Student Probability Seminar

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Roots of random polynomials: a gentle introduction to the Kac-Rice formula

Speaker: Tim Kunisky, CIMS

Location: TBA

Date: Wednesday, October 14, 2020, 10 a.m.

Synopsis:

How many real roots does a polynomial with gaussian coefficients usually have? How are they distributed along the real line? I will introduce an elegant method of calculating the answers to these classical questions, called the Kac-Rice formula. In particular, we will see why the choice of variances of the gaussian coefficients of a random polynomial can dramatically affect the number of real roots and their positions. Only calculus and basic probability knowledge required! But, time permitting, I will also mention how more advanced applications of the same idea have been used to great effect recently to understand the critical points of high-dimensional optimization landscapes, as arise in machine learning and the statistical physics of spin glasses.