My research generally lies in the realm of algebraic geometry, but aspects of my work also touch on ideas from algebraic combinatorics, representation theory, symplectic geometry, and mathematical physics.

I am especially interested in tools coming from mirror symmetry, a phenomenon first observed by string theorists which relates the algebro-geometric structure of one space to the symplectic structure of a "mirror" space. The Gross-Siebert program uses tropical geometry (a sort of piecewise-linear skeleton of geometry) to create a combinatorial framework for understanding the structures arising in mirror symmetry. Much of my work is based on applying these tools from the Gross-Siebert program towards problems concerning cluster algebras.

Cluster algebras are a class of combinatorially defined commutative algebras which admit many different local coordinate systems, called clusters, related to each other via certain birational maps called mutations. They were defined by Fomin-Zelevinsky with the goal of better understanding certain canonical bases and positivity phenomena observed by Lusztig in the context of quantum groups and representation theory. Work of Gross-Hacking-Keel-Kontsevich applied the Gross-Siebert program to construct canonical "theta bases" for cluster algebras, thus settling many important conjectures in cluster theory. Most of my research is centered around better understanding every aspect of these theta bases, incluing their general properties, quantization, tropicalization, and connections to Gromov-Witten theory (counting holomorphic curves) and quiver DT-theory (e.g., Euler characteristics of certain spaces of quiver representations).