Research
My advisor is Stephen Bigelow. My research is in diagrammatic algebra, quantum algebra, and quantum topology. I am involved in two main research projects right now. I love discussing my research, so if you would like any more details on either of the below projects, please feel free to email me.
Project 1: Give diagrammatic presentations by generators and relations for subfactor planar algebras of index 4.
Synopsis: The category of representations of a finite group is an example of a fusion category. Within this type of category we can take linear combinations of morphisms, and tensor products and direct sums of objects. A planar algebra can be constructed from a unitary fusion category with the advantage that the structure is now conveniently encoded by diagrams in the plane. Conversely, given a subfactor planar algebra, we can construct a fusion category of projections. The Kuperberg program is a proposal to give a presentation by generators and relations for every subfactor planar algebra. For my eventual thesis I hope to give diagrammatic presentations for the subfactor planar algebras of index 4. I have currently finished the \(\tilde{A}_{2n-1}\) and \(\tilde{D}_\infty\) cases and am in the process of writing this paper. For more details on the project you can click here.
Project 2: Recovering knot polynomials from graphs.
Synopsis: The goal of this project is to obtain knot polynomials from the use of graph theory. This project is through the UCSB REU program where I serve as a graduate mentor. You can read a full description of the project here.