|Date: Tuesday, February 16, 2016
Title: Configurations, arithmetic groups, cohomology, and stability
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 n-tuples 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 n-tuple.) 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 FI-modules, 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, Castelnuovo-Mumford regularity) persist in the category of FI-modules.
The work discussed will include joint work with Tom Church, Benson Farb, Rohit Nagpal, and John Wiltshire-Gordon, as well as results of Andy Putman, Andrew Snowden, and Steven Sam.
Some relevant papers:
Speaker: Jordan Ellenberg
Institution: University of Wisconsin