Braided Vector Spaces and Arithmetic Statistics Over Function Fields

The past few years have seen rapid development in arithmetic statistics over function fields over finite fields, questions like:  what is the distribution of the ell-primary part of the class group of a random quadratic extension of F_q(t)?  (Cohen-Lenstra heuristics) or:  how many G-covers of P^1/F_q are there with bounded discriminant?  (Malle's conjecture.)  Many of these results can be thought of in a unified way as questions about cohomology of the braid group with coefficients in a braided vector space.  I'll explain how this works, talk about some new results with Mark Shusterman in this vein which give novel results on prime discriminants, and point at many questions that remain only partially understood.