Carl Mautner

I am a graduate student of mathematics at UT Austin studying with David Ben-Zvi. Recently I have been working with geometric methods in representation theory. More specifically, I have been studying `modular' perverse sheaves - perverse sheaves on complex algebraic varieties with coefficients in a field of positive characteristic - and connections with the modular representation theory of reductive groups and their Weyl groups.

In a project, joint with my collaborators Daniel Juteau and Geordie Williamson, we have defined a class of objects, called `parity sheaves.' Parity sheaves behave in many ways like intersection cohomology complexes and appear to play an important role in the behaviour of modular perverse sheaves. A preliminary version of part of our work is now available.

I am also working on a project that gives a geometric interpretation of the Schur functors - classical functors relating the modular representation theory of the general linear groups with that of the symmetric groups. More on both of these projects should appear in the not-so-distant future.

Papers:

Let X be a stratified complex variety, whose strata satisfy a cohomological parity vanishing condition with respect to a field or complete discrete valuation ring k. Parity sheaves are a class of objects in the constructible derived category of sheaves on X with coefficients in k. We find that parity sheaves share many of the good properties of intersection cohomology sheaves and also appear to carry interesting information about representation theory over the field k. For example, we prove that parity sheaves with coefficients in k on the affine Grassmannian for the complex reductive group G are closely related to tilting modules for the split form of the Langlands dual group over k. We show that, unlike the intersection cohomology sheaves, even when k is of positive characteristic, the parity sheaves satisfy a decomposition-type theorem for a class of proper morphisms which we call `even.'

The first half of this article is a brief survey of three connections between modular perverse sheaves and representation theory. In the second half of the paper, we compute a number of stalks of modular perverse sheaves on nilpotent cones to give the reader a sense of the intricacies and techniques involved.

Some past activities:
In Fall 2008 and Spring 2009, I co-organized a reading seminar on papers in algebra and geometry.

In Fall 2007, I organized a learning seminar on etale cohomology and the Weil conjectures.

I co-organized a graduate student workshop on knot theory and representations in January 2007.

Teaching:
Fall 2007, Math 427L-AP (TA for D. Freed).
Spring 2007, Math 427K-H (TA for K. Uhlenbeck).
Fall 2006, Math 427K (TA for D. Khosla).

Here are some rapidly aging pictures.