mordell--Weil rank of the Jacobian of C is less than the genus, the Chabauty--Coleman method can be used to find rational points, using the construction of certain p-adic line integrals. Nevertheless, the situation in higher rank is still rather mysterious. I will describe the quadratic Chabauty method (developed in joint work with N. Dogra, S. Müller, J. Tuitman, and J. Vonk), which can apply when the rank is equal to the genus. I will also highlight some examples of interest, from the time of Diophantus to the present day.

__ October 21 (Thursday), 4:00pm__ -

*In person*

**Speaker:**Alan Reid, Rice University

**Title:**

*The geometry, topology and arithmetic of Bianchi orbifolds and their finite covers*

**Abstract:**Let d be a square-free positive integer, and let O

_{d }denote the ring of integers in Q(sqrt(−d)) . Then the groups PSL(2,O

_{d}) are known as the Bianchi groups, and are a natural generalization of the modular group PSL(2,Z). The Bianchi groups are discrete subgroups of PSL(2,C), and as such act discontinuously on H

^{3}. The quotient Bianchi orbifolds, Q

_{d }= H

^{3}/PSL(2,O

_{d}) are non-compact finite volume hyperbolic 3-orbifolds with h

_{d}(class number of O

_{d }) cusps. These represent the totality of commensurability classes of non-compact arithmetic hyperbolic 3-orbifolds. This talk will survey some recent and not so recent work on understanding the geometry and topology of these orbifolds, their covers and connections to number theory.

__ October 26 (Tuesday), 4:00pm__ -

*Virtual,*AWM Speaker Series

**Speaker:**Deanna Needell, University of California Los Angeles

**Title:**

*On the topic of topic modeling: enhancing machine learning approaches with topic features*

**Abstract:**In this talk we touch on several problems in machine learning that can benefit from the use of topic modeling. We present topicmodeling based approaches for online prediction problems, computer vision, text generation, and others. While these problems have classical machine learning approaches that work well, we show that by incorporating contextual information via topic features, we obtain enhanced and more realistic results. These classical methods include non-negative matrix and tensor factorization, generative adversarial networks, and even traditional epidemiological SIR models for prediction. In this talk we provide a brief overview of these problems and show how topic features can be used in these settings. We include supporting theoretical and experimental evidence that showcases the broad use of our approaches.

__ November 11 (Thursday), 4:00pm__ -

*In person, Distinguished Lecturer Series*

**Speaker:**Thomas Hales, University of Pittsburgh

**Title:**

*Formal Proof*

**Abstract:**A formally verified proof is a proof that has been checked by software to be free of errors. The software is based on the foundational axioms and rules of logic of mathematics. This talk will give a survey of this field of research, including some of the recent theorems that have been formally verified (such as the Kepler conjecture about sphere packings and the Continuum Hypothesis). I will describe some recent advances in the technology aimed at bringing these tools to a larger mathematical audience.

__ November 18 (Thursday), 4:00pm__ -

*Virtual*

**Speaker:**Charles Smart, Yale University

**Title:**

*Localization and unique continuation on the integer lattice*

**Abstract:**I will discuss results on localization for the Anderson-Bernoulli model. This will include my work with Ding as well as work by Li-Zhang. Both develop new unique continuation results for the Laplacian on the integer lattice.

__ December 2 (Thursday), 4:00pm__ -

*In person*

**Speaker:**Mason Porter, University of California Los Angeles

**Title:**

*Topological Data Analysis of Spatial Complex Systems*

**Abstract:**From the venation patterns of leaves to spider webs, roads in cities, social networks, and the spread of COVID-19 infections and vaccinations, the structure of many systems is influenced significantly by space. In this talk, I'll discuss the application of topological data analysis (specifically, persistent homology) to spatial systems. I'll discuss a few examples, such as voting in presidential elections, city street networks, spatiotemporal dynamics of COVID-19 infections and vaccinations, and webs that were spun by spiders under the influence of various drugs.

__ December 14 (Tuesday), 4:00pm__ -

*In person*

**Speaker:**Rachel Skipper, Ohio State University

**Title:**

*From simple groups to symmetries of surfaces*

**Abstract:**We will take a tour through some families of groups of historic importance in geometric group theory, including self-similar groups and Thompson’s groups. We will discuss the rich, continually developing theory of these groups which act as symmetries of the Cantor space, and how they can be used to understand the variety of infinite simple groups. Finally, we will discuss how these groups are serving an important role in the newly developing field of big mapping class groups which are used to describe symmetries of surfaces.

## Summer 2021 - past

__ July 6 (Tuesday), 4:00pm__ · (Online)

**Speaker:**Li-Cheng Tsai, Rutgers University

**Title:**

*When particle systems meet PDEs*

**Abstract:**Interacting particle systems are models that involve many randomly evolving agents or particles. These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems: the law of large numbers, random fluctuations, and large deviations (the study of rare events). Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems.