In this article, I aim to establish some essential properties of the tensor 2-product for sl(2), including: (i) functoriality of the 2-product in the second argument, (ii) definitions for the 2-product in reverse order, (iii) a functorial ‘braid’ action on products L(1)×(L(1)×L(n)), (iv) details for the relation to Ebert-Rouquier’s 2-product L(1)×U
In the mid-90’s, Varchenko discovered a relation between crystal bases, Gaudin Hamiltonians, and asymptotics of solutions to the Knizhnik-Zamolodchikov system over configuration spaces. I seek an algebraic interpretation of this relation. My goal is a better understanding of the asymptotic structure of the category of quantum group representations. Eventually, this may illuminate the new 3-manifold invariant of Sergei Gukov et al. which returns q-series having the appearance of expansions around q = 0.
This project examines the effectiveness of custom web resources offered for students studying to pass the Pre-Calculus Mastery Program that constitutes part of the Calculus I and Calculus II curriculum in the Center for Applied Mathematics at UVA.
with Sabando Alvarez, C., Li, M., Abramenko, M., Pisano, S., Morris, D., Dizaji, F., Provat, S.
Abstract:
Let W be a complex reflection group and X the De Concini-Procesi wonderful model of its reflection arrangement. Inside Etingof’s sheaf of Cherednik algebras on X we define a subsheaf generated by the Dunkl operators that extend the standard ones on the hyperplane complement. We conjecture that at t = 0 the image of the logarithmic vector fields in a natural module over this subsheaf is the sheaf of Gaudin algebras on X defined by Aguirre, extending a construction of Bonnafé and Rouquier from the hyperplane complement to X. We prove the conjecture for W = S3.
Comment:
I hope this will contribute a new approach to the study of the monodromy of Gaudin Hamiltonians, and that in turn will contribute to the conjecture of Bonnafé-Rouquier that Calogero-Moser cells associated to complex reflection groups generalize Kazhdan-Lusztig cells associated to Coxeter groups. My main tool, the KZ system on a de Concini-Procesi wonderful model, also features in my study of quantum group asymptotics.
Submitted. Abstract:
A formal framework is suggested for Leibniz’s system of monadic expression using the language of mathematical categories, and this model is compared with the Structural Representation model of expression that has been developed by various authors. Leibniz’s distinctive theses are modelled using categorical notions, including the Identity of Indiscernibles, Universal Expression, Pre-established Harmony, and the theses that every monad is characterised by its point of view and that all points of view are expressions of God. A perspective on Leibniz’s distinction between natural and arbitrary expression is offered.
The preprint is unavailable before publication according to journal policy.
arXiv | Download 20pp | Tutorial
An article documenting a system of tools and workflows that allows instructors to build and modify course websites very efficiently. These sites have the merit of being easily adapted to generate ADA-compliant “digitally accessible” content, as is now required of public universities in the US. Since content source lives in local Markdown files, this system is trivial to integrate with standard AI workflows such as Claude Code. Many “content management” features come along for free, such as batch publishing of solutions or internal linking to reference material. These are (i) not too complicated for instructors to implement, and (ii) improve courseware organization and navigation for students.
The paper also presents extremely encouraging student outcomes data. Presumably the students performed better because they could study more effectively using the site.
Tool list: Obsidian (Markdown-based document authoring, content management), Quartz (static site generator), Git (collaboration and version control), and Cloudflare Pages (build and host static sites without exposing all files). One can add a Cloudflare Worker to implement password protection. All of these tools are currently free.
with Eli Boyden
ASEE 2026 Annual Conference & Exposition
PhilArchive | The Leibniz Review
A translation of Leibniz’s Specimen Analyseos Figuratae in Elementis Geometriae (1686): ‘A Sample of Figurate Analysis in the Elements of Geometry’.
with David Jekel
The Leibniz Review 35, 45-59 (2025).
A rather technical proof that my abelian tensor 2-product construction works for 2-representations of the full algebra sl(2), not just the positive half.
This is the first construction of an operation of tensor product for higher representations of a full Lie algebra in the abelian setting.
We construct an explicit abelian model for the operation of tensor 2-product of 2-representations of sl(2)+, specifically the product of a simple 2-representation L(1) with a given abelian 2-representation V taken from the 2-category of algebras. We study the case V = L(1) in detail, and we show that the 2-product in this case recovers the expected structure. Our construction partially verifies a conjecture of Rouquier from 2008.
BPhil Thesis, as submitted. Abstract:
Leibniz’s Analysis Situs was a programme to reform geometry. He wanted to provide a new characteristic, or artificial technical language, for geometric quality understood by some kind of contrast with geometric quantity. This thesis is a study of Leibniz’s concept of geometric quantity aimed to help us understand the Analysis Situs.
The few works of critical scholarship on the Analysis Situs employ a naïve concept of geometric quantity as a simple distance relation predicable of abstract mathematical figures and their parts. It is supposed that Leibniz thought that quantities and qualities are mutually independent and jointly exhaustive classes of attributes of abstract figures, and it is supposed that Leibniz’s plan with the Analysis Situs was to develop a characteristic for geometric quality in contradistinction to quantity. There is no indication that Leibniz was moving in the direction of projective geometry or modern topology, which in their own ways eliminate geometric quantity, so these scholars conclude that his project was a failure.
I offer a novel analysis of the logic of geometric quantity with key features: (1) Geometric quantity is a species of a more general metaphysical concept of quantity that is ‘the affection of a composite relative to its parts’. (2) Geometric quantity is aimed at bodies or figures with indeterminate place and extension in an abstract homogeneous continuum, and place and extension are abstracted from the determinate ‘situs’ and ‘complexion’ of a prior phenomenal reality that I call the ‘perceptual field’. (3) Since geometric quantity requires the abstraction and the concrete perceptual composite, it is not attributable to abstract figures.
This account shows Leibniz’s concept of geometric quality to be close to our concept of abstract structure, where identity of quality (similarity) is just isomorphism. The separation between the abstraction and the perceptual field combines with the dependence of quantity on the perceptual field to yield a very advanced notion of mathematical space. The account also shows that critics were wrong to think geometric quantity is an attribute of an abstract figure and were wrong to think Leibniz aimed for a geometry without quantity.
Essay in the philosophy of science.
Essay in the philosophy of science.
Cambridge Part III ‘Essay’ for the MASt degree, written under Jacob Rasmussen.
This paper is the result of a summer RTG at UC Berkeley, guided by Dan Cristofaro-Gardiner. It developed into my undergraduate honors thesis Embedded Contact Homology and Symplectic Capacities of Toric Domains.
with Landry, M., Tsukerman, E.
Involve, a Journal of Mathematics 8-4, 665-76 (2015).
I made a parallelized 3D computational model of neutral beam injection into non-symmetric fusion confinement devices (stellarators) and resultant plasma heating.
The code is routinely used to study neutral beam physics in Wendelstein 7-X (a stellarator), where it has been validated against some experiments. It is used to model stellarator confinement more generally. Results have been compared to experimental data from ASDEX-U (a tokomak) as well.
Cf. e.g. Gyro orbit simulations and references therein.
APS-DPP 2012: poster
Subsequent EPS posters:
EPS 2015: poster | 4p statement
EPS 2016: poster | 4p statement
Github page as part of the STELLOPT suite.
with Samuel Lazerson
Plasma Physics and Controlled Fusion 56-9, 095019 (2014).
Undergrad research project led to a paper. I had the main idea during a course on differential geometry. We show that the volume of a “tube”, suitably defined with symmetric cross-section, does not depend on the curve traced by the centroid of the cross-section, provided the curvature is limited such that the boundary does not kink. A similar result holds for the surface area, but the required symmetry condition is stricter.
It turned out this fact already had some history. In 1939 Weyl wrote “On the Volume of Tubes” which describes a relation between the volume of a tube in some manifold, of fixed radius around a 2D surface, to the area of this surface. In 1969 Goodman discussed the extension we had in mind for more general cross-sections, and he noticed how volume and surface area behave differently, but he considered only tubes in 3D space. Finally in 2000, Gray-Miquel obtained all the results that we did. So, our paper supplies a different proof of a known theorem. Our method is essentially an application of Stokes’ Theorem.
with Adams, C., Lovett, S.
Involve, a Journal of Mathematics 8-5, 771-85 (2015).
Designed, obtained, tested, implemented, calibrated custom signal conditioning electronics for plasma diagnostics: arrays of integrating amplifiers for magnetic field probes; arrays of transimpedance amplifiers for photodiode cameras. Analysis on resulting data. (Paper in preparation by the lab; APS-DPP posters available here for 2010 and 2011.)
with Craig, D., Blasing, D., Cartolano, M., Adams, C.
Mechanics project of fall 2010 eventually became a short AJP article.
with Blasing, D., Whitney, H.
American Journal of Physics 81.9, 682-7 (2013).