Georgia Institute of Technology College of Sciences

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.