Research Statement: Combinatorial Methods in Algebra and Geometry

My research is at the intersection of algebra, geometry, and combinatorics. My dissertation work under Pramod Achar involved a deep study of the singularities of certain topological spaces through the lens of perverse sheaves. Concurrently, I began working with Greg Muller on cluster algebras, a subject that is related to a large number of areas in mathematics, but which has a somewhat easier point of entry due to its computable examples. This early blend of, almost diametrically opposed, styles of math shaped my research direction. My work gravitated toward interpreting complicated topological or algebraic questions as combinatorial ones, which lend themselves to explicit computations. Each of my projects has this theme in one way or another. My research statement is divided into three broad areas: matroids, cluster algebras, and representation theory. The projects described in each subsection are outlined below.