# Brian Hall (University of Notre Dame)

Date

Friday March 1, 20242:30 pm - 3:30 pm

Location

Jeffery Hall, Room 234## Math & Stats Department Colloquium

**Friday, March 1st, 2023**

**Time:** 2:30 p.m. **Place: **Jeffery Hall, Room 234

**Speaker:** Brian Hall (University of Notre Dame)

**Title:** Heat flow on polynomials with connections to random matrix theory

**Abstract:** Here is a simple question with a surprisingly rich answer: How do the roots of a polynomial change as the polynomial evolves according to the heat equation? The simplest case of this question concerns the backward heat flow on polynomials with all real roots. In this case, an old theorem of P´olya and Benz says that the roots will remain real. Much more recent results tell us how these real roots evolve in the high-degree limit—and there is a surprising connection to random matrix theory. Things become even more interesting when we study the heat flow (forward or backward) on polynomials with complex roots. I will present some recent conjectures and rigorous results on this question. This is joint work with Ching Wei Ho, Jonas Jalowy, and Zakhar Kabluchko. The talk will be self-contained and have lots of pictures and animations.

**Bio:** Brian Hall received his Ph.D. from Cornell University in 1993 under Leonard Gross and is currently a professor at the University of Notre Dame. He is the author of two textbooks in the Graduate Texts in Mathematics series, Quantum Theory for Mathematicians and Lie Groups, Lie Algebras and Representations. His recent interests concern random matrices, random polynomials, and free probability.