Fox-Neuwirth cells, quantum shuffle algebras, and Malle’s conjecture for function fields

Geometry Topology Seminar
Monday, September 19, 2016 - 14:00
1 hour (actually 50 minutes)
Skiles 006
University of Minnesota
I will describe new techniques for computing the homology of braid groups with coefficients in certain exponential coefficient systems.  An unexpected side of this story (at least to me) is a connection with the cohomology of certain braided Hopf algebras — quantum shuffle algebras and Nichols algebras — which are central to the classification of pointed Hopf algebras and quantum groups. We can apply these tools to get a bound on the growth of the cohomology of Hurwitz moduli spaces of branched covers of the plane in certain cases.  This yields a weak form of Malle’s conjecture on the distribution of fields with prescribed Galois group in the function field setting.  This is joint work with Jordan Ellenberg and TriThang Tran.