Welcome to my homepage. I am a fourth year student of Pavel Etingof, studying representation theory and tensor categories. You can learn a few things about me here.

## Research Interests

I study various algebraic systems which lie between representation theory and physics. More precisely, I study tensor categories in their various flavors (fusion, braided, symmetric, ...), and also quantization problems involving quantum groups and their differential geometry, specifically via the algebra of quantum differential operators. I am currently trying to develop a braided generalization of quiver varieties.

## Research Publications

Here is a chronological list of my work to date, with links to the arXiv versions of each paper: Click here for a slightly more detailed infromal description of each work.**Lower central series of free algebras in a symmetric tensor category**, with Asilata Bapat. arXiv:1001.1375.**Quantum symmetric pairs and representations of double affine Hecke algebras of type C^C_n**, with Xiaoguang Ma. arXiv:0908.3013, to appear in Selecta Mathematica.**New results on the lower central series quotients of a free associative algebra**, with Noah Arbesfeld. arXiv:0908.3013, published in Journal of Algebra, 2009.**On the classification of certain fusion categories**, with Eric Larson. arXiv:0812.1603, published in Journal of Noncommutative Geometry. 2008**Quantum D-modules, elliptic braid groups, and double affine Hecke algebras**. 2008 arXiv:0805.2766, published in International Math Research Notes.

## Teaching

I have had several opportunities to teach at MIT. I have:- TA'ed Project Lab in Mathematics, 18.821. In this class, undergraduates choose from a list of open-ended problems in pure and applied mathematics, and try their hands at developing solutions. My job was to oversee their work and help them edit their papers and prepare two talks.
- TA'ed Multivariable Calculus 18.02. This is MIT's standard introduction to vector calculus. I led weekly recitations, held office hours, and graded exams.
- Taught Calculus III - Multivariable Calculus - at the Interphase Program. This program is run by the Office of Minority Education at MIT, and is an intensive introduction to the MIT curriculum. I had about 24 students, and the program lasted approximately seven weeks. This was far and away the most rewarding teaching experience I have had to date! My students were very motivated and clever, and I hope several of them will become math majors (I guess I'm supposed to say "Course 18" here...).
- Mentored high school students at the Research Science Institute program, which is hosted by MIT. One of my students, Sana Raoof, won a Grand Prize at the International Science and Engineering Fair for her project. The next year, both my students were finalists in the Intel Science Talent Search, with Eric Larson finishing in first place, and Noah Arbesfeld in 6th nationally. Eric also finished second nationally in the Siemens Science competition. More importantly, they and my other students learned a great deal and I think they have had fun doing it. I am very proud of them all!
- Mentored an MIT undergraduate, Asilata Bapat, in the summer program SPUR, and then the fall program UROP in her research. She received a Hartley Rogers prize for her work this summer, and is working this fall to extend her results to more general settings.
- Taught "Math Circle", a program for young children (ages 6-9) in the Boston area which introduces mathematical concepts, puzzles, and games at an early age.
- Tutored students weekly in mathematics at the "Student Success Jobs Program" at Brigham and Women's Hospital, in Boston.

## My CV

Click here. Details about travel funding are included because MIT's GSC request this sometimes when reqeusting travel grants.

In the course of my work, I had to write two computer algebra packages: The first, written in MAGMA Computational Algebra programming language, constructs many of the vector spaces comprising the W_n modules from the papers with Noah and Asilata above, and can be used to hunt for identities, and also to compute partial Hilbert series. Anyone interested in the combinatorial structure of lower central series of free (or at least graded) algebras, please drop me an email and I'll send the code along. (I only don't post them online because there are many different files which would require a few words of explanation anyways). The second package allows for the construction of algebras with FRT type relations as FPAlg's (MAGMA for finitely presented algebras). These are very useful for exploring centers, characters, flatness properties, etc, of these algebras in a concrete way. They are written primarily for the case sl_2 since most interesting computations are already interesting there. They are probably inferior to any other package a professional has written, but if you don't have an alternative, email me and I'll share mine.