## Higher representation theory

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.

arXiv | Download 60pp | Journal of Algebra

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.

*with Laurent Vera*

We show that the 2-product of L(1) and L(n) has the expected structure. We might show that the n-fold iterated product of L(1) has the expected structure.

To show:

- Functoriality of the 2-product in the second argument

- Definitions for the 2-product in reverse order

- A functorial ‘braid’ action on products L(1)x(L(1)xL(n))

## Quantum algebra

Bonnafé-Rouquier have a method of extracting Gaudin algebras from Cherednik algebras. I interpret these algebras sheaf-theoretically, using Etingof’s generalized definition of the latter, and extend the BR method to the de Concini-Procesi model of the complement of a hyperplane arrangement arising from a complex reflection group.

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 model, also features in my study of quantum group asymptotics.

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.

## Geometry

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*, which is available by email request to anyone wanting a fast and accessible introduction to ECH and its capacities. (But most of these people should read Hutchings’s notes instead.)

*with Landry, M., Tsukerman, E.Involve, a Journal of Mathematics 8-4, 665-76 (2015).*

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 more strict. It turned out this theorem 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 the divergence between the results for volume and for surface area, 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).*

## Physics

I created 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 therin.

Subsequent EPS publications:

EPS 2015: poster | 4p statement

EPS 2016: poster | 4p statement

Github page as part of the STELLOPT suite.

*with Samuel LazersonPlasma Physics and Controlled Fusion 56-9, 095019 (2014).*

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).*