Random Matrices

Abstract

Random matrix theory (RMT) has emerged as a central object of study across physics, computer science, and mathematics. Originally introduced by Wigner to describe energy levels in heavy nuclei, RMT reveals universal statistical properties of eigenvalues, dependent solely on the matrix symmetry class. Over the past three decades, it has found applications in diverse fields, from growth models and number theory to multivariate statistics and data science.

Research in RMT follows two interconnected paths: integrability, which seeks explicit distributions for specific models, and universality, which extends these properties to broader classes of probabilistic systems. Recent breakthroughs include the proof of the Anderson transition for band matrices in any dimension, universality for non-Hermitian matrices and $d$-regular graphs, and the resolution of long-standing conjectures in number theory.

Key open challenges persist, particularly in random Schrödinger operators (e.g., proving delocalization in physical dimension 3), Coulomb gases (crystallization phenomena), large deviations, and random tensors. The interplay between universality and integrability continues to drive innovation, with new connections to machine learning, extreme value theory, and quantum field theory.

This workshop brought together outstanding researchers from a variety of mathematical backgrounds whose areas of research are linked to random matrices. While there are strong links between their motivations, the techniques used by these researchers span a large swath of mathematics, ranging from purely algebraic techniques to stochastic analysis, classical probability theory, operator algebra, supersymmetry, orthogonal polynomials, etc.