I am working in algebro-geometric and combinatorial aspects of representation theory.

One topic I have been investigating is the p-adic theory of Kac-Moody groups. A special case of this is a notion of a double loop group. This is very closely related to the geometry of double affine Schubert varieties. Conjecturally, these double affine Schubert varieties should also be closely related to instanton spaces as well as Coulomb branches for quiver gauge theories.

A closely related topic is the geometric Satake equivalence and Mirković-Vilonen (MV) cycles. In this area I have been studying both the MV basis as well as MV polytopes and their affine generalizations. Finally, I have been recently thinking about Chern-Mather classes and characteristic cycles for Schubert varieties.