Broadly speaking, my research is in the area of representation theory. The general idea is to study the mathematics of symmetry. For example, taken together, the transformations of the plane
which leave the background of this page unchanged is a mathematical
object called a group. One can study this group to better
understand the pattern in the background or, vice versa, use the
symmetries of the background to study that group. As one might
imagine, this field involves algebra, geometry, combinatorics, and many
other areas of mathematics (which makes it interesting!).
My research interests include Lie theory, algebraic combinatorics,
crystal/canonical bases, representations of finite and algebraic
groups, Lie algebras, cohomology and support varieties, and the super
analogue of these topics. This naturally leads to
questions in algebraic geometry, quantum groups, finite dimensional
algebras, homological algebra and derived categories, and myriad other
topics. Below is a two dimensional shadow of the root system of type E7 (borrowed from John Stembridge). The combinatorics and symmetries of this picture reflect the rich theory of the associated Lie group and Lie algebra.