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Stefan Steinerberger: A PDE describing roots of polynomials under differentiation

Submitted by Jorge Cisneros on April 1, 2021 - 12:00pm
Stefan Steinberger

Speaker: Stefan Steinerberger, University of Washington 

Date: April 1, 2021

Title:  A PDE describing roots of polynomials under differentiation

Abstract:  Suppose you have a polynomial of degree n = 10^10 on the real line whose roots are distributed roughly like a Gaussian. Suppose now you differentiate this polynomial t*n times (where 0< t < 1 plays the role of time), where are the roots of that (t*n)-th derivative located on the real line? I will describe an extremely simple nonlinear evolution equation which describes the evolution of these densities. A recent result of Dimitri Shlyakhtenko & Terence Tao shows that the equation can also be interpreted in terms of free probability theory. There will be many pretty pictures and open problems!

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