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.

• Hall algebras and the Fukaya category, I
(with B. Cooper)
(Submitted.) (63 pages.)

This paper is intended to be the first in a series studying the Fukaya Hall algebra of a surface $S$, which is defined as the derived Hall algebra of the Fukaya category of $S$. For most surfaces, we give an explicit presentation of an algebra which surjects onto the Fukaya Hall algebra, and conjecture this is an isomorphism.

We also follow a hint from Kontsevich's homological mirror symmetry, combined with the paper with Morton below, which together suggest that the Hall algebra of the Fukaya category of the torus may be related to the skein algebra of the torus. We show that such a relationship actually holds for surfaces in general by showing that as elements of the Hall algebra, (certain) curves satisfy a graded version of the HOMFLYPT skein relation.

• The HOMFLYPT skein algebra of the torus and the elliptic Hall algebra.
(with H. Morton)
(Duke, 166 (2017) no. 5, 805-854.)

In the early 90's, Turaev showed that the $q \to 1$ limit of the skein algebra $Sk_q(S)$ of a surface determines Goldman's Poisson structure on the $\mathrm{GL}_n$ character variety of $S$, and he also asked for a description of the algebra structure for general $q$. In this paper we provide an explicit answer for the torus, and show that in this case, the skein algebra is isomorphic to (a specialization of) the elliptic Hall algebra $\mathcal E$.

One rather mysterious corollary is that every knot produces a module over the Hall algebra of an elliptic curve. We show that for the unknot, this module is a "double" of a module constructed geometrically by Schiffmann and Vasserot using convolution operators on equivariant $K$-theory of Hilbert schemes of points on $\mathbb C^2$.

• The elliptic Hall algebra and the deformed Khovanov Heisenberg category.
(with S. Cautis, A. Lauda, A. Licata, J. Sussan)
(Submitted.) (49 pages.)

The Heisenberg category $H$ was defined by Khovanov (and deformed by Licata-Savage) with the goal of realizing the Heisenberg algebra as the the $K$-theory of $H$. The trace (or Hochschild homology) $Tr(H)$ is also an algebra, and we prove an explicit description of it. A corollary is that it is isomorphic to (a specialization of) the elliptic Hall algebra, and in particular, it contains the Heisenberg algebra, but is much larger.

A key step in this proof is the construction of an algebra structure on the sum $\oplus_n \mathrm{HH}_0(\dot H_n)$ of the Hochschild homology of the affine Hecke algebras, which turns out to have a surprisingly simple description coming from the skein algebra $Sk_q(T^2)$. Finally, we prove that the action of $\mathcal E$ on symmetric functions arises in this context as the trace of the categorical representation of $H$ on modules over Hecke algebras.

• Factorization homology and the elliptic Hall algebra
(with D. Jordan and H. Morton)
(In preparation.)

The $GL_n$ character variety of the torus can be constructed using quasi-Hamiltonian reduction. This statement is expected to quantize: in particular, an algebra $D_q(GL_n)$ is conjectured to surject onto the $GL_n$ double affine Hecke algebra.

In this paper we (expect to) prove the $n=\infty$ version of this conjecture, where the DAHAs are replaced by their limit, the elliptic Hall algebra. To do this, we construct a new, skein-theoretic model of the DAHA, and use this to define a surjective map from the skein algebra of the punctured torus to the elliptic Hall algebra. We then compose this with a map from the algebra $D_q(GL_\infty)$ to the skein algebra of the punctured torus, which comes from earlier work of Ben-Zvi, Brochier, and Jordan.

### 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.

• Double affine Hecke algebras and generalized Jones polynomials
(with Y. Berest)
(Compositio 152 (2016), no. 7, 1333-1384.)

In this paper we study modules over the DAHA at $t=1$ associated to knots via the skein module construction. We conjecture that there is a canonical deformation of this module structure to arbitrary $t$, and prove this in several examples. The conjecture implies some integrality properties and recursion relations for the Jones polynomials, which we confirm independently using theorems of Habiro, and of Garoufalidis and Le.

The deformation conjectured above is constructed using certain divided-difference operators. For the unknot, this deformation is well-known and is responsible for the existence of Macdonald polynomials. However, for nontrivial knots this deformation is quite delicate - it can be viewed as a (conjectural) nonobvious symmetry of skein modules of knot complements.

• Affine cubic surfaces and character varieties of knots
(with Y. Berest)
(Journal of Algebra, to appear, 2017), (30 pages.)

In this paper we study the (commutative) $q=1$ case of our conjecture above, which reduces to a question about character varieties of knot complements. We show our conjecture is implied by a conjecture of Brumfiel and Hilden, and confirm both conjectures for torus knots, 2-bridge knots, and connect sums of such.

• Iterated torus knots and double affine Hecke algebras
(IMRN to appear, 2017) (29 pages)

This paper has two main goals (both in the $SL_2$ case). First, we use a modification of the skein module construction to associate a module over the DAHA $H_{q,t}$ to any knot (and for any $q,t$). Second, we use the DAHA to construct $q,t$ polynomials associated to iterated cables of the unknot, and use a cabling formula to show these specialize to the Jones polynomials of the knot. This generalizes a similar construction by Cherednik for torus knots.

• Character algebras of decorated SL_2(C)-local systems
(with G. Muller)
(Algebr. Geom. Topol. 13 (2013) no. 4, 2429-2469.)

Bullock, Przytycki, and Sikora showed that the $q=-1$ specialization of the Kauffman bracket skein module of a 3-manifold is the ring of functions on the $SL_2$ character variety of the manifold. In this paper, we generalize this construction for surfaces with boundary by allowing arcs in the surface which end on marked points on the boundary. We show that this parameterizes the moduli space of $SL_2$-local systems on the surface together with a choice of section at each marked point.

### Earlier Work

• Rational Cherednik alegbras and quasi-invariants of complex reflection groups
(with Y. Berest)
(Mathematical aspects of quantization, Contemp. Math. 583, AMS, 2012.)

This is a survey paper about rational Cherednik algebras associated to complex reflection groups, which are degenerations of the double affine Hecke algebras mentioned above. We also discuss the space of quasi-invariants, which are certain interesting modules over these algebras.

One natural question in geometric group theory is "which groups act properly discontinuously on a non-positively curved metric space?" Groups of the form $F_2\rtimes \mathbb Z$ are known to have such an action (Brady), but some groups of the form $F_3\rtimes \mathbb Z$ do not (Gersten). In this paper we construct such an action for a certain class of groups of the form $F_3\rtimes \mathbb Z$.