Speaker:  Will Sawin (Columbia University)

Title:  How do special polynomials typically factor?

Abstract:  How large are the prime factors of a random number, the  cycles of a random permutation, or the irreducible factors of a random  polynomial over a finite field? If we measure the size in a  sufficiently coarse way, these questions are answered in each case by  the same elegant probability distribution. If we seek more precise  information, the answers become more complicated, and are different in  each case. In this talk I will explain why, in the case of  polynomials, the answer doesn’t change as we restrict to polynomial satisfying congruence conditions. The proof uses deep tools in  algebraic geometry to estimate the cohomology of certain spaces and a  simple estimate on the original probability distribution.