Mathematics

I’m a mathematician, philosopher, writer... maybe inventor?

In spring 2023 I finished a PhD in mathematics from UCLA, where I studied under Raphaёl Rouquier. Now I’m a professor at UVA.

Math

I work on algebraic machinery that is designed to classify the smooth-topological phenomena discovered with PDE-based manifold invariants (i.e., gauge theory) that emerged originally from quantum field theory and string theory. These phenomena were discovered using analysis, but in order to understand them we must use algebra.

My dissertation was in ‘higher quantum algebra’. I found a model construction of an operation of tensor product for higher representations. This gives a solution to a conjecture of Rouquier from 2008 which hypothesized the existence of an abelian tensor 2-product. I hope to provide a totally new construction of Khovanov homology using this operation by extending it to more general settings. The project is part of a program to build a non-trivial topological quantum field theory in dimension 4.

In a second project, I am developing a theory of quantum group asymptotics for |q| ≪ 1. I hope it will illuminate Sergei Gukov’s recently proposed q-series 3-manifold invariant. Quantum groups are interesting here because of their role in the Reshetikhin-Turaev invariant that Gukov’s construction recovers at a root of unity. My approach involves Varchenko’s hypergeometric functions, as well as an analogy with Cherednik algebras (for |t| ≪ 1) from the world of complex reflection groups.

A third project has grown out of the second. Bonnafé-Rouquier conjectured a generalization of the Kazhdan-Lusztig cells defined for Coxeter groups to ‘Calogero-Moser cells’ defined for complex reflection groups. The CM and KL cells have been shown to coincide in type A and in rank 2. It appears possible to study these cells using Varchenko’s method: a KZ-type monodromy on the complement of the reflection hyperplanes is analyzed in ‘asymptotic zones’ (resolving crossings with a de Concini-Procesi model), and the limit as t → 0 is related to the CM cells arising at t = 0. The KZ system involves Gaudin Hamiltonians that feature in Cherednik algebras, but Gaudin Hamiltonians are also related to the q = 0 ‘crystallization’ phenomenon in quantum groups. This explains the analogy of the previous paragraph.

Philosophy, etc.

I also read and write philosophy. In one project I aim to portray and clarify the nature of mathematics as a living discipline. I wish to cultivate a new approach to the philosophy of math that starts from an analogy with music, languages, architecture, and other rule-systems we indwell and within which we build or compose works of art, and that we also apply to create technologies with extrinsic purposes. My ideas have applications to ‘criticism’ and ‘pedagogy’ of math, and I write about these, but the ideas do not consist in the applications.

In addition, I have some historical philosophy research going: a few projects on Leibniz, some old work on Berkeley, and some essays on Wittgenstein.

I think a lot about symmetry, duality, and layers. I think about symmetry as a matter of mathematics, of perception, of aesthetics, and of fundamental physics, and (more than I would like) as a matter of politics. I maintain another pet interest in the concept of ‘expertise’ as a matter of epistemology (and politics).

My interest in Leibniz stems from an underlying fascination with language and space from philosophical and mathematical points of view. My perspective compares to Couturat’s, although his writing about mathematics was mainly restricted to ‘logico-foundational’ topics, and he did not share my concern for geometry.

I study and practice foreign languages as a hobby, and currently have some facility with Russian, German, Latin, Hebrew, Spanish, and Italian. Having tried various methods and tools, I finally settled happily with the LingQ software invented by Steve Kaufmann. It is the preferred tool for many polyglots. Perhaps in a similar vein I enjoyed learning to read electronic circuit diagrams during childhood. (Building the circuits was a hobby of a different nature.) In the past I trained in classical music performance, on piano and trumpet; that has given way to concert and radio music appreciation in the last decade. I count music as a linguistic kind of endeavor, and indeed our musical notation exemplifies a well-adapted symbolic algebra for Western music, although it is not without shortcomings.