My work involves studying the properties of reductive algebraic groups. Algebraic groups are groups equipped with the Zariski topology such that the multiplication and inverse maps are maps of varieties. They behave like Lie groups, except that there is the freedom to work over any field.
Specifically, I study the objects that arise when trying to understand the representation theory of algebraic groups, especially nilpotent orbits and affine Weyl groups. Recently I have been thinking about the connection between nilpotent orbits, Borel-stable ideals in the nilradical, Kazhdan-Lusztig cells, and certain duality maps.