The Picard rank conjecture predicts the vanishing of the rational Picard group of the Hurwitz space parameterizing simply branched covers of 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 stabilizes and compute their stable value. Using the same homological stability results, we prove a version of Malle's conjecture over F_q(t), which predicts the number of G extensions of F_q(t), for G a finite group.