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Mathematics Colloquium


Date:  Tuesday, September 24, 2013

Title:  Combinatorial stability and representation stability

Abstract:  If you choose a squarefree polynomial f(T) in F_q[T] uniformly at random, it will have slightly less than one linear factor on average. The exact value of this expectation depends on the degree of f(T), but as deg f(T) goes to infinity, the expectation stabilizes and converges to 1 - 1/q + 1/q^2 - 1/q^3 + ... = q / (q+1). In joint work with J. Ellenberg and B. Farb, we proved that the stabilization of this combinatorial formula is equivalent to a representation-theoretic stability in the cohomology of braid groups. I will describe how combinatorial stability for statistics of squarefree polynomials, of maximal tori in GL_n(F_q), and other natural geometric counting problems can be converted to questions of representation stability in topology, and vice versa. This talk will assume no background, and is intended for a general mathematical audience.

Speaker:  Tom Church
Institution:  Stanford University


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