# Caltech/UCLA Joint Analysis Seminar

Suppose you have a polynomial of degree p_n whose n real roots are roughly distributed like a Gaussian (or some other nice distribution) and you differentiate t*n times where 0<t<1. What's the distribution of the (1-t)*n roots of that (t*n)-th derivative? How does it depend on t? We identify a relatively simple nonlocal evolution equation (the nonlocality is given by a Hilbert transform); it has two nice closed-form solutions, a shrinking semicircle and a family of Marchenko-Pastur distributions (this sounds like random matrix theory and we make some remarks in that direction). Moreover, the underlying evolution satisfies an infinite number of conservation laws that one can write down explicitly. This suggests a lot of questions: Sean O'Rourke and I proposed an analogous equation for complex-valued polynomials. Motivated by some numerical simulations, Jeremy Hoskins and I conjectured that t=1, just before the polynomial disappears, the shape of the remaining roots is a semicircle and we prove that for a class of random polynomials. I promise lots of open problems and pretty pictures.