Title: Configurations, arithmetic groups, cohomology, and stability
Speaker: Jordan Ellenberg
Speaker Info: University of Wisconsin, Madison
Consider the following two objects:Date: Wednesday, April 23, 2014
* 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 of S_n for all sufficiently large n. This implies, for instance, that the dimensions of these cohomology groups are (for sufficiently large n) polynomials in n. In the congruence subgroup context, our results refine a 2012 theorem of Putman, and have been refined by 2013 results of Frank Calegari. In the configuration space context, the result here refines a 2011 theorem of Church.
The main ingredient is the theory of FI-modules, developed by myself, Tom Church, and Benson Farb, together with a Noetherianness theorem proved by the three of us and Rohit Nagpal: