Tom Church, Chicago
Representation theory and homological stability
Homological stability is a 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 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 is known about H_i(X_n); indeed it's possible there is no nice "closed form" for the answers.
While doing some homology computations for the Torelli group, we found what looked like the shadow of an overarching pattern. In order to explain it and to formulate a specific conjecture, we came up with the notion of "representation stability" for a sequence of representations of groups. 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. This work is joint with Benson Farb.
In this talk I will explain our broad picture and give two major applications. One is a surprisingly strong connection between representation stability for certain configuration spaces and arithmetic statistics for varieties over finite fields, joint with Jordan Ellenberg and Benson Farb. The other is representation stability for the homology of the configuration space of n distinct points on a manifold M.
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