- Extension Theory of Tensor Categories over Non–Algebraically Closed Fields:
- In a 2009 paper, Etingof, Nikshych and Ostrik establish an obstruction theory method for classifying extensions of fusion categories by groups.Their paper focuses on the case of fusion categories over algebraically closed fields. The goal of this project is to understand the changes thatmust be made in order to adapt these techniques to the non-semisimple, and non-algebraically closed cases.
- This is the topic of my dissertation.
- Non-Split Tambara-Yamagami Categories Over the Reals:
- In 1998, Daisuke Tambara and Shigeru Yamagami investigated a simple set of fusion rules, and proved under which circumstances those rules could be given a coherent associator. In this project, we are investigating a generalization of such fusion rules to the setting where the simpleobjects are no longer required to be absolutely simple. Over the real numbers, this means that objects are either real, complex or quaternionic.
- This joint work with Julia Plavnik and Dalton Sconce is the outgrowth of an REU in Summer 2021 for which I was a mentor.
- Universal Traces:
- For finite-dimensional vector spaces, the coend of the Hom-functor is isomorphic to the base field, and the corresponding universal cowedge is precisely the trace map. More generally, this coend defines an important trace-like invariant T(C) for every (suitably nice) category C. This has applications to the theory of modified traces, which are essential to the definition of 3-manifold invariants coming from nonsemisimple TQFTs.