This is not my most recent research statement! It has been deprecated since 2008-09-21. Please see my current research summary page.
I like to think about interesting examples of 2-, 3- and 4-categories with duals. Another way of saying this is that I like to do research that lets me draw pictures!

Khovanov homology and homological TQFTs

Publications »
Fixing the functoriality of Khovanov homology with David Clark and Kevin Walker. Accepted by Geometry and Topology, to appear.
On Khovanov's cobordism theory for su_3 knot homology with Ari Nieh, Journal of Knot Theory and its Ramifications Vol. 17, No. 9 (2008).
The Karoubi Envelope and Lee's Degeneration of Khovanov Homology with Dror Bar-Natan, Algebraic & Geometric Topology 6 (2006) 1459-1469.
Blob homology
(joint with Kevin Walker, see his Oberwolfach abstract)

We define the "blob complex" B_*(M, C) associated to an n-manifold M and a (suitable) n-category C. This is a simultaneous generalisation of two interesting gadgets. When n=1, M = S^1 and C is an algebra, the homology of the blob complex is the Hochschild homology of the algebra. On the other hand, the zero-th homology of the blob complex is the usual skein module of "pictures from C drawn on M". In this sense the blob complex is a "derived" version of a TQFT.

We can prove several interesting properties of the blob complex. It's a functorial construction, and diffeomorphisms of the manifold act on the complex. Moreover, there's an action of chains of diffeomorphisms as well. Thus for example on the torus we get not only an action of the mapping class group, but also a compatible action of rotations along rational slopes. There appears to be a good gluing formula, expressed in terms of A_∞ bimodules. (The details here are still being worked out.)

We hope to apply blob homology to tight contact structures (for n=3) and Khovanov homology (for n=4). In both theories exact triangles play an important role. These exact triangles don't interact well with the gluing structure of the usual TQFTs, however. One of our motivations for considering blob homology is to work around these difficulties.

The 2-point and 4-point Khovanov categories
(joint with David Clark and Kevin Walker)

Fusion categories and planar algebras

Publications »
Skein theory for the D_2n planar algebras with Emily Peters and Noah Snyder. Submitted to Journal of Pure and Applied Algebra.
Knot polynomial identities and quantum group coincidences
(joint with Emily Peters and Noah Snyder)
Making use of the skein theory of the D_2n planar algebras, we prove a series of new identities relating quantum knot invariants (often relating different groups, and different roots of unity). Further, we make use of the D_2n planar algebras to explain SO(3)-SO(n) level-rank duality. Armed with this (along with Kirby-Melvin symmetry and a few coincidences of small quantum groups), we give separate proofs of these knot invariant identities, by lifting them to equivalences of braided tensor categories.
Atlas of subfactors (wiki »)
(joint with Emily Peters and Noah Snyder)
We aim to implement algorithms to enumerate and construct subfactors of small index and rank (and also singly generated fusion categories). There are four principal steps:
  1. Enumerate principal graphs up to a given index and rank. (done: examples »)
  2. Use the methods of annular and quadratic tangles to eliminate impossible graphs, or impossible graph pairs. (partially implemented)
  3. Construct subfactors (or prove they don't exist), using bipartite graph planar algebras. (getting started!)
  4. Identify subfactors in our list coming from known constructions of small subfactors. (very little done)

Diagrammatic representation theory

Publications »
A Diagrammatic Category for the Representation Theory of U_q(sl_n). (UC Berkeley Ph.D. thesis, 2007, synopsis).
Representations of U_q(sl_n)

The representation theory of a quantum group forms a planar algebra (equivalently, a spider or pivotal category). For U_q(sl_2) and U_q(sl_3), there are nice combinatorial models (that is, finite presentations by generators and relations) of the planar algebras. These are the Temperley-Lieb algebra, and Kuperberg's A_2 spider, respectively.

I've made some progress extending these ideas to treat all U_q(sl_n). It's easy now to find a set of generators for the planar algebra. The difficulty is in finding all the relations amongst them. We can understand these relations by thinking about the inclusion of SU(n) inside SU(n+1), and thence the way that representations of U_q(sl_{n+1}) break up as representations of U_q(sl_n). This theory of "branching" and "Gelfand-Tsetlin bases" can be described combinatorially in terms of the planar algebra, and using this we can lift relations from one level to the next. My Berkeley Ph.D. thesis (see below) gives the details. It remains a conjecture that the relations I describe are complete.

Kashaev-Reshetikhin knot invariants

With Noah Snyder, I'm working on some computations, and a paper, about the Kashaev-Reshetikhin knot invariants.

We first explain details from the papers of Kashaev and Reshetikhin to give a fully rigorous construction of their new knot invariants. This invariant is a function on the space Hom(G_m(K), SL_2)/SL_2 where G_m(K) is the generalized knot group, and the action of SL_2 is by conjugation. We prove several basic results, for example, that the value of the function on the trivial point is |J_m(\zeta_m)|^2.

Further, we give the first computations of this invariant for nontrivial knots, based on a Mathematica package we've written. For several small 2-bridge knots we compute explicitly the entire Kashaev-Reshetikhin knot invariant at a third root of unity and at a fifth root of unity. We're planning to compute the knot invariants evaluated at the finite volume hyperbolic point for a larger collection of 2-bridge knots (again at a third and fifth root of unity).