**Jordan Ellenberg** (Wisconsin)

* Random matrices, random permutations, conjectures of
arithmetic distribution over function fields, topology of Hurwitz
spaces*

A Hurwitz space *H*_{G,n}
is an algebraic variety parametrizing
branched covers of the projective line with some fixed finite Galois
group *G*. We will prove that, under some hypotheses on *G*, the
rational *i*th homology of the Hurwitz spaces stabilizes when the
number of branch points is sufficiently large compared to *i*.

This purely topological theorem has some interesting number-theoretic
consequences. It implies, for instance, a weak form of the Cohen-
Lenstra conjectures over rational function fields, and some
quantitative inverse Galois results over function fields. For
instance, we show that the average size of the *p*-part of the class
number of a hyperelliptic genus-g curve over **F**_{q} is bounded
independently of g, when q is large enough relative to p; the key
point here is q can be held fixed while g grows.

I will try to give a general overview of the dictionary between
conjectures about topology of moduli spaces, on the one hand, and
arithmetic distribution conjectures (Cohen-Lenstra, Bhargava, Malle,
inverse Galois....) on the other. In particular, I will explain how
*vanishing* statements in cohomology imply that a natural Frobenius
action behaves like a random matrix, a random permutation, or a
random element of some other more exotic monodromy group.

(joint work with Akshay Venkatesh and Craig Westerland.)