## Spring 2022

### A Dynamical Approach to Multiplicative Number Theory

- Florian Richter, École Polytechnique Fédérale de Lausanne
- February 2, 2-3 p.m., Virtual Meeting

Abstract: One of the fundamental challenges in number theory is to understand the intricate way in which the additive and multiplicative structures in the integers intertwine. In my talk we will explore a dynamical approach to this topic. More precisely, we will introduce a new framework for treading questions in multiplicative number theory using ideas from ergodic theory. This leads to a new proof of the Prime Number Theorem and to a formulation of an extended form of Sarnak's Mobius randomness conjecture.

### Symmetry in Deep Neural Networks

- Robin Walters, Northeastern University
- February 16, 2-3 p.m., Virtual Meeting

Abstract: Deep neural networks have had transformative impacts in many fields including computer vision, computational biology, and dynamics by allowing us to learn functions directly from data. However, there remain many domains in which learning is difficult due to poor generalization or the need for enormous model sizes. We'll explore two applications of representation theory to neural networks which help address these issues. Firstly, consider the case in which the data represent an $G$-equivariant function. In this case, we can consider spaces of equivariant neural networks which may more easily be fit to the data using gradient descent. Secondly, we can consider symmetries of the function space as well. Exploiting these symmetries can lead to models with fewer free parameters, faster convergence, and more stable optimization. This is joint work with Rui Wang, Jinxi Li, Rose Yu, and Iordan Ganev.

### Convex relaxations for variational problems arising from pairwise interaction problems

- David Shirokoff, New Jersey Institute of Technology
- March 30, 2-3 p.m., Virtual Meeting

Abstract: We examine the problem of minimizing a class of nonlocal, nonconvex variational problems that arise from modeling a large number of pairwise interacting particles in the presence of thermal noise (i.e., molecular dynamics). Although finding and verifying local minima to these functionals is relatively straightforward, computing and verifying global minima is much more difficult. Global minima (ground states) are important as they characterize the structure of matter in models of self-assembly, as well as (1st order) phase transitions. We discuss how minimizing the functionals can be viewed as testing whether an associated bilinear form is co-positive. We then develop sufficient conditions for global optimality (which in some cases are provably sharp) obtained through a convex relaxation related to the cone of co-positive functionals. The advantage of the convex relaxation is that it results in a conic variational problem that is "computationally tractable", and may be solved using modern numerical techniques. The solutions provide fundamental information on the shapes of self-assembled materials in the corresponding models and phase transitions at zero temperature.

### Knot Invariants, Categorification, and Representation Theory

- Arik Wilbert, University of South Alabama
- April 13, 2-3 p.m., Virtual Meeting

Abstract: This talk provides a survey highlighting connections between representation theory, low-dimensional topology, and algebraic geometry related to my current research. I will begin by recalling basic facts about the representation theory of the Lie algebra sl2 and discuss how these relate to the construction of knot invariants such as the well-known Jones polynomial. I will then introduce certain algebraic varieties called Springer fibers and explain how they can be used to geometrically construct and classify irreducible representations of the symmetric group. These two topics turn out to be intimately related. More precisely, I will demonstrate how one can study the topology of certain Springer fibers using the combinatorics underlying the representation theory of sl2. On the other hand, I will show how Springer fibers can be used to categorify certain representations of sl2. As an application, one can upgrade the Jones polynomial to a homological invariant which distinguishes more knots than the polynomial invariant. At the end of the talk, I will discuss future research directions and explore how this picture might generalize to other Lie types beyond sl2.

### Recent Progress on Mahler’s Problem in Diophantine Approximation

- Osama Khalil, University of Utah
- April 27, 2-3 p.m., Virtual Meeting

Abstract: A classical result of Khintchine’s provides a zero-one law for the Lebesgue measure of points in Euclidean space with a given quality of approximation by rational points. In 1984, Mahler asked whether a similar law holds for Cantor’s middle thirds set. His question is part of a long history of results and conjectures aiming at showing that unlikely intersections between Diophantine sets and natural subsets of Euclidean space only occur for well-understood algebraic reasons. Some of these elementary Diophantine questions ultimately lead to difficult problems at the interface of ergodic theory and spectral theory of automorphic forms. I will describe recent joint work with Manuel Luethi leading to progress towards Mahler’s problem and how it is linked it to a notion of “sparse Hecke operators”.