Past Representation Theory Seminar

I'll present the work of Gaitsgory arXiv:1108.1741. In it he uses Beilinson-Drinfeld factorization techniques in order to uniformize the moduli stack of G-bundles on a curve. The main difference with the gauge theoretic technique is that the the affine Grassmannian is far from being contractible but the fibers of the map to Bun(G) are contractible.

I will discuss some of new concepts and objects of two-dimensional number theory: 

how the same object can be studied via low dimensional noncommutative theories or higher dimensional commutative ones, 

what is higher Haar measure and harmonic analysis and how they can be used in representation theory of non locally compact groups such as loop groups and Kac-Moody groups, 

how classical notions split into two different notions on surfaces on the example of adelic structures, 

what is the analogue of the double quotient of adeles on surfaces and how one

could approach automorphic functions in geometric dimension two.

The class of finite dimensional algebras over an algebraically closed field K

may be divided into two disjoint subclasses (tame and wild dichotomy).

One class

consists of the tame algebras for which the indecomposable modules

occur, in each dimension d, in a finite number of discrete and a

finite number of one-parameter families. The second class is formed by

the wild algebras whose representation theory comprises the

representation theories of all finite dimensional algebras over K.

Hence, the classification of the finite dimensional modules is

feasible only for the tame algebras. Frequently, applying deformations

and covering techniques, we may reduce the study of modules over tame

algebras to that for the corresponding simply connected tame algebras.

We shall discuss the problem concerning connection between the

tameness of simply connected algebras and the weak nonnegativity of

the associated Tits quadratic forms, raised in 1975 by Sheila Brenner.

This is based on joint work with Dave Jorgensen. Given a Gorenstein algebra,

one can define Tate-Hochschild cohomology groups. These are defined for all

degrees, non-negative as well as negative, and they agree with the usual

Hochschild cohomology groups for all degrees larger than the injective

dimension of the algebra. We prove certain duality theorems relating the

cohomology groups in positive degree to those in negative degree, in the

case where the algebra is Frobenius (for example symmetric). We explicitly

compute all Tate-Hochschild cohomology groups for certain classes of

Frobenius algebras, namely, certain quantum complete intersections.

The category of perverse sheaves, Perv_X, on a stratified space X plays an

important role in the Intersection cohomology of Goresky-MacPherson and on the theory of

D-modules. It is defined as a subcategory of the derived category of

sheaves. Hence a usual complaint is that there are not very concrete

objects. A lot of work has been done to describe Perv_X more explicitly.

Hence many methods had been develop to describe Perv_X as a category of

quiver representations. An important property of perverse sheaves is that they can be viewed as a

stack, it means that a perverse sheaf can be defined up to isomorphism from

the data of perverse sheaves on an open cover of X plus some glueing data.

In this talk we show how the theory of stacks and more precisely the

notion of constructible stacks can be used in order to glue a description

due to Galligo, Granger and Maisonobe of the category Perv_X when X is C^n

stratified by a normal crossing stratification. Thanks to this we will

obtain a description of Perv_X on smooth toric varieties stratified by the

torus action.

Subfactor theory provides a framework for studying modular invariant partition functions in conformal field theory, and candidates for exotic modular tensor categories and almost Calabi-Yau algebras. I will survey some joint work with Terry Gannon and Mathew Pugh.

Last lecture of Bloc meeting

Second lecture of Bloc meeting

First lecture of Bloc meeting

In joint work with Karin Baur (ETH, Zurich) and Karin Erdmann (Oxford),

we study certain Delta-filtered modules for the Auslander

algebra of k[T]/T^n\rtimes C_2 where C_2 is the cyclic group

of order two.

The motivation of this lies in the problem of describing the $P$-orbit

structure for the action of a parabolic subgroup $P$ of a linear algebraic

group on its nilradical \mathfrak{n}. In general, there are

infinitely P-orbits in \mathfrak{n} and it is a ``wild'' problem to describe them.

However, in the case of a parabolic subgroup of SL_N, there

exists a bijection between P-orbits in the nilradical and

certain (Delta-filtered) modules for the Auslander algebra of k[T]/T^n,

due to work of Hille and Rohrle and Brustle et al..

Under this bijection, the Richardson orbit (i.e. the

dense orbit) corresponds to the Delta-filtered module without

self-extensions.

It has remained an open problem to describe such

a correspondence for other classical groups.

A finite p-group is said to be of Gorenstein-Kulkarni type if the set of all elements of non-maximal order is a subgroup of index p. This notion is motivated from the fact that 2-groups of Gorenstein-Kulkarni type arise naturally in the study of group actions on compact Riemann surfaces. In this talk, we introduce the notions relevant to describe group actions on Riemann surfaces, in particular the notion of the genus spectrum of a finite group, show how the Gorenstein-Kulkarni property arises in this framework. We then proceed to towards a classification of groups of Gorenstein-Kulkarni type.

We show that an adaptation of Landrock's Lemma holds for cellular algebras

and BGG algebras. As an application, we show that BGG reciprocity

respects Loewy structure.