## Seminars and Colloquia by Series

### Erdős covering systems

Series
School of Mathematics Colloquium
Time
Thursday, April 8, 2021 - 11:00 for 1 hour (actually 50 minutes)
Location
https://us02web.zoom.us/j/87011170680?pwd=ektPOWtkN1U0TW5ETFcrVDNTL1V1QT09
Speaker
Rob MorrisNational Institute for Pure and Applied Mathematics, Rio de Janeiro, Brazil

A covering system of the integers is a finite collection of arithmetic progressions whose union is the integers. The study of these objects was initiated by Erdős in 1950, and over the following decades he asked a number of beautiful questions about them. Most famously, his so-called "minimum modulus problem" was resolved in 2015 by Hough, who proved that in every covering system with distinct moduli, the minimum modulus is at most $10^{16}$.

In this talk I will present a variant of Hough's method, which turns out to be both simpler and more powerful. In particular, I will sketch a short proof of Hough's theorem, and discuss several further applications. I will also discuss a related result, proved using a different method, about the number of minimal covering systems.

Joint work with Paul Balister, Béla Bollobás, Julian Sahasrabudhe and Marius Tiba.

### Symmetry and uniqueness via a variational approach

Series
School of Mathematics Colloquium
Time
Thursday, April 1, 2021 - 11:00 for 1 hour (actually 50 minutes)
Location
https://us02web.zoom.us/j/87011170680?pwd=ektPOWtkN1U0TW5ETFcrVDNTL1V1QT09
Speaker
Yao YaoGeorgia Institute of Technology

For some nonlocal PDEs, its steady states can be seen as critical points of an associated energy functional. Therefore, if one can construct perturbations around a function such that the energy decreases to first order along the perturbation, this function cannot be a steady state. In this talk, I will discuss how this simple variational approach has led to some recent progresses in the following equations, where the key is to carefully construct a suitable perturbation.

I will start with the aggregation-diffusion equation, which is a nonlocal PDE driven by two competing effects: nonlinear diffusion and long-range attraction. We show that all steady states are radially symmetric up to a translation (joint with Carrillo, Hittmeir and Volzone), and give some criteria on the uniqueness/non-uniqueness of steady states within the radial class (joint with Delgadino and Yan).

I will also discuss the 2D Euler equation, where we aim to understand under what condition must a stationary/uniformly-rotating solution be radially symmetric. Using a variational approach, we settle some open questions on the radial symmetry of rotating patches, and also show that any smooth stationary solution with compactly supported and nonnegative vorticity must be radial (joint with Gómez-Serrano, Park and Shi).

### Neural network and finite element functions

Series
School of Mathematics Colloquium
Time
Thursday, March 18, 2021 - 11:00 for 1 hour (actually 50 minutes)
Location
https://us02web.zoom.us/j/87011170680?pwd=ektPOWtkN1U0TW5ETFcrVDNTL1V1QT09
Speaker
Jinchao XuPennsylvania State University

Piecewise polynomials with certain global smoothness can be given by traditional finite element methods and also by neural networks with some power of ReLU as activation function. In this talk, I will present some recent results on the connections between finite element and neural network functions and comparative studies of their approximation properties and applications to numerical solution of partial differential equations of high order and/or in high dimensions.

### Linear multistep methods for learning dynamics

Series
School of Mathematics Colloquium
Time
Thursday, March 11, 2021 - 11:00 for 1 hour (actually 50 minutes)
Location
https://us02web.zoom.us/j/87011170680?pwd=ektPOWtkN1U0TW5ETFcrVDNTL1V1QT09
Speaker
Qiang DuColumbia University

Numerical integration of given dynamic systems can be viewed as a forward problem with the learning of unknown dynamics from available state observations as an inverse problem. The latter has been around in various settings such as the model reduction of multiscale processes. It has received particular attention recently in the data-driven modeling via deep/machine learning. Indeed, solving both forward and inverse problems forms the loop of informative and intelligent scientific computing. A natural question is whether a good numerical integrator for discretizing prescribed dynamics is also good for discovering unknown dynamics. This lecture presents a study in the context of Linear multistep methods (LMMs).

### Impossibility results in ergodic theory and smooth dynamical systems

Series
School of Mathematics Colloquium
Time
Thursday, February 25, 2021 - 11:00 for 1 hour (actually 50 minutes)
Location
https://us02web.zoom.us/j/87011170680?pwd=ektPOWtkN1U0TW5ETFcrVDNTL1V1QT09
Speaker
Matthew ForemanUniversity of California, Irvine

The talk considers the equivalence relations of topological conjugacy and measure isomorphism on diffeomorphisms of compact manifolds of small dimension. It is shown that neither is a Borel equivalence relation.  As a consequence, there is no inherently countable method that,  for general diffeomorphisms $S$ and $T$, determines whether $S\sim T$. It is also shown that the Time Forward/Time Backward problem for diffeomorphisms of the 2-torus  encodes most mathematical questions, such as the Riemann Hypothesis.

This work is joint with B Weiss and A Gorodetski.

### Mathematical modeling of the COVID-19 pandemic: an outsider's perspective

Series
School of Mathematics Colloquium
Time
Thursday, February 18, 2021 - 11:00 for 1 hour (actually 50 minutes)
Location
https://us02web.zoom.us/j/87011170680?pwd=ektPOWtkN1U0TW5ETFcrVDNTL1V1QT09
Speaker
Wesley PegdenCarnegie Mellon University

In this talk we will discuss epidemic modeling in the context of COVID-19.  We will review the basics of classical epidemic models, and present joint work with Maria Chikina on the use of age-targeted strategies in the context of a COVID-19-like epidemic.  We will also discuss the broader roles epidemic modeling has played over the past year, and the limitations it as presented as a primary lens through which to understand the pandemic.

### Applications of Ergodic Theory to Combinatorics and Number Theory

Series
School of Mathematics Colloquium
Time
Thursday, February 11, 2021 - 11:00 for 1 hour (actually 50 minutes)
Location
Online via Zoom
Speaker
Florian RichterNorthwestern University

This talk will focus on the multifaceted and mutually perpetuating relationship between ergodic theory, combinatorics and number theory. We will begin by discussing Furstenberg’s ergodic approach to Szemerédi’s Theorem and how it has inspired a recent solution to a long-standing sumset conjecture of Erdős. Thereafter, we will explore a new dynamical framework for treating questions in multiplicative number theory. This leads to a variant of the ergodic theorem that contains the Prime Number Theorem as a special case, and reveals an intriguing new connection between the notion of entropy in dynamical systems and the distribution of the number of prime factors of integers.

### Nielsen realization problems

Series
School of Mathematics Colloquium
Time
Friday, December 4, 2020 - 15:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Bena TshishikuBrown University

Please Note: This is the opening talk of the 2020 Tech Topology Conference http://ttc.gatech.edu

For a manifold M, the (generalized) Nielsen realization problem asks if the surjection Diff(M) → π_0 Diff(M) is split, where Diff(M) is the diffeomorphism group. When M is a surface, this question was posed by Thurston in Kirby's problem list and was addressed by Morita. I will discuss some more recent work on Nielsen realization problems with connections to flat fiber bundles, K3 surfaces, and smooth structures on hyperbolic manifolds.

### Insights on gradient-based algorithms in high-dimensional non-convex learning

Series
School of Mathematics Colloquium
Time
Thursday, November 12, 2020 - 11:00 for 1 hour (actually 50 minutes)
Location
https://us02web.zoom.us/j/89107379948
Speaker
Lenka ZdeborováEPFL

Gradient descent algorithms and their noisy variants, such as the Langevin dynamics or multi-pass SGD, are at the center of attention in machine learning. Yet their behaviour remains perplexing, in particular in the high-dimensional non-convex setting. In this talk, I will present several high-dimensional and non-convex statistical learning problems in which the performance of gradient-based algorithms can be analysed down to a constant. The common point of these settings is that the data come from a probabilistic generative model leading to problems for which, in the high-dimensional limit, statistical physics provides exact closed solutions for the performance of the gradient-based algorithms. The covered settings include the spiked mixed matrix-tensor model and the phase retrieval.

### An Introduction to Gabor Analysis

Series
School of Mathematics Colloquium
Time
Thursday, October 29, 2020 - 11:00 for 1 hour (actually 50 minutes)
Location
ONLINE at https://us02web.zoom.us/j/89107379948
Speaker
Kasso OkoudjouTufts University

In 1946, Dennis Gabor claimed that any Lebesgue square-integrable function can be written as an infinite linear combination of time and frequency shifts of the standard Gaussian.  Since then, decomposition methods for larger classes of functions or distributions in terms of various elementary building blocks have lead to an impressive body of work in harmonic analysis. For example, Gabor analysis, which originated from Gabor's claim, is concerned with both the theory and the applications of the approximation properties of sets of time and frequency shifts of a given function. It re-emerged with the advent of wavelets at the end of the last century and is now at the intersection of many fields of mathematics, applied mathematics, engineering, and science. In this talk, I will introduce the fundamentals of the theory highlighting some applications and open problems.