Malle’s conjecture over function fields

For G a finite group, Malle’s conjecture predicts the asymptotic growth of the number of G extensions of a fixed global field. In joint work with Ishan Levy, we compute the asymptotic growth of the number of Galois G extensions of F_q(t), for q sufficiently large and relatively prime to |G|.  In the first part of the talk we will introduce Malle’s conjecture and explain how to reduce the above result to a homological stability result for spaces of G bundles. In the second part of the talk we will explain some of the key ideas in the proof of this homological stability result, which uses tools from higher algebra.