TBA

TBA

In recent years, significant progress has been made in our understanding of the quantitative behavior of random matrices. One research direction of continued interest has been the estimation of the smallest singular value. A measurement of matrix’s ``invertibility’’, quantitative bounds on the smallest singular value are important for a variety of tasks including establishing a circular law for a non-Hermitian random matrix and for proving stability of numerical methods. In view of the universality phenomena of random matrices, one tries to prove these estimates for more general matrix ensembles satisfying weaker assumptions.

In the geometric approach to proving smallest singular value estimates a key ingredient is the use of a 'distance theorem', which is a small ball estimate for the distance between a random vector and subspace. In this talk we will discuss a new distance theorem and its application to proving smallest singular value estimates for inhomogeneous random rectangular matrix with independent entries. We will also discuss how the recent resolution of the Slicing Conjecture, due to Klartag, Lehec, and Guan, implies smallest singular values estimates for a number of log-concave random matrix ensembles. In some cases, independent entries are no longer necessary!

Random matrix statistics appear across diverse fields and are now recognized as exhibiting universal behavior. However, even within random matrix theory itself, the mechanism underlying the universality of local eigenvalue statistics beyond the mean-field setting remains poorly understood.

In this talk, we consider symmetric and Hermitian random matrices whose entries are independent random variables with an arbitrary variance pattern. Under a novel Short-to-Long Mixing condition—sharp in the sense that it excludes any deterministic correction at the spectral edge—we establish GOE/GUE edge universality for such inhomogeneous random matrices, which may be sparse or far from the classical mean-field regime. This condition reduces the universality problem to verifying the mixing properties of Markov chains defined by the variance profile matrix. This talk is based on joint work with Dang-Zheng Liu.

Schubert coefficients are nonnegative integers that arise in Algebraic Geometry and play a central role in Algebraic Combinatorics. It is a major open problem whether they have a combinatorial interpretation. A related open problem seeks an algorithm to determine when a Schubert coefficient is positive. In this talk we explore this problem and present an algorithmic solution. This is joint work with Igor Pak.