Representation theory and homological stability
Homological stability is the remarkable phenomenon where for certain sequences X_n of groups or spaces -- for example SL(n,Z), the braid group B_n, or the moduli space M_n of genus n curves -- it turns out that the homology groups H_i(X_n) do not depend on n once n is large enough. But for many natural analogous sequences, from pure braid groups to congruence matrix groups to Torelli groups, homological stability fails horribly. In these cases the rank of H_i(X_n) blows up to infinity, and in the latter two cases almost nothing was known about H_i(X_n); indeed it's possible there is no nice "closed form" for the answers. Representation stability is a notion which takes into account the action of certain symmetries to meaningfully talk about "the stable homology of the pure braid group" or "the stable homology of the Torelli group", even though the homology never stabilizes. In this talk I will explain our broad picture of representation stability and describe a number of connections to other areas of math. In particular, I will consider various sequences of integers a_n arising in topology (e.g. Betti numbers of spaces on configurations of points, of n-pointed curves, of matrices of rank at most n, etc.) and in algebra/combinatorics (e.g. dimensions of spaces of harmonic polynomials, of coinvariant algebras, of free Lie algebras, etc.), and explain how to use representation stability to prove that for each of these sequences (and many more) there is a polynomial P(n) with a_n = P(n) for all n big enough. Joint work with Benson Farb and Jordan Ellenberg.