Sam Gunningham - q-Character sheaves and Springer theory
The category of equivariant D_q(G)-modules sits at the interface of low-dimensional topology and geometric representation theory. It appears naturally in the context of factorization homology and skein theory, but it may also be thought of as a q-deformation of the category of equivariant modules for the ring D(G) of differential operators on a complex reductive group G. In this talk I will survey various aspects of an ongoing program to adapt the powerful tools available in the D-module setting (e.g. character sheaves, parabolic induction, Springer theory) to the realm of D_q-modules. As an example, I will present a new derivation of a recent result of Carrega and Gilmer on the 9-dimensionality of the skein module for a 3-torus. This is joint work with (combinations of) David Jordan, Monica Vazirani, and Pavel Safronov.