Autumn 2023

Seminar time: Thursdays 10:20-11:15am.
Mathematics Tower (MW) 154.

August 31:

September 7:

September 14:

September 21: Alex Mason (University of Washington)
Title: h-vector inequalities for weak maps of matroids
The study of matroids and their invariants has undergone remarkable developments in recent years. In particular, many long-standing conjectures concerning inequalities that are satisfied between certain invariants, such as the number of flats of a given rank, associated to a given matroid, have been resolved. We take a different perspective and consider inequalities between invariants of different matroids. By employing combinatorial and algebraic methods, we prove several results, with our main result being that the flag h-vector is nonincreasing under weak maps. The talk is intended to be accessible to graduate students.

September 28: Xiaoqin Guo (University of Cincinnati)
Title: Optimal homogenization rates in the stochastic homogenization in a balanced random environment
Abstract: In this talk we consider the stochastic homogenization of elliptic non-divergence from equations on the integer lattice and the corresponding model of random walks in random environment (RWRE) which is a martingale. We will derive the optimal rates of the homogenization for the Dirichlet problem. We will also discuss the correlation structure of the invariant measure and quantitative estimates for the quenched central limit theorem of the RWRE. Joint work with Hung V. Tran (UW-Madison).

October 5: Shu Kanazawa (Ohio State University)
Title: Central limit theorem for linear eigenvalue statistics of the adjacency matrices of random simplicial complexes
We consider the (higher-dimensional) adjacency matrix of the Linial-Meshulam complex model, which is a higher-dimensional generalization of the Erdős-Rényi graph model. Recently, Knowles and Rosenthal proved that the empirical spectral distribution of the adjacency matrix is asymptotically given by Wigner’s semicircle law in a diluted regime. In this talk, I will present a central limit theorem for the linear eigenvalue statistics for test functions of polynomial growth that is of class C2 on a closed interval. The proof is based on higher-dimensional combinatorial enumerations and concentration properties of random symmetric matrices. Furthermore, when the test function is a polynomial function, we obtain the explicit formula for the variance of the limiting Gaussian distribution. This is joint work with Khanh Duy Trinh (Waseda University).

October 12: Fall Break,  no seminar

October 19: Jiaming Xu (University of Wisconsin-Madison)
Title: Additions of Beta-ensembles and their aymptotic behaviors
Abstract: The additions of independent random matrices with certain symmetry is a classical topic, and has various connections with free probabilty.  We define the addition operation of rectangular matrices for general Beta>0 by an algebraic approach, and prove a Law of Large Numbers in high temperature regime when Beta goes to 0. In the other direction, we turn to additions of self-adjoint Beta ensembles defined and studied by Benaych-Georges, Cuenca, Gorin and Marcus, and prove an edge universality result among a class of such additions.  This talk is based on two projects, and the second one is joint with David Keating.

October 26: No seminar.

November 2: Adam Waterbury (Denison University)
Title: Large Deviations for Empirical Measures of Self-Interacting Markov Chains
Abstract: Self-interacting Markov chains arise in a range of models and applications. For example, they can be used to approximate the quasi-stationary distributions of irreducible Markov chains and to model random walks with edge or vertex reinforcement. The term self-interacting Markov chain is something of a misnomer, as such processes interact with their full path history at each time instant, and therefore are non-Markovian. Under conditions on the self-interaction mechanism, we establish a large deviation principle for the empirical measure of self-interacting chains on finite spaces. In this setting, the rate function takes a strikingly different form than the classical Donsker-Varadhan rate function associated with the empirical measure of a Markov chain; the rate function for self-interacting chains is typically non-convex and is given through a dynamical variational formula with an infinite horizon discounted objective function. This is based on joint work with Amarjit Budhiraja and Pavlos Zoubouloglou.

November 9: David Herzog (Iowa State University)
Title: Laws of the iterated logarithm for hypoelliptic diffusions at time zero.
Abstract:  We discuss the method of stochastic characteristics to help motivate laws of the iterated logarithm. This method is used to solve second-order linear boundary-valued PDEs, such as the Dirichlet or Poisson Problem in a bounded domain in R^n. If the operator L defining the equation is uniformly elliptic in a neighborhood of the domain and the boundary is smooth, then we can always use this method to find an expression for the unique classical solution, provided the data is smooth enough. However, if we drop the ellipticity assumption, say L is only hypoelliptic (i.e. the external randomness is degenerate but the associated Markov process has a smooth transition density), then the candidate expression typically makes sense, but it may no longer satisfy the equation. In order to determine when it does satisfy the equation, we establish laws of the iterated logarithm for the diffusion at time zero.  This talk covers joint work with Marco Carfagnini and Juraj Foldes.

November 15, 1:00pm — 2:00pm: Mateusz Piorkowski (KU Leuven). NOTE SPECIAL DAY AND TIME!!!
Title: Doubly periodic models of the Aztec diamond
Doubly periodic tiling models have gained considerable attention in recent years due to their deep connections with random matrices, combinatorics, algebraic geometry, orthogonal polynomials and various other fields of mathematics. In this talk I will report on recent developments in the study of the doubly periodic Aztec diamond, and present some new results obtained in a collaboration with Arno Kuijlaars concerning the arctic curve. Our methods are based on the analysis of certain matrix-valued orthogonal polynomials introduced by Duift and Kuijlaars in 2021.

November 16: Jonathan Husson (University of Michigan)
Title: Generalized empirical covariance matrices and large deviations
In many applications of random matrix theory, such as Principal Component Analysis or the study of random landscapes, the behaviour of the largest eigenvalue is of particular importance. In this talk, we will consider a  model of generalized empirical covariance matrix and we will state a large deviation principle for its largest eigenvalue. The main tool of the proof is the use of a spherical integral of rank one as a proxy for this largest eigenvalue. This makes it possible to tackle not only Gaussian entries but also so-called “sharp sub-Gaussian” entries such as Rademacher random variables. We then have a universality phenomenon – which is rather surprising in the large deviation regime – as well as an elegant representation for the rate function. This talk is based on a collaboration with Ben McKenna.

November 23: Thanksgiving, no seminar

November 30:

December 7: