ISRMT Seminar: Roots of polynomials under differential flows, with connections to random matrix theory

I will discuss how the roots of polynomials evolve under differential flows, focusing on two main examples: heat flow and repeated differentiation. In the case of real roots, the situation is well understood and the evolution of the roots is described by constructions from random matrix theory. In the case of complex roots, I will present a general PDE-based conjecture, together with a random matrix interpretation. Finally, I will describe recent rigorous results for random polynomials.

The talk will be self-contained and have lots of pictures and animations.