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.

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