Thomas Church

Abstract: Homological stability is the remarkable phenomenon where for certain sequences X_n of groups or spaces -- e.g. the special linear group SL(n,Z), the braid group B_n, or the moduli space M_n of genus n Riemann surfaces -- the homology groups H_i(X_n) do not depend on n once n is large enough. But for many analogous sequences, from pure braid groups to congruence matrix groups to Torelli groups, homological stability fails horribly, and H_i(X_n) blows up to infinity. In many of these cases we know very little concretely about H_i(X_n), and it's possible there is no nice "closed form" for the answers.

Representation stability is a program that extends homological stability to situations like these where it's false, by taking the action of symmetry groups into account. This makes it possible 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. However, it turns out that representation stability is also useful elsewhere in math: it lets us prove and explain a polynomial behavior in the dimensions of vector spaces arising in algebra/combinatorics (coinvariant algebras, free Lie algebras, spaces of harmonic polynomials, etc.) as well as in topology (cohomology of spaces of configurations in a manifold, of pointed curves, of bounded-rank matrices, etc.). Joint work with Benson Farb, Jordan Ellenberg, and Rohit Nagpal.