Description:

Title: Homological stability for Hurwitz spaces

Abstract: I will explain work joint with Aaron Landesman where we prove that for a finite group G and conjugacy invariant subset c, Hurwitz spaces parameterizing connected G-covers of the complement of a configuration of points on a disk with monodromy in c satisfy homological stability. We moreover describe the stable homology after inverting finitely many primes in terms of simpler Hurwitz spaces. This has applications to Malle’s conjecture over function fields, the Cohen—Lenstra—Martinet heuristics over function fields, as well as to the Picard rank conjecture. If time permits, I will indicate some future directions.