Date: Tuesday, February 16, 2016
Title: Configurations, arithmetic groups, cohomology, and stability
Abstract:
Consider the following two objects:
* The congruence subgroup of level p in SL_n(Z); that is, the group of integral matrices congruent to 1 mod p;
* The ordered configuration space of n points on a manifold M, which is to say, the space parametrizing ordered ntuples of distinct points on M;
Each of these objects carries a natural action of the symmetric group S_n on n letters. (In the first case, this is by permuting the elements of the standard basis; in the second case, by permuting the points in the ntuple.) What's more, each one is naturally described by cohomology groups H^i, which inherit the action and thus become representations of S_n.
Although these examples are quite different, it turns out there is a general notion of stability which applies to both of these cases (and many other examples in representation theory, algebraic geometry, and combinatorics.) In some sense, each H^i is "the same representation" but of different groups! The goal of the talk is to explain a framework, the category of FImodules, in which this notion actually makes sense, and to use this framework to show (for example) that the dimensions of these cohomology groups are polynomials in n for sufficiently large n. It turns out that the stability theorems one wants to get are typically consequences of showing that appealing features of the category of modules over a polynomial ring (e.g. Noetherian properties, CastelnuovoMumford regularity) persist in the category of FImodules.
The work discussed will include joint work with Tom Church, Benson Farb, Rohit Nagpal, and John WiltshireGordon, as well as results of Andy Putman, Andrew Snowden, and Steven Sam.
Some relevant papers:
http://arxiv.org/abs/1204.4533
http://arxiv.org/abs/1210.1854
http://arxiv.org/abs/1508.02430
Speaker: Jordan Ellenberg
Institution: University of Wisconsin
