Title: Polynomial statistics over the little disks operad
Speaker: Joel Specter
Speaker Info: Northwestern
The statistics of polynomials over finite fields is controlled by topology. This philosophy, which dates back to Weil, has found recent application in the works of Church-Ellenberg-Farb, Farb-Wolfson, Ellenberg-Vankatesh-Westerland, Chen, and others, who have examined how the number of polynomials subject to certain constraints varies by studying the singular homology of certain families of varieties. Each of these works deduces strong statistical results. However, the only cases where these methods have yielded exact formulae are those in which the cohomology consist entirely of Hodge classes, and, in each of those cases, the statistics can be deduced by elementary methods.Date: Monday, May 02, 2016
In this talk, I will observe that families associated to certain natural counting problems carry the homotopy type of algebras over the little disks operad. As such, their homology caries the structure of a Gerstenhaber algebra. The corresponding coalgebra structure will be shown to persist in etale cohomology and be Galois equivariant. As an application, I will give exact formulae for the distribution of discriminants over finite fields. The cohomology of the corresponding varieties arises (explicitly) in a product of Fermat curves and is not of Tate type. This work is joint in part with Richard Moy, Arnav Tripathy, and Anastassia Etropolski.