Event

Talk
Representation stability in cohomology and asymptotics for families of varieties over finite fields
Prof. Benson Farb (University of Chicago)
6 Jan 2014, 16:15 – 17:00

In this talk Prof. Benson Farb will consider two families $X_n$ of varieties on which the symmetric group $S_n$ acts: the configuration space of $n$ points in $\C$ and the space of $n$ linearly independent lines in $\C^n$. He will explain via these two beautiful examples how non-experts can use the (twisted) Grothendieck-Lefschetz Fixed-Point Theorem in \'{e}tale cohomology as a machine to convert information, as follows:
Input: How the multiplicity of a given irreducible representation $V$ of $S_n$ in $H^*(X_n;\Q)$ varies with n
Output: Formulas for the number of polynomials over $\F_q$ (resp.\ maximal tori in $\GL_n(\F_q)$) with specified properties related to $V$.
In particular we explain how representation stability of $H^*(X_n;\Q)$ corresponds to asymptotic stability of various point counts as $n\to \infty$.

http://www.math.uchicago.edu/~farb/