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.

Inspired by recent work on the quantile-randomized Kaczmarz method (qRK) for solving a linear system of equations, we propose an acceleration of the randomized Kaczmarz method using quantile information. We show that the method converges faster than the randomized Kaczmarz algorithm when the linear system is consistent. In addition, we demonstrate how this new acceleration may be used in conjunction with qRK, without additional computational complexity, to produce both a fast and robust iterative method for solving large, sparsely corrupted linear systems. Finally, we provide new error horizon bounds for qRK in the setting where the corruption may not be sparse.

SoCalDM is a friendly, day-long research conference designed for discrete mathematicians in Southern California. For more details please visit the conference website:

https://sites.google.com/view/socaldm2025/home

Here is the schedule for the event: 

9:00-9:30 Welcome coffee

9:30-10:00 Jonathan Davidson (Cal State LA), A Combinatorial Design Approach to a Multicolor Bipartite Ramsey Problem

10:10-10:40 Lenny Fukshansky (Claremont McKenna College), On a new absolute version of Siegel’s lemma

10:40-11:00 Coffee Break

11:00-11:30 Mason Shurman (UCI), Covering Random Digraphs with Hamilton Cycles

11:40-12:10 Claire Levaillant (USC), Solutions to the Diophantine equation $\sum_{i=1}^n\frac{1}{x_i}=1$ in integers of the form p^a*q^b with p and q two distinct primes. 

12:10-2:00 Lunch

2:00-2:30 Sehun Jeong (Claremont Graduate University), Integral quadratic forms and lattice angles

2:40-3:10 Justin Troyka (Cal State LA), Growth rates of permutations with given descent or peak set

3:20-3:50 Yizhe Zhu (USC), CLTs for linear spectral statistics of inhomogeneous random graphs