I am interested in the broad area of representation theory and tensor categories, which sits at the intersection of algebra, topology, combinatorics, model theory, and mathematical physics. More specifically, I have been working on questions involving representation categories of quantum groups, crystal bases, and related tensor cateogies. I am also interested in the applications of braided tensor categories to topological quantum field theories, invariants of links and manifolds, and topological quantum computing. Recently, I have also become quite fascinated by the rich world of oligomorphic tensor categories and their many peculiar properties.
Projects I am actively pursuing at the moment include:
Harman-Snowden measures on Fraïssé classes and oligomorphic tensor categories.
Web categories arising from crystal limits of quantum group representations.
Drinfeld centers of tensor categories arising from quantum group representations.
Papers and Preprints
Presentations for Categories of Crystals. Joint with David He and Dani Tubbenhauer. Submitted, 2026. arXiv:2606.02249
Abstract: We give generators and relations for the monoidal categories of crystals generated by the fundamental crystals of a simple complex Lie algebra. We also spell out several small-rank examples.
The Drinfeld Center of the Generic Temperley–Lieb Category. Submitted, 2026. arXiv:2603.28970
Abstract: We show that the Temperley–Lieb category \(\mathrm{TL}(q;\mathbb{C})\) embeds in an ultraproduct of modular tensor categories when \(q\) is not a root of unity. As a result, we show that its Drinfeld center is semisimple and describe its simple objects. The canonical functor \[\mathrm{TL}(q;\mathbb{C})\boxtimes \mathrm{TL}(q;\mathbb{C})^{\mathrm{rev}} \boxtimes \mathrm{Rep}(\mathbb{Z}/2\mathbb{Z}) \to \mathcal Z(\mathrm{TL}(q;\mathbb{C})),\] induced by the braiding and the \(\mathbb{Z}/2\mathbb{Z}\)–grading on the Temperley–Lieb category, is thus shown to be a monoidal equivalence, which becomes a braided equivalence upon twisting the braiding by a certain bicharacter. Along the way, we formalize some general results about ultraproducts of tensor categories and tensor functors, building on earlier works of Crumley, Harman, and Flake–Harman–Laugwitz. We also discuss the center at some exceptional values of \(q\).
A Coboundary Temperley–Lieb Category for \(\mathfrak{sl}_{2}\)-Crystals. Joint with Mateusz Stroiński. J. London Math. Soc., 112: e70283, 2025.
Abstract: By considering a suitable renormalization of the Temperley–Lieb category, we study its specialization to the case \(q=0\). Unlike the \(q\neq 0\) case, the obtained monoidal category, \(\mathrm{TL}_0(\mathbb{K})\), is not rigid or braided. We provide a closed formula for the Jones–Wenzl projectors in \(\mathrm{TL}_0(\mathbb{K})\) and give semisimple bases for its endomorphism algebras. We explain how to obtain the same basis using the representation theory of finite inverse monoids, via the associated Möbius inversion. We then describe a coboundary structure on \(\mathrm{TL}_0(\mathbb{K})\) and show that its idempotent completion is coboundary monoidally equivalent to the category of \(\mathfrak{sl}_{2}\)-crystals. This gives a diagrammatic description of the commutor for \(\mathfrak{sl}_{2}\)-crystals defined by Henriques and Kamnitzer and of the resulting action of the cactus group. We also study fiber functors of \(\mathrm{TL}_0(\mathbb{K})\) and discuss how they differ from the \(q\neq 0\) case.
Upcoming papers:
Measures on Wreath Products of Fraïsseé Classes. (In preparation). Joint with Ibrahim Abdelaziz and Maria Madrugo.
Abstract: Given a Harman–Snowden measure on a ;class \(\mathfrak F_1\) and a compatible weak measure on another class \(\mathfrak F_2\), we give a measure on their wreath product \(\mathfrak F_1\wr\mathfrak F_2\). We show that all measures on \(\mathfrak F_1\wr\mathfrak F_2\) arise in this way, thus completely classifying measures on the wreath product in terms of those on the original classes. We illustrate this classification theorem in several examples including iterated wreath products of the classes of finite sets and finite linear orders, producing infinite families of new pre-Tannakian tensor categories of super exponential growth.
Broad topics/problems I’ve recently thought about and would like to understand better and/or work on in the near future include:
Tensor categories arising from (pro)oligomorphic groups.
Classification of incompressible symmetric tensor categories in positive characteristic.
Drinfeld centers (and associated link invariants) of categories with monoidal presentations (e.g. Soergel bimodules).
Topological and conformal quantum field theories.
Computability questions in fusion categories (Topological quantum computing).
Constructions and existence conditions for abelian envelopes of non-abelian tensor categories.
Coding Projects
As part of my research, I sometimes write code to do computations for myself. You can find some of these computational tools on my github. You are welcome to use them as well, though they are unfortunately poorly documented.
Undergraduate Work
Distinct Distances with \(\ell_p\) Metrics. Joint with Polymath REU. Computational Geometry 100:101785, 2022. This paper received the CGTA Young Researchers Award for best paper written by authors under 35 years old!
We study Erdős’s distinct distances problem under \(\ell_p\) metrics with integer \(p\). We prove that, for every \(\varepsilon>0\) and \(n\) points in \(\mathbb{R}^2\), there exists a point that spans \(\Omega(n^{6/7-\varepsilon})\) distinct distances with the other \(n-1\) points. This improves upon the previous best bound of \(\Omega(n^{4/5})\). We also characterize the sets that span an asymptotically minimal number of distinct distances under the \(\ell_1\) and \(\ell_\infty\) metrics.
I wrote an expository undergraduate thesis on the fundamental group of wild (non-semilocally simply-connected) spaces. My thesis was supervised by Michel Hébert.