# Analysis Seminar

#### An evolution equation modeling roots of polynomials under differentiation

**Speaker:**
Stefan STEINERBERGER, Yale

**Location:**
Warren Weaver Hall 1302

**Date:**
Thursday, April 16, 2020, 11 a.m.

**Synopsis:**

Let p_n be a polynomial of very large degree n such that all

its roots lie on the real line. Suppose the roots are roughly

distributed like random variables coming from, say, a Gaussian. What

can you say about the roots of, say, the (n/2)-th derivative of the

polynomial? We propose the underlying dynamical system might indeed

have a mean field limit and identify a remarkably simple nonlinear,

nonlocal transport equation on the line governed by the Hilbert

transform. This equation has at least two very nice closed-form

solutions: a shrinking semi-circle and a family of distributions

evolving in the Marchenko-Pastur family of probability distributions; we

also show that these solutions satisfy an infinite number of

conservation laws. Many open problems are being discussed.