Random Matrices, Intrinsic Freeness, and Sharp Non-Asymptotic Inequalities

Random matrix theory has played a major role in several areas of pure and applied mathematics, as well as statistics, physics, and computer science. This lecture aims to describe the intrinsic freeness phenomenon and how it provides new easy-to-use sharp non-asymptotic bounds on the spectrum of general random matrices. We will also present a couple of illustrative applications in high dimensional statistical inference. This article accompanies a lecture that will be given by the author at the International Congress of Mathematicians in Philadelphia in the Summer of 2026.