I am interested in the broad area of representation theory and tensor categories, which sits at the intersection of algebra, topology, combinatorics, and mathematical physics. More specifically, I have recently been researching questions about representation categories of quantum groups, crystal bases, and related (diagrammatically presented) 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.
Problems I am actively researching at the moment include:
Web categories arising from crystal limits of quantum group representations.
Drinfeld centers of tensor categories arising from quantum group representations.
Papers and Preprints
The Drinfeld Center of the Generic Temperley–Lieb Category. Submitted, 2026. arXiv:2603.28970
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.
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
A Crystal Spider for \(\mathfrak{sl}_3\).
We give a diagrammatic monoidal presentation of the category of \(\mathfrak{sl}_3\)-crystals. We study this category, its Jones-Wenzl projectors, and cactus group action.
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 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.