My research mainly focuses on interactions between representation theory and low dimensional topology. For me, the meeting point of these two fields is the character variety of a surface or 3-manifold, which parameterizes representations of the fundamental group into a fixed Lie group $G.$

This variety is a powerful invariant of a 3-manifold which has been studied extensively in topology. On the other hand, the ring of functions on the character variety of a surface has a natural noncommutative deformation (or `quantization'), and the representation theory of this deformation is very rich but still relatively unexplored.

Hall algebras

The Hall algebra of an abelian category $\mathcal A$ is an algebra whose multiplication "counts extensions in $\mathcal A$." Hall algebras tend to have very rich representation theory; as a recent example, Burban and Schiffmann gave a beautiful description of $\mathcal E$, the Hall algebra of the category of sheaves over an elliptic curve. This algebra has become a central object in a broad and developing framework involving Hilbert schemes, Khovanov-Roszansky knot homology, (double affine) Hecke algebras, positivity conjectures in algebraic combinatorics, the AGT conjectures in mathematical physics, and more.

The works below expand the above framework in several ways. The first initiates the study of the Hall algebra of Fukaya categories of surfaces. The next three show that $\mathcal E$ is related to a quantization of the character variety of the torus, to the Hochschild homology of a monoidal category defined using Hecke algebras, and also to the factorization homology of the punctured torus.

Hecke algebras and knot invariants

One construction of quantizations of $SL_2$ character varieties of surfaces uses the Kauffman bracket skein module, which associates algebras to surfaces and modules to 3-manifolds with boundary. The skein module of a knot complement is a knot invariant which determines, in particular, the Jones polynomial of the knot.

Cherednik's double affine Hecke algebra is an algebra $H_{q,t}$ depending on parameters $q,t$ that controls Macdonald polynomials. Work of Frohman and Gelca shows that the Kauffman bracket skein algebra of the torus is $H_{q,t=1}$, which shows each knot provides module over the DAHA when $t=1$. Very little is known about these modules, except in some small examples.

Earlier Work