Week of December 4, 2022

Mon Dec 5, 2022
12:00pm - zoom - Probability and Analysis Webinar
Ramon van Handel - (Princeton)
Universality in nonasymptotic random matrix theory

Nonasymptotic random matrix theory aims to estimate spectral statistics (such as the extreme eigenvalues) of rather general random matrix models in a quantitative fashion. Such results are often first established under Gaussian or sub-Gaussian assumptions, and much work is then devoted to extending such bounds to more general situations. In this talk I will discuss a very different perspective on such problems: under remarkably weak structural assumptions, one can show in a precise nonasymptotic manner that the behavior of random matrices is accurately captured by that of an associated Gaussian model, regardless of the behavior of the Gaussian model itself. When combined with recent developments in the understanding of Gaussian random matrices, this nonasymptotic universality principle yields a powerful "black box" tool for understanding the behavior of extremely general nonhomogeneous and non-Gaussian random matrix models. If time permits, I will discuss applications to random graphs, spiked models, sample covariance matrices, and/or free probability theory. (Based on joint work with Tatiana Brailovskaya.)

https://sites.google.com/view/paw-seminar/

4:00pm to 5:00pm - RH306 - Applied and Computational Mathematics
Qing Du - (Columbia University)
From nonlocal variational problems to nonlocal conservation laws

Nonlocal interaction has become increasingly noticeable in nature. The modeling and simulation of it presence and its impact lead to new development of mathematical theory. In this lecture, we briefly review some nonlocal models with a finite horizon of interactions, including both nonlocal variational problems and nonlocal conservation laws, motivated by applications ranging from mechanics to traffic flows. We also present examples to illustrate some new elements of the mathematical framework for the analysis and computation of nonlocal models.

Wed Dec 7, 2022
2:00pm to 3:00pm - 510R Rowland Hall - Combinatorics and Probability
Gwen McKinley - (UCSD)
Colorings of the Middle Layers of the Hamming Cube

There is a rich and growing body of literature dedicated studying typical colorings, independent sets, and more general homomorphisms in regular bipartite graphs. Much of this literature has been devoted to the Hamming cube and the discrete torus, where very strong structural and enumerative results are known. However, a number of the techniques that have been used rely heavily on the specific structure of these graphs. Here, we consider the middle two layers of the Hamming cube, which have slightly less "nice structure" than the entire Hamming cube, and ask for the typical structure of a q-coloring (where q is any constant). When q is even, we prove analogous structural and enumerative results to those known for the Hamming cube. In this talk, I will discuss some of our techniques, and future directions to generalize this work to other graphs. This project is joint with Lina Li and Jinyoung Park.

Thu Dec 8, 2022
4:00pm to 4:50pm - RH 306 - Inverse Problems
Teemu Saksala - (NC State)
Reconstruction of a manifold from travel time data

In this talk I will introduce two geometric datasets given by the distance function on a Riemannian manifold with boundary. For each of these data sets I will provide geometric conditions that are sufficient to determine the isometry class of the manifold producing the data.  This talk is based on joint works with Maarten V. de Hoop, Joonas Ilmavirta, Matti Lassas and Ella Pavlechko.

5:00pm to 5:50pm - RH 306 - Inverse Problems
Boya Liu - (NC State)
Travel time inverse problems on simple Riemannian manifolds

We provide new proofs based on the Myers--Steenrod theorem to confirm that travel time data, travel time difference data and the broken scattering relations determine a simple Riemannian metric on a disc up to the natural gauge of a boundary fixing diffeomorphism. Our method of the proof leads to a Lipschitz-type stability estimate for the first two data sets in the class of simple metrics. This is joint work with Joonas Ilmavirta and Teemu Saksala.

Fri Dec 9, 2022
2:00pm to 2:50pm - RH 510R - Inverse Problems
Li Li - (UC Irvine)
Inverse problems for fractional operators

I will talk about several uniqueness results for inverse problems. I will first review the classical Calderón problem. Then I will focus on the fractional Calderón problem and its evolutionary and nonlinear variants. The goal is to determine nonlinearities/coefficients in fractional equations from exterior partial measurements of the Dirichlet-to-Neumann map.