Abstract:  In work of Mathieu and Fernando the modules of the enveloping algebra with finite dimensional weight spaces are understood. These conditions can be translated into a support condition for the associated graded or some singular support condition for some sheaves on G/B. This singular support is given by taking the union of W copies of the conditions for category O.

In work of Kazhdan and Laumon the construct a category by glueing W copies of the category of perverse sheaves on G/U. This category was studied by Bezrukavnikov, Polishchuk and Morton-Ferguson. In particular some subcategory known as Kazhdan-Laumon category O was related to the representation theory of the small quantum group u_q.

In joint work with Morton-Ferguson we relate the Kazhdan-Laumon category O to some subcategory of weight modules. This connection should explain the relation to the representation theory of u_q.

In this talk I will discuss Kazhdan-Laumon’s category O and its connection to weight modules and time permitting the connection to u_q via the joint work with Bezrukavnikov, McBreen and Yun and the the geometry of affine Springer fibers

Abstract: Springer fibers are subvarieties of the flag variety defined as the fixed-point loci of nilpotent elements. They play a central role in geometric representation theory: their cohomology and Grothendieck groups provide geometric constructions of modules for Weyl groups and Hecke algebras.

Asymptotic Hecke algebras, introduced by Lusztig, can be viewed as certain limits of Hecke algebras $H_q$ that captures much of the structure and representation theory of $H_q$. They also encode the structure of Kazhdan–Lusztig cells, a fundamental combinatorial feature of Kazhdan–Lusztig theory.

In this talk, we discuss realizations of asymptotic Hecke algebras in terms of Springer fibers. If time permits, we will also explain discretizations of Springer fibers and their connections to cell modules and modular representation theory.

Abstract: In joint work with Savage, we introduced two categories, the

spin Brauer category and the quantum spin Brauer category. One idea of

the construction is to provide an interpolating category for

representations of (quantum) spin groups. This talk will give an

overview of the construction, then focus on discussing natural areas

of future exploration.

Abstract: The G-skein module for the 3-torus is finite dimensional by a result of

Gunningham-Jordan-Safronov. For G= GL_N and SL_N, there is a

lovely combinatorial formula for these dimensions in terms of the

partitions of N. Our formula arises from studying skeins on the 2-

torus via the representation theory of the double affine Hecke

algebra (DAHA) of type A and its connection to quantum D-modules.

Using the DAHA, we also prove that all tangles in the relative N-point

skein algebra are in fact equivalent to linear combinations of braids,

modulo skein relations. I will discuss these results. This is joint work

with Sam Gunningham and David Jordan.

Abstract: Khovanov homology is a very powerful modern topological invariant of knot and links, but it is quite hard to compute. In particular, Khovanov homology of torus knots is not known. I will describe a spectral sequence computing stable Khovanov homology of torus knots, in terms of an explicit Koszul complex. The main tool is the link-splitting deformation, or y-ification, of link homology; in the y-ified context, the relevant spectral sequence collapses and we explicitly compute the y-ified  stable Khovanov  homology of torus knots. I will also outline a connection to Hilbert schemes of points and centralizer schemes. 

This is a joint work with William Ballinger, Matt Hogancamp and Joshua Wang.

Abstract: Modular functors are collections of vector bundles with flat connections on (twisted) moduli spaces of curves, individually known as conformal blocks, that satisfy strong compatibility conditions with respect to natural maps between these moduli spaces. Such structures arise naturally in the representation theory of affine Lie algebras and quantum groups, where the conformal blocks are known to be semisimple.

Recently, Hodge structures on the genus-0 conformal blocks associated to affine Lie algebras have been studied by Belkale, Fakhruddin, and Mukhopadhyay through a motivic construction. In particular, they computed genus-0 Hodge numbers for $sl_n$.

I will discuss an axiomatic proof of the existence and uniqueness of such Hodge structures and of the semisimplicity of conformal blocks, for any modular functor (i.e. any modular fusion category). If the flat bundles of conformal blocks were rigid and semisimple, a result of Simpson in non-Abelian Hodge theory would imply that they support Hodge structures. However, this is not the case in general. I will explain how a different form of rigidity for modular fusion categories—Ocneanu rigidity—can be used, together with non-Abelian Hodge theory, to tackle these questions. Finally, I will discuss an application to the computation of Hodge numbers for $sl_2$ modular functors of odd level in higher genus and how these numbers are part of (new) cohomological field theories (CohFTs).

Abstract:  In this talk I will show that the quantized K-theoretic Coulomb branch of a 3d N=4 quiver gauge theory without 1-cycles is isomorphic to a quantum cluster variety. The former is a non-commutative algebra, defined by Braverman, Finkelberg, and Nakajima as an equivariant K-theory of a certain variety with convolution. It is determined by a reductive group G and its representation V, and is called "a quiver theory" when G is a product of general linear groups, and V is a tensor product of their fundamental, co-fundamental, and adjoint representations. The "no 1-cycles" condition excludes the adjoint ones. In that latter case, I will construct a cluster variety, isomorphic to the quantized Coulomb branch. This is a joint work with Gus Schrader, based on our preprint from several years ago.

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