Bhargava’s conjecture over function fields

Bhargava’s conjecture predicts the number of degree d extensions of $\mathbb Q$. In joint work with Ishan Levy, we prove a version of this conjecture over $\mathbb F_q(t)$, for $q$ sufficiently large relative to $d$ and prime to $d!$. The key new input is a refined understanding of the stable homology of Hurwitz spaces, and more generally an understanding of the stable homology of Hurwitz space modules. Time permitting, we may also describe how these ideas can also be used to compute the average size of Selmer groups in quadratic twist families of elliptic curves over function fields.