Limits of differential operators via finite free probability

Universality is a concept in probability and mathematical physics, where certain widespread emergent behaviors of a system are observed in numerous different contextsthat is, these behaviors are universal and do not depend on the exact details of its construction. A classic example is the law of large numbers in probability, where the average of a large number of independent samples is very close to the true mean, regardless of the underlying distribution from which they are drawn. In the last two decades, a huge number of universality results have been proven in the field of random matrix theory, which lies firmly in the intersection of both probability and mathematical physics. We aim to use random matrix theory as a framework for proving universality results in areas of mathematics which initially appear quite far from probability. On one hand, connections between random matrix theory and other areas of mathematics, such as combinatorics and analytic number theory, have been explored for decades. On the other hand, these connections are often formulated for problems involving randomness or hold only heuristically. We will use tools and techniques from the newly developed "finite free probability", where operations on common mathematical objects are described by averaging over random matrices, to establish universality principles for the root dynamics for polynomials and analytic functions.