Jordan Ellenberg, mathematician at the University of Wisconsin, gives this Joint IAS/Princeton Number Theory seminar on recent progress in arithmetic statistics over function fields. He frames questions like the distribution of class groups of random quadratic extensions of F_q(t) under Cohen-Lenstra heuristics, and counting G-covers of the projective line with bounded discriminant under Malle's conjecture, as instances of a single structure: cohomology of the braid group with coefficients in a braided vector space. Ellenberg walks through how this framework unifies the two problems, then presents new joint work with Mark Shusterman yielding results on prime discriminants. He closes by laying out open questions that remain only partially resolved. The talk assumes graduate-level familiarity with number theory and algebraic topology, and runs just over an hour at Simonyi Hall.