Daniele Garzoni, USC

Abstract: Given a group G acting on a set, an element of G is called a derangement if it acts without fixed points. Luczak-Pyber and Fulman-Guralnick showed that if G is a finite simple group acting transitively, then the proportion of derangements is bounded away from zero absolutely. I will discuss a conjugacy-class version of this result for groups of Lie type, obtained in joint work with Sean Eberhard. I would like to discuss mainly two things: (i) why derangements are interesting, and (ii) explain some interesting connections between the proof of the result and the subject of "anatomy of polynomials", which essentially studies divisors of random polynomials.