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Mathematics Colloquium

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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 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: 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

 

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