Research

My research interests are jointly in the cluster combinatorics of knot theory and in finding applications of techniques from mathematical logic in understanding the complexities of various properties 0f (both finite and infinite) quivers, particularly those properties which are mutation-invariant and/or hereditary.

My primary project lives at the intersection of knot theory, representation theory, and cluster algebras. Recently, Bazier-Matte and Schiffler have introduced a representation-theoretic and a cluster-algebraic method to reproduce the Alexander polynomial of a prime knot or link. Their method provides a very strong connection between knot theory and cluster algebras. My work is a modification of theirs which allows the use of tools from the theory of cluster algebras associated to triangulations of marked surfaces and recent skein-theoretic results about them. My long-term objective is to use these tools to reproduce different knot invariants from cluster algebras.

Another direction of my research includes looking at a pair of spaces that parametrize countably infinite quivers, both of which are (non-canonically) homeomorphic to the product topology on countably many copies of a countably infinite discrete space. Mutations give rise to continuous group actions on these spaces, so we may use results from topological dynamics and descriptive set theory to understand various properties of infinite quivers. Moreover, one can also use the topologies on these spaces to make sense of infinite mutation sequences. My goal is to be able to use this framework to initiate a systematic study of the various descriptive complexity hierarchies of quiver properties, to provide a uniform method for considering infinite sequences of mutations, and to contextualize recent work by a number of other authors on infinite-rank cluster algebras, particularly those of surface type.

Aside from these two projects, I am also working collaboratively on problems in the combinatorics of snake graphs and fence posets, webs and plabic graphs, and an original adaptation of chip-firing on quivers incorporating their mutations.