Combinatorial stability and representation stabilityAlgebra & Discrete Mathematics
|Speaker:||Thomas Church, Stanford|
|Start time:||Mon, Jan 26 2015, 5:10PM|
How many roots does a random squarefree polynomial f(T) in Fq[T] have? On average, it's a bit less than one root per polynomial. The precise answer 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/q2 - 1/q3 + ... = 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 give a general picture of this representation stability for sequences of Sn-representations, and describe how combinatorial stability for statistics of squarefree polynomials, of maximal tori in GLn(Fq), and other natural geometric counting problems can be converted to questions of representation stability in topology, and vice versa.