SCIENTIFIC PROGRAMS AND ACTIVITIES

David Jordan
How to integrate the quantum group over a surface

I'll explain joint work with D. Ben-Zvi and A. Brochier, in which we construct a partially defined 4D ``Quantum Geometric Langlands" TFT from the quantum group U_q(g) associated to a reductive group G, and develop completely explicit computations of the QGL theory of arbitrary surfaces, using factorization homology of Francis. For once-punctured tori we recover algebras of quantum differential operators on G. For closed tori and G=GL_N, we expect to recover the double affine Hecke algebra associated to G.

The QGL theory of 3-manifolds yields knot invariants valued in modules for the DAHA, and hence is closely related to several recently conjectured knot invariants of Cherednik, Gukov, Oblomkov-Rasmussen-Shende, and Berest-Samuelson.

Manish Patnaik
Casselman-Shalika Formula for Loop Groups

The usual Casselman-Shalika formula relates the Weyl character to unramified Whittaker functions on the p-adic points of a reductive group. We will explain how to define Whittaker functions on a p-adic loop group (i.e, the group of p-adic points of an affine Kac-Moody group), present the analogue of the Casselman-Shalika formula in this context, and then sketch its proof.
Based on joint work with A. Braverman, H. Garland, and D. Kazhdan.

Sachin Gautam
Tensor isomorphism between Yangians and quantum loop algebras

The Yangian and the quantum loop algebra of a simple Lie algebra g arise naturally in the study of the rational and trigonometric solutions of the Yang--Baxter equation, respectively. These algebras are deformations of the current algebra g[s] and the loop algebra g[z,z^(-1)] respectively.
The aim of this talk is to establish an explicit relation between the finite--dimensional representation categories of these algebras, as meromorphic braided tensor
categories. The notion of meromorphic tensor categories was introduced by Y. Soibelman and finite--dimensional representations of Yangians and quantum loop algebras are among the first non--trivial examples of these.
The isomorphism between these two categories is governed by the monodromy of an abelian difference equation. Moreover, the twist relating the tensor products is a solution of an abelian version of the qKZ equations of Frenkel and Reshetikhin.
These results are part of an ongoing project, joint with V. Toledano Laredo.

Brad Hannigan-Daley
Derived equivalences of hypertoric varieties

Given a representation of a complex torus, there is a natural hyperplane arrangement A whose faces correspond to the stability conditions defining the hypertoric varieties associated to this representation. We describe ongoing work on constructing a representation of the Deligne groupoid of A on the derived categories of the hypertoric varieties corresponding to the chambers of A.

Jonathon Fisher
Cohomology rings of quiver moduli from the Harder-Narasimhan stratification

Let Q be a quiver with dimension vector d. The Harder-Narasimhan filtration on the category Rep(Q) gives an algebraic stratification of the vector space Rep(Q,d) of representations of fixed dimension. When the ground field is C, this stratification agrees with the Morse stratification coming from the norm-square of a moment map. I will give an explicit description of this stratification, and show that it leads to an inductive procedure for computing the cohomology ring of the moduli space of stable representations of Q. I will give a few worked examples, and time-permitting, I will describe some applications to Nakajima quiver varieties.

Jenna Rajchgot
Type A quiver loci and Schubert varieties

I'll describe a closed immersion from each representation space of a type A quiver with bipartite (i.e., alternating) orientation to a certain opposite Schubert cell of a partial flag variety. This "bipartite Zelevinsky map" restricts to an isomorphism from each orbit closure to a Schubert variety intersected with the above-mentioned opposite Schubert cell. For type A quivers of arbitrary orientation, I'll discuss a similar result up to some factors of general linear groups.
These identifications allow us to recover results of Bobinski and Zwara; namely we see that orbit closures of type A quivers are normal, Cohen-Macaulay, and have rational singularities. We also see that each representation space of a type A quiver admits a Frobenius splitting for which all of its orbit closures are compatibly Frobenius split.
This work is joint with Ryan Kinser.

Alex Weekes
Yangians and the affine Grassmanian

We will discuss ongoing research linking Yangians and some subvarieties of the affine Grassmannian, namely transverse slices to Schubert varieties. The geometry of these spaces hints that the representation theory of the quantizations should be very interesting, and tied to the representation theory of simple Lie algebras. We will present a summary of our work so far on these links, highlighting the case of SL(2, C).

Chris Dodd
Cycles of Algebraic D-modules in positive characteristic

I will explain some ongoing work on understanding algebraic D-moldules via their reduction to positive characteristic. I will define the p-cycle of an algebraic D-module, explain the general results of Bitoun and Van Den Bergh; and then discuss a new construction of a class of algebraic D-modules with prescribed p-cycle.

We give a diagrammatic presentation of the representation category of SLn (and its quantum version). Our main tool is an application of quantum skew Howe duality.

Daniele Rosso
The mirabolic Hecke algebra

The Iwahori-Hecke algebra of the symmetric group is the convolution algebra arising from the variety of pairs of complete flags over a finite field. Considering convolution on the space of triples of two flags and a vector we obtain the mirabolic Hecke algebra, which had originally been described by Solomon. We will see a new presentation of this algebra which shows that it is a quotient of a cyclotomic Hecke algebra. This lets us recover Siegel's results about its representations, as well as proving new 'mirabolic' analogues of classical results about the Iwahori-Hecke algebra.

Kashiwara's equivalence, saying that the category of D-modules on a variety X supported on a smooth, closed subvariety Y is equivalent to the category of D-modules on Y, is a key result in the theory of D-modules. In this talk I will explain how one can generalize Kashiwara's result to modules for deformation-quantization algebras on a conic symplectic manifold. As an illustrative application, one can use this result to calculate the additive invariants such as the K-theory and Hochschild homology of these module categories. This is based on joint work C. Dodd, K. McGerty and T. Nevins.

Tsao-Hsien Chen
Quantization in positive characteristic and Langlands duality

I will first explain a version of geometric Langlands correspondence in positive characteristic for a reductive group. Then I will explain how to use the method of quantization in positive characteristic to construct generic parts of the correspondence. (This is a joint work with Xinwen Zhu.)

Peter Samuelson
Double affine Hecke algebras and Jones polynomials

For a (reductive) group G, let O(K; G) be the ring of functions on the variety of G-representations of the fundamental group of the complement of a knot K in S^3. There is an algebra map from the spherical double affine Hecke algebra H^+(G; q=1,t=1) to O(K; G), which leads to the question "does O(K) deform to a module over H^+(G; q,t)?" We give a conjectural positive answer for G=SL_2(C) and discuss some corollaries of this conjecture involving the SL_2(C) Jones polynomials of K. (This is joint work with Yuri Berest.)

Yuri Berest
Representation homology and strong Macdonald conjectures

In the early 1980s, I. Macdonald discovered a number of highly non-trivial combinatorial identities related to a semisimplecomplex Lie algebra g. These identities were under intensive study for a decade until they were proved by I.Cherednik using representation theory of his double affine Hecke algebras.One of the key identities in the Macdonald list - the so-called constant term identity - has a natural homological interpretation: it formally follows from the fact that the Lie algebra cohomology of a truncated current Lie algebra over g is a free exterior algebra with generators of prescribed degree (depending on g). This last fact (called the strong Macdonal conjecture) was proposed by P.Hanlon and B.Feigin in the 80s and proved only recently by S. Fishel, I.Grojnowski and C. Teleman (2008).

In this talk, I will discuss analogues (in fact, generalizations) of strong Macdonald conjectures arising from homology of derived representations schemes.

Dinakar Muthiah
Double Mirkovi\'c-Vilonen Cycles and the Naito-Sagaki-Saito Crystal

The theory of Mirkov\'c-Vilonen (MV) cycles associated to a complex reductive group $G$ has proven to be a rich source of structures related to representation theory. I investigate double MV cycles, which are analogues of MV cycles in the case of an affine Kac-Moody group.

I will shortly review some aspects of the theory of MV cycles for finite-dimensional groups. The story gives rise to MV polytopes and a surprising connection with Lusztig's canonical basis. Then I will discuss double MV cycles. Here the finite-dimensional story does not naively generalize. Nonetheless, in type A, I will present a method to parameterize double MV cycles. This method gives rise to exactly the combinatorics of the Naito-Sagaki-Saito crystal. If I have time, I will discuss some related work and some open problems.

Omar Ortiz Branco
On the torus-equivariant cohomology of p-compact flag varieties

p-compact groups are the homotopy analogues of compact Lie groups. The torus-equivariant cohomology of p-compact flag varieties can be described as a quotient ring of polynomials. I will give another description of this cohomology via moment graph theory and its relation with the polynomial description, generalizing results of Goresky-Kottwitz-MacPherson from classical Schubert calculus.

Leonid Rybnikov
Cactus group and monodromy of Bethe vectors

The cactus group is the fundamental group of the real locus of the Deligne-Mumford moduli space of stable rational curves. This group appears naturally as an analog of braid group in coboundary monoidal categories; the main example of this is the category of crystals where the cactus group acts on tensor product of crystals by crystal commutors. We define an action of the cactus group on the set of Bethe vectors of the Gaudin magnet chain (for Lie algebra sl_2) and prove that this action is isomorphic to the action of cactus group on the tensor product of sl_2-crystals. We also relate this to the Berenstein-Kirillov group of piecewise linear transformations of the Gelfand-Tsetlin polytope. Some conjectures generalizing our construction will be discussed.

Yanagida Shintaro
K-theoretic AGT conjecture

I will explain the AGT conjecture and its K-theoretic analogue. The conjecture states (deformed) W-algebras should act on the equivariant cohomology groups (or K-theory) of instanton moduli spaces over CP^2. If have enough time, I will also explain combinatorical conjectures on the Whittaker vectors of deformed W-algebras.

Chia-Cheng Liu
Quantum loop algebras and the conjectural monoidal categorification of cluster algebras

Beyond the case of sl_2, the tensor structure of the category of finite-dimensional representations of a quantum loop algebra is still not very well-understood. For instance, we do not have a general factorization theorem (in terms of prime objects) for simple objects. We will explain a conjecture by Hernandez and Leclerc relating this tensor structure to the combinatorics in some cluster algebra of geometric type. Special cases of this conjecture are proved by Hernandez-Leclerc and Nakajima.