UBC Math Department Colloquium: Craig Westerland

The inverse Galois problem asks whether every finite group occurs as the Galois group of an extension of the rational numbers. In 2002, Malle made this existential question more numerical, by conjecturing an asymptotic formula on the growth of the number of fields with a given Galois group, as a function of discriminant. One may reformulate this question in a function field context, replacing the rational numbers with the field of rational functions in positive characteristic. In joint work with Jordan Ellenberg and TriThang Tran, we show that Malle’s conjectured formula does give an upper bound on that distribution. Our methods are very topological, relying on new tools for computing the homology of Hurwitz moduli spaces of branched covers. We will elide these technicalities in this talk — they will be the focus of the topology seminar earlier this week — and focus on the number theoretic results and how to reformulate them in topological terms.