Lecture Event

**SFB Colloquium Representation Stability (Research Project C1 and C3)**

Prof. Benson Farb (University of Chicago)

7 Jan 2014, 16:00 – 18:00

Talk

**Representation stability in cohomology and asymptotics for families of varieties over finite fields**

Prof. Benson Farb (University of Chicago)

7 Jan 2014, 17:15 – 18:00

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 $n\to \infty$.