Seminars Fall 2008

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Fall 2009

Friday, September 26, 2008

Catharina Stroppel, Bonn

Cohomology of Springer fibres in representation theory

The aim of the talk is to describe how cohomology rings of flag varieties Springer fibres occur in representation theory of Lie algebras as centres of categories, and explain the role of their equivariant version as deformations of categories. As an application we get Springers construction of the irreducible representations of the symmetric group on the top cohomology of the Springer fibre in a categorical version as the top degree of the centre of a category with the symmetric group action induced from a braid group action on the derived category.

Monday, September 29, 2008

Catharina Stroppel, Bonn

Finite dimensional modules for super gl(m|n) via diagrams

The aim of the talk is to give a description of the category of finite dimensional modules over the Lie superalgebra $\mathbf{gl}(m|n)$ in terms of modules over some combinatorially defined algebra $A$. The proof uses a categorification of certain $\mathbf{gl}(\infty)$-modules with their crystal structure. I will briefly recall the notion of categorification, crystals and describe the resulting algebra $A$. The main idea of the proof will be sketched, since we expect to be applicable in many other contexts.

Tuesday, October 7, 2008

John Francis, Northwestern


We will discuss some foundational aspects of geometry and deformation theory in the less commutative, or $E_n$, setting. In particular, we establish a relation between the $E_n$-tangent complex and $E_n$-Hochschild cohomology, proving a version of a conjecture of Kontsevich. Finally, we will sketch how a fuller development of $E_n$-geometry offers an algebro-geometric counterpart to categorical structures of topological field theory.

Thursday, October 23, 2008

Yongbin Ruan, University of Michigan

Integrable hierarchies and singularity theory

Almost twenty years ago, Kontsevich proved an amazing theorem (conjectured by Witten) that the intersection theory of Deligne-Mumford moduli space of curves is governed by KDV-hierarchy. In many ways, this is a mysterious theorem that we don't really understand why it should be true. In the talk, I will described a generalization (again conjectured by Witten). It will take us into the context of integrable hierarchies, Kac-Moody algebra and singularity theory. Now, we understand that Witten-Kontsevich theorem is just a special case of larger correspondence between representation theory and singularities.

Tuesday, November 04, 2008

Zhiwei Yun, Princeton

A global analogue of Springer representations

Classical Springer theory originated from the study of representations of groups over a finite field. I will replace the finite field by the function field of a curve (or a global function field), as people often do in number theory. I will describe how to get generalizations of Springer representations in this "global" situation by considering Hitchin moduli spaces with parabolic structures. I will discuss an example in $\text{SL}(2)$ to convince you that these representations are highly nontrivial, and from this example we can see general properties of such representations. Time permitting, I will mention the relation between global Springer representations and Langlands duality.

Tuesday, November 18, 2008

A. J. Tolland, UC Berkeley

Thursday, November 20, 2008

Sucharit Sarkar, Princeton

Computing Heegaard Floer Homology

We will review the definition and a few properties of Heegaard Floer homology for three-manifolds and knots inside three-manifolds. We will then describe an algorithm to compute the invariant and give an application in terms of arc presentations for knots. Time permitting, we will show how this homology theory can be extended to a homotopy theory in certain cases.

Tuesday, December 2, 2008

Dmytro Shklyarov, Kansas State University

Equivariant Topological Field Theories

I will give an overview of the notion of 2-dimensional equivariant topological field theory due to V. Turaev and describe a class of examples coming from non-commutative orbifolds.

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