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.