Probability and Mathematical Physics Seminar

---

Random Matrices and the Roots of (Random) Polynomials

Speaker: Andrew Campbell, Lafayette College

Location: Warren Weaver Hall 1302

Date: Friday, November 6, 2026, 11 a.m.

Synopsis:

How does a given differential operator affect the distribution of a polynomial's roots? When the operator is just the derivative operator, then the Gauss--Lucas theorem guarantees that the critical points are in the convex hull of the original roots. This provides a general answer to where the roots are distributed, but not how they are distributed. It turns out answering how roots are distributed, even for just one derivative, can be a significantly more difficult task. In this talk we will discuss some motivations and partial history behind this question for both deterministic and random polynomials, and then see how random matrix theory can help us understand how the roots are distributed.

Based on joint work with Jonas Jalowy (Universität Paderborn), Sean O'Rourke (CU Boulder), and David Renfrew (Binghamton University).