**Title:** Pairing between zeros and critical points of random polynomials

**Program Website: Large N limit problems in Kahler Geometry**

**Speaker:** Boris Hanin, MIT

**Date:** Tuesday, June 02, 2015

**Time:** 10:00am – 11:00am

**Place:** Seminar Room 313, Simons Center

[box, type=”download”]Watch the Video.

[/box]

**Abstract:** Consider a polynomial p_N(z) in one complex variable. The Gauss-Lucas Theorem says that the critical points of p_N lie inside the convex hull of its zeros. But are critical points actually distributed inside the convex hull if p_N is chosen at random?

The purpose of this talk is to explain that in fact each critical point of p_N typically comes paired with a single zero. The distance between a critical point and its paired zero is on the order of N^{-1}, which is much smaller than the typical N^{-1/2} spacing between order of N iid points on the sphere.

In the first part of my talk, I will give a heuristic interpretation for this pairing by relating zeros and critical points to electrostatics on the Riemann sphere. In the second part, I will explain how to use Bergman kernel asymptotics to prove rigorous theorems about this pairing.