Research

My research lies primarily in the areas of representation theory and quantum algebra. More specifically, I study tensor categories and representation theory of quantum groups. I am also interested in their applications to the study of topological quantum field theory, topological quantum computing, and invariants of links and manifolds.

My current projects include:

Papers and Preprints

  • The Drinfeld Center of the Generic Temperley–Lieb Category. Preprint, 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

  • Crystal Spiders for Rank 2 Lie Algebras.

    We give diagrammatic monoidal presentations of the categories of \(\mathfrak g\)-crystals, where \(\mathfrak{g}\) is a rank 2 Lie algebra. We study those categories, their Jones-Wenzl projectors, cactus actions, etc.

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.