I will talk about the beta orbital corners process, a natural interpolation (on the dimension of the base field) of orbital measures from random matrix theory. The new result is the convergence in probability of the orbital corners process to a Gaussian process. Our approach is based on the study of asymptotics of the multivariate Bessel functions via explicit formulas. I will also discuss some variations of the problem by means of multivariate special functions.

Free probability provides a unifying framework for studying random multi-matrix models in the large $N$ limit. Typically, the purview of these techniques is limited to invariant or mean-field ensembles. Nevertheless, we show that random band matrices fit quite naturally into this framework. Our considerations extend to the infinitesimal level, where finer results can be stated for the $\frac{1}{N}$ correction. As an application, we consider the question of outliers for finite-rank perturbations of our model. In particular, we find outliers at the classical positions from the deformed Wigner ensemble. No background in free probability is assumed.

Let s_n(M) denote the smallest singular value of an n x n random (possibly complex) matrix M. We will discuss a novel combinatorial approach (in particular, not using either inverse Littlewood--Offord theory or net arguments) for obtaining upper bounds on the probability that s_n(M) is smaller than a given number for quite general random matrix models. Such estimates are a fundamental part of the non-asymptotic theory of random matrices and have applications to the strong circular law, numerical linear algebra etc. In several cases of interest, our approach provides stronger bounds than those obtained by Tao and Vu using inverse Littlewood-Offord theory. 

Stochastic iterative algorithms have gained recent interest for solving large-scale systems of equations, Ax=y. One such example is the Randomized Kaczmarz (RK) algorithm, which acts only on single rows of the matrix A at a time. While RK randomly selects a row, Motzkin's algorithm employs a greedy row selection; meanwhile the Sampling Kaczmarz-Motzkin (SKM) algorithm combines these two strategies. In this talk, we present an improved convergence analysis for SKM which interpolates between RK and Motzkin's algorithm.  

This talk will focus on applications of hashing. We use Leftover Hash Lemma to count linearly independent polynomials defined on a given set. From this we will derive a recent result of Abbe, Shpilka and Wigderson on linear independence of random tensors. Unfortunately, methods based on hashing only work over finite fields. A totally different approach to random tensors was found by Pierre Baldi and myself, which I will explain in detail.