Title: The asymptotic Picard rank conjecture
Abstract: The Picard rank conjecture predicts the vanishing of the rational picard group of the Hurwitz space parameterizing simply branched covers of $\mathbb P^1$ of degree $d$ and genus $g$. In joint work with Ishan Levy, we prove the Picard rank conjecture when $g$ is sufficiently large relative to $d$. The main input is a new result in topology where we prove that the homology of Hurwitz spaces stabilize and compute their dominant stable value. This is, in some sense, a continuation of my talk in the number theory seminar earlier today, though nothing from that talk will be assumed.