Noncommutative Algebra Seminar

The transfer conjecture of Langlands-Shelstad (which is now a theorem due to Waldspurger and Ngo) asserts that for every function $f$ on a p-adic group $G$ there is a function $f^H$ on its endoscopic group $H$ such that $f$ and $f^H$ have "matching orbital integrals".

The famous "fundamental lemma", which was conjectured by Langlands and recently completed by Ngo, describes function $f^H$ in the case when $G$ is unramified and $f$ is bi-invariant under a hyperspecial subgroup of $G$.

Though for a general function $f$ one does not expect to construct $f^H$ explicitly, it would be desirable to have a greater supply of functions $f$ for which $f^H$ is known. In mid 90'th Kottwitz gave a conjectural description of $f^H$ in the case when $f$ is an inflation of the character of a Deligne-Lusztig representation.

In our joint work with David Kazhdan we prove this conjecture (under some mild restriction on the residue characteristic).

In the first part of my talk I will give an introduction to the theory of endoscopy, formulate the transfer conjecture and try to indicate its significance.

In the second part of my talk I will describe Kottwitz' conjecture and outline its proof.

Fusion categories are a certain type of semisimple abelian categories equipped with a tensor product. They arise naturally as representation categories of finite dimensional Hopf algebras and in many other areas of mathematics. Fusion categories can be considered as the categorical notion of a ring, (or "categorification" of a ring). In this language, module categories are the categorification of the notion of a module.

Tambara-Yamagami (TY) fusion categories are fusion categories whose set of simple objects contains a finite group and one additional object. Tambara has classified all equivalences between TY categories and categories of the form Rep-H, where H is a finite dimensional semisimple Hopf algebra. In the language of module categories, he classified all module categories of rank 1 over TY categories.

In this talk we will discuss a generalization of Tambara's work by exhibiting a classification of all module categories over TY categories. We will show that up to equivalence there are two types of module categories: the first arises from representations of certain algebras of specific form inside the Tambara Yamagami categories, and the second arises from module categories of rank 1 over another TY categories (these were classified by Tambara).

The goal of my talk is to construct a functor from the category of representations of an affine Kac-Moody algebra to a certain category of D-modules.

To this end I shall consider some non-highest weight "vacuum" modules of the Kac-Moody algebra (in a sense these modules are as close to highest-weight modules as possible). I shall construct an embedding of certain rings of differential operators into endomorphisms of above modules, then the required functor is basically the functor of coinvariants.

The embedding of differential operators into the endomorphism rings is obtained via free field realization of the modules as irregular Wakimoto modules.

In the simplest case the above functor gives rise to DMT connection.

I shall also explain how the quasi-classical limit of this story is related to isomonodromic deformation.

I will begin with a review of Atiyah's notion of a topological quantum field theory. I will then sketch the definition of a more elaborate structure, called an extended topological quantum field theory. In some sense, the notion of an extended field theory is much simpler than the more classical notion. This idea was articulated by John Baez and James Dolan, who gave a conjectural classification of extended theories (the "Baez-Dolan cobordism hypothesis"). I will explain what the cobordism hypothesis says, and (time permitting) describe a few applications.

Cherednik and Suzuki introduced infinite periodic tableaux on skew diagrams and used it to describe those irreducible modules for the double affine Hecke algebra and rational Cherednik algebra that are diagonalizable with respect to a certain commutative subalgebra.

In this talk we'll see how to do something similar for the standard and irreducible modules for rational Cherednik algebras attached to the family G(r,p,n) of complex reflection groups. Applications include the construction of some new finite dimensional rational Cherednik algebra modules, an answer to a question of Iain Gordon on the "right" ordering for the highest weight structure of category O, and---we expect---a classification of the unitary irreducible modules in category O.

This is joint work with Charles Dunkl and Emanuel Stoica, and a portion of the talk is based on the paper "Unitary representations of rational Cherednik algebras" by Pavel Etingof and Emanuel Stoica.

In recent years, categorification has generated a great deal of interest in knot theory, following the development of Khovanov's categorification of the Jones polynomial and the appearance of knot Floer homology. On the hand, similar techniques have been used in geometric representation theory for decades, in particular in Kazhdan-Lusztig theory and character sheaves. I'll describe recent work of myself and Geordie Williamson which shows how one particular homological knot invariant, the HOMFLYPT homology of Khovanov and Rozansky can be constructed in terms of a geometric categorifcation of the Hecke algebra.

I will talk on my joint work with A.Davydov, M.Mueger and D.Nikshych. The classical Witt group of a field K is a quotient of monoid of quadratic forms over K (with direct sum as an operation) by the hyperbolic quadratic forms. In this talk I will describe a generalization of this group where quadratic forms are replaced by certain braided tensor categories. Namely, we consider a monoid formed by equivalence classes of non-degenerate braided fusion categories (where non-degeneracy is a condition equivalent to modularity in the presence of spherical structure) with operation induced by external tensor product. The Witt group is by definition a quotient of this monoid by the relation which says that class of a Drinfeld center is zero. In this talk I will explain basic results about this group (which generalize basic properties of the classical Witt group) and discuss some open questions.

This talk is based on a joint project with A.Borodin. We study discrete isomonodromy transformations of discrete equations; discrete Painleve equations provide an important example of such transformations. In this talk, we approach the discrete equations geometrically as discrete connections on vector bundles on the Riemann sphere. This approach clarifies many features of isomonodromy trasnformations (such as spaces of solutions, tau functions, and symmetries).

In the first hour of this talk, we recall the Lie-theoretic construction of representations of the degenerate affine Hecke algebras of the type A (given by T. Arakawa and T. Suzuki), the dDAHA of type A (given by D. Calaque, B. Enriquez and P. Etingof) and the dAHA and dDAHA of type BC (given by P. Etingof, R. Freund and M.). In the second hour, we describe recent joint work giving a quantum analogue of the above constructions. In type A, we use quantum D-modules to construct representations of the affine and double affine braid groups and Hecke algebras. In the type BC_n case, we use quantum symmetric pairs developed by G. Letzter and S. Kolb, and twisted quantum D-modules to construct representations of the affine and double affine braid groups and affine and double affine Hecke algebras. All terms above will be defined in the talk.