# Departmental Colloquium 2023-2024

*The schedule for last year, 2022-2023, can be found here.*

## Spring 2024

__ January 9 (Tuesday), 4pm__ -

*In person, JWB 335, and online*

**Speaker:**Mathilde Gerbelli-Gautheir, McGill University

**Title:**

*Growth of cohomology in towers of manifolds: a topological application of the Langlands program*

**Abstract:**How complicated can successive manifolds get in a tower of covering spaces? Specifically, how large can the dimension of the first cohomology get? We will begin with a tour of possible behaviors for low-dimensional spaces, and then focus on arithmetic manifolds. Specifically, for towers of complex-hyperbolic manifolds, I will describe how to bound the rates of growth using known instances of Langlands functoriality.

__ January 11 (Thursday), 4pm__ -

*In person, JWB 335, and online*

**Speaker:**Nicholas Miller, University of Oklahoma

**Title:**

*Superrigidity theorems: old and new*

**Abstract:**In the 1970s, seminal work of Margulis showed that higher rank lattices have superrigid representations, which in particular implies that all such lattices are arithmetic. Since then Gromov--Piatetski-Shapiro and Deligne--Mostow have shown that a similar superrigidity theorem cannot hold in the hyperbolic setting. In this talk, we will survey the work of Margulis on superrigidity and then go on to discuss how one can prove certain superrigidity/arithmeticity theorems for hyperbolic manifolds provided that the associated manifolds satisfy additional geometric hypothesis.

__ January 16 (Tuesday), 4pm__ -

*In person, JWB 335, and online*

**Speaker:**Lucas Mason-Brown, University of Oxford

**Title:**

*Unitary representations of semisimple Lie groups and conical symplectic singularities*

**Abstract:**One of the most fundamental unsolved problems in representation theory is to classify the set of irreducible unitary representations of a semisimple Lie group. In this talk, I will define a class of such representations coming from filtered quantizations of certain graded Poisson varieties. The representations I construct are expected to form the "building blocks" of all unitary representations.

__ January 18 (Thursday), 4pm__ -

*In person, JWB 335, and online*

**Speaker:**Francisco Arana-Herrera, University of Maryland

**Title:**

*Closed geodesics on surfaces: topology, geometry, arithmetic*

**Abstract:**The study of closed geodesics on hyperbolic surfaces is a subject that shares deep connections with many areas of mathematics. We survey classic and recent results in the subject, emphasizing the relations between three different perspectives: topology, geometry, and arithmetic. In particular, we discuss recently discovered connections with mapping class groups and Teichmüller dynamics. The talk assumes no previous knowledge on the subject and is aimed at a wide mathematical audience.

__ March 14 (Thursday), 4pm__ -

*In person, JWB 335*

**Speaker:**Carolyn Abbot, Brandeis University

**Title:**

*Boundaries, boundaries, and more boundaries*

**Abstract:**It is possible to learn a lot about a group by studying how it acts on various metric spaces. One particularly interesting (and ubiquitous) class of groups are those that act nicely on negatively curved spaces, called hyperbolic groups. Since their introduction by Gromov in the 1980s, hyperbolic groups and their generalizations have played a central role in geometric group theory. One fruitful tool for studying such groups is their boundary at infinity. In this talk, I’ll discuss two generalizations of hyperbolic groups, relatively hyperbolic groups and hierarchically hyperbolic groups, and describe boundaries of each. I’ll describe various relationships between these boundaries, and explain how the hierarchically hyperbolic boundary characterizes relative hyperbolicity among hierarchically hyperbolic groups. This is joint work with Jason Behrstock and Jacob Russell.

__ March 19 (***Tuesday***), 4pm__ -

*In person, JWB 335*

**Speaker:**Ami Radunskaya, Pomona College

**Title:**

*Can a function tell us how immune cells kill?*

**Abstract:**The immune system is able to fight cancer by mustering and training an army of effector “killer” cells. Mathematical models of tumor-immune interactions must describe the proliferation, recruiting and killing rates of immune cells. Earlier work surprisingly showed that the functions describing the kill rates distinguish between two types of immune cells. The mechanisms behind these differences have been a mystery, however. In an attempt to unravel this mystery, we have created a cell-based fixed-lattice model that simulates immune cell and tumor cell interaction involving tumor recognition and two killing mechanisms. These mechanisms play a big role in the effectiveness of many cancer immunotherapies. Results from model simulations, along with theories developed by ecologists, can help to illuminate which mechanisms are at work in different conditions.

__ April 11 (Thursday), 4pm__ -

*In person, JWB 335*

**Speaker:**Jodi Mead, Boise State University

**Title:**

*Variational Data Assimilation and Regularization for Ill-posed Problems: A Common Framework*

**Abstract:**Data assimilation and inverse methods for ill-posed problems find optimal estimates of states or parameters. Methods for both combine observations with a model, which here we assume is a partial differential equation (PDE). Finding a compromise between observations and model is challenging because the actual observations often have values significantly different than the corresponding PDE estimates. Neither the observations nor the PDE exactly characterize the state because each has error, and in the case of the PDE, this can be due to unknown forcings, initial or boundary conditions. State estimates from data assimilation can vary significantly depending on specified errors in the PDE. In this work we estimate PDE errors by developing a common framework between variational data assimilation and regularization for ill-posed problems. This framework arises when weakly constrained variational data assimilation is viewed as regularizing the severely underdetermined data fitting problem in data assimilation. Within this framework we derive error estimates for data assimilation using regularization parameter selection methods including the L-curve, Generalized Cross Validation (GCV) and the Chi-squared method. Data assimilation results will be shown from a one dimensional transport model with simulated data, where the resulting state estimates can be viewed as air quality estimates.

## Fall 2023

__ October 26 (Thursday), 4pm__ -

*In person, JWB 335*

**Speaker:**Melody Chan, Brown University

**Title:**

*Combinatorial methods in the study of moduli spaces*

**Abstract:**I will discuss some combinatorial methods used to study the geometry of moduli spaces: spaces which parametrize geometric objects. These spaces play a central role in the field of algebraic geometry, the study of spaces that are patched together from zero sets of systems of polynomial equations. This talk will be accessible to all, including undergraduate math students and graduate students working in other fields.

__ November 9 (Thursday), 4pm__ -

*In person, JWB 335*

**Speaker:**Matt Menickelly, Argonne National Laboratory

**Title:**

*Exploiting Structure in (Derivative-Free) Composite Nonsmooth Optimization*

**Abstract:**We present new methods for solving a broad class of bound-constrained nonsmooth composite minimization problems. These methods are specially designed for objectives that are some algebraically specified (nonsmooth) mapping of a vector of outputs from a computationally expensive (black-box) function. We provide rigorous convergence analysis and guarantees, and test the implementations on synthetic problems, and on motivating problems from a wide range of applications relevant to the Department of Energy Office of Science. For this particular presentation, I will also provide introduction to the larger field of (model-based) black-box optimization.

__ November 16 (Thursday), 4pm__ -

*In person, JWB 335*

**Speaker:**Alejandro Maas, University of Chile

**Title:**

*Expansivity for the action of general groups*

**Abstract:**S. Schwartzman in his Ph.D. thesis in the 1950s observed a deep result in dynamical systems. It states that a homeomorphism of a compact metric space always has two points whose iteration into the future remain closer than an arbitrary small distance. The same occurs when we consider the past. A notable extension to multidimensional dynamics is due to M. Boyle and D. Lind in 1997. It claims that any Zd-action (that is, d-commuting homeomorphisms) admits a half-space and two different points whose iterations under elements of the half-space remain arbitrarly close. While Schwartzman's result ensures this asymptotic property for both possible half-spaces of the integers, Boyle and Lind's result guarantees this property for only one of these half-spaces.

In this talk we develop a geometric framework to address asymptoticity and the related property of nonexpansivity in topological dynamics when the acting group is second countable and locally compact. As an application, we show extensions of Schwartzman's theorem in this context. Also, we get new results when the acting groups is Zd: any half-space of Rd contains a vector defining a (oriented) nonexpansive direction in the sense of Boyle and Lind.

__ November 30 (Thursday), 4pm__ -

*Online*

**Speaker:**Cory Hauck, Oak Ridge National Laboratory

**Title:**

*Kinetic models of particle systems*

**Abstract:**Kinetic models are used to simulate the collective behavior of particle systems. They provide a mesoscopic description that forms a link between continuum fluid models, which are not valid in non-equilibrium settings, and molecular dynamics models, which are often too expensive for practical purposes. In this talk, I will introduce the basic formalism of kinetic theory and present some relevant applications. I will then discuss some of the challenges of solving kinetic equations numerically, and present some of the tools being developed to address these challenges.