Seminar Talk

The inverse Galois problem, a foundational question in number theory, asks whether every finite group $G$ can be realized as the Galois group of 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.