Departmental Colloquium
Spring 2022
Regular Day: Thursday
Regular Time: 4:00PM - 5:00PM
Regular Location: JWB 335
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January 3 |
Michael Lindsey
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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
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January 6 |
Spencer Leslie
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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.
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January 6 |
Jingni Xiao
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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.
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January 10 |
Junshan Lin
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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.
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January 11 |
Geordie Williamson
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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.
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January 13 |
Alice Nadeau
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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.
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January 13 |
Wenyu Pan
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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.
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January 20 |
Rebecca Bellovin
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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.
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January 25 |
Jody Reimer
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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.
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February 24 |
Robin Pemantle
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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.
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March 17 |
Nicolas Garcia Trillos
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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.
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March 24 |
Mary Silber
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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.
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March 31 |
Jayadev Athreya
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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.
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April 7 |
Elizabeth Munch
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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.
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April 14 |
Jeff Calder
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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.
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July 6 |
Li-Cheng Tsai
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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.
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