Current Projects
- Invertible Fusion Categories:
- A tensor category C over a field K is said to be invertible if there is a tensor category D such that C[x]D is Morita equivalent to Vec_K. When K is algebraically closed, it is well-known that the only invertible fusion category is Vec_K, and any invertible multi-fusion category is Morita equivalent to Vec_K. In general, the invertible multi-fusion categories over K are classified (up to Morita equivalence) by H^3(K;G_m), the third Galois cohomology of the absolute Galois group of K. This is a categorified analogue of the Brauer group H^2(K;G_m). As an application, we show that the Morita classification of fusion categories by their Drinfel’d canters breaks when this cohomology group is nontrivial.
- Braidings for Non-Split TY categories over R:
- Continuing the investigation of the categories discovered in https://arxiv.org/abs/2303.17843, we follow the outline of Siehler in https://arxiv.org/abs/math/0011037 and classify all possible braidings that these categories can be equipped with. We discover that all possible braidings on real/quaternionic TY categories are necessarily symmetric, and that some real/complex TY categories appear as the Drinfel'd centers of real/quaternionic TY categories.
- This is joint with David Green and Yoyo Jiang
- Pivotal Extension Theory:
- A pivotal fusion category is a fusion category equipped with a monoidal natural isomorphism from the identity functor to the double dual functor. Pivotal structures are desirable because they allow for the definition of powerful computational tools like quantum traces. This project aims to upgrade the extension theory outlined in https://arxiv.org/abs/0909.3140 to the setting of pivotal fusion categories.
- This is joint with Agustina Czenky, David Jaklitsch, Dmitri Nikshych, Julia Plavnik, David Reutter, and Harshit Yadav
Older Projects
- 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:
- Current preprint available here
- 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.