My current research interests are in representation theory and algebraic geometry. My undergraduate research also focused on combinatorics.
Recently, I have been interested in differential geometry in positive characteristic and deformation quantization. I have also been thinking about classical questions about the representation theory of the general linear group.
In tropical geometry, I studied tropical linear spaces. They are certain weighted polyhedral complexes: in that language, tropical linear spaces are tropical cycles of degree 1. They may also be described algebraically as certain modules over the tropical semifield; the analogous algebraic object over the Boolean semifield is exactly a matroid. I hope to further develop this perspective by creating more analogies between matroids, tropical linear spaces, and generally matroids over hyperfields through the lens of module theory. I am also interested in using tropical linear algebra to develop tropical ideals and tropical schemes.
“A Module-Theoretic Approach to Matroids.” AMS Spring Sectional Meeting, Special Session on Algebraic and Combinatorial Aspects of Tropical Geometry, March 17, 2018. Slides from this talk