Malle's conjecture over function fields

The inverse Galois problem asks whether every finite group \(G\) can be realized as the Galois group of a field extension of the rational numbers. Malle's conjecture is a refined version of the inverse Galois problem which predicts the asymptotic number of such extensions. In joint work with Ishan Levy, we prove a version of Malle's conjecture, computing the asymptotic growth of the number of Galois \(G\) extensions of \(\mathbb F_q(t)\), for \(q\) sufficiently large and relatively prime to \(|G|\). We use tools from algebraic geometry to relate this conjecture to a question in topology about the cohomology of certain Hurwitz spaces. We then complete the proof by solving the topological question using techniques from homotopy theory.