In 1985, Harer proved the homology of the moduli space of curves stabilizes, and in 2007, Madsen and Weiss computed the stable value of its homology. Hurwitz spaces, parameterizing branched covers of the projective line, are of comparable importance to the moduli space of curves, yet almost nothing is known about their stable homology.
In a 2016 Annals paper, Ellenberg, Venkatesh, and Westerland proved the homology of dihedral group Hurwitz spaces  stabilizes, but the stable value of this homology remained open. In joint work with Ishan Levy, we compute the stable value of this homology. As some applications, we are able to compute the stable rational Picard groups of these Hurwitz spaces, asymptotically verifying the Picard rank conjecture, and verify predictions of the Cohen-Lenstra heuristics from number theory.