My research area is Algebraic combinatorics. My recent research located at the intersection of Combinatorics and Representation Theory with natural links to Theoretical Physics. For some families of groups, it is not possible to find a satisfactory classification of irreducible representations (or equivalently conjugacy classes). Most recently, I defined normal supercharacter theory, which is a mechanism to replace conjugacy classes by normal subgroups in such a way that this 'coarsening' simulates many desirable properties of representation theory. This construction gives new tools to construct algebraic structures isomorphic to combinatorial structures. For instance, we build a supercharacter theory for unipotent upper-triangular matrices in which Dyck paths index supercharacters.
global connections related to teaching and scholarly work (in recent years)