Matthew Bennett (UC Riverside, USA)

A BGG type duality for the current algebra of $\mathfrak {sl}_2$

Abstract.
In the 1970s, Bernstein, Gelfand and Gelfand proved a very
important reciprocity relation in the category $\mathscr O$ of modules for a simple Lie
algebra. Their result related projective modules, Verma modules and
irreducible modules in the categroy.
In this talk we will be concerned with an analogy of BGG duality for for
the current algebra $\mathfrak{sl}_2[t]$ of $\mathfrak{sl}_2$, or equivalently the
maximal parabolic subalgebra of the affine algebra. We define the analog of
the category $\mathscr O$ in this context, roughly speaking, the category of modules
for $\mathfrak{sl}_2[t]$ which are sums of finite dimensional modules of
$\mathfrak{sl}_2$.
We shall introduce the simple modules and their
projective covers, define the appropriate analogs of the Verma modules and
finally obtain the required duality.
This is based on joint work with V. Chari and N. Manning.

Ghislain Fourier (Universität zu Köln, Germany)

Weyl modules for equivariant map algebras

Abstract.
Weyl modules were defined and studied over the last two decades in various cases and from various perspectives. In this talk, I'll extend the definition of local Weyl modules to equivariant map algebras (EMA). By showing that there exists an isomorphism between the category of finitedimensional modules for the EMA and the analog for the generalized current algebras, one can translate every reasonable question about this category in the EMA setting into the context of generalized current algebras. This applies especially to Weyl modules and irreducibles.

Yiqiang Li (Virginia Tech, USA)

Tensor product varieties, perverse sheaves and localization conditions

Abstract.
A new class of simple perverse sheaves was defined recently by Zheng in categorifying the tensor product of simple modules of a quantized enveloping algebra. In this talk, I will present my recent work on the interaction between certain class of tensor product varieties (in the spirit of Lusztig, Malkin and Nakajima) and this class of simple perverse sheaves. I'll discuss in details the structures of the classes of varieties and perverse sheaves involved. This leads to the equivalence between the localization process using support of complexes and the localization process using singular support of complexes.

Erhard Neher (University of Ottawa, Canada)

Finitedimensional representations of equivariant map algebras

Abstract.
Suppose a finite group acts on an algebraic variety $X$ and
a finitedimensional Lie algebra $\mathfrak g$. Then the space of equivariant
algebraic maps from $X$ to $\mathfrak g$ is a Lie algebra under pointwise
multiplication. Examples of such equivariant map algebras include
current algebras, twisted and untwisted loop algebras and their
multivariable versions, and the (generalized) Onsager algebra.
In this talk I will present a classification of all
finitedimensional irreducible representations of equivariant map
algebras (joint work with Alistair Savage and Prasad Senesi) and
describe their extensions (joint work with Alistair Savage). The
latter result allows us to determine the block decomposition of the
category of all finitedimensional representations.

Raphaël Rouquier (Oxford University, United Kingdom, and UCLA, USA)

Higher representation theory

Abstract.
We will discuss the theory developed with Joe Chuang of
2representation theory. We will explain how representations of
KacMoody algebras are replaced by actions on categories. We will
explain the theory of highestweight 2representations.

Valerio Toledano Laredo (Northeastern University, USA)

Yangians, quantum loop algebras and trigonometric connections

Abstract.
I will describe monodromy representations of affine braid
groups arising from a flat connection with values in the Yangian of
a simple Lie algebra $\mathfrak g$. These representations are related to those
arising from the quantum Weyl group operators of the quantum loop
algebra of $\mathfrak g$. Matching these two classes of representations involves
in particular the construction of a functor relating finitedimensional
modules of those two quantum groups.
This is based on joint work with Sachin Gautam.

Benjamin Webster (University of Oregon, Eugene, USA)

Categorification, Lie algebras and topology

Abstract.
It's a long established principle that an interesting way to think
about numbers as the sizes of sets or dimensions of vector spaces, or
better yet, the Euler characteristic of complexes. You can't have a
map between numbers, but you can have one between sets or vector
spaces. For example, Euler characteristic of topological spaces is not
functorial, but homology is.
One can try to extend this idea to a bigger stage, by, say, taking a
vector space, and trying to make a category by defining morphisms
between its vectors. This approach (interpreted suitably) has been a
remarkable success with the representation theory of semisimple Lie
algebras (and their associated quantum groups). I'll give an
introduction to this area, with a view toward applications in
topology; in particular to replacing polynomial invariants of knots
that come from representation theory with vector space valued
invariants that reduce to knot polynomials under Euler characteristic.

Eliana Zoque (UC Riverside, USA)

Partitions, polynomials and current algebras

Abstract.
While studying the category of finite dimensional representations of the affine Lie algebra associated to $\mathfrak{sl}_2$ and trying to develop a theory of highest weight categories, Chari and Greenstein found that one of the results required would be to prove that a certain module for the ring of symmetric functions was free of rank equal to a Catalan number. This module is indexed by a family of polynomials indexed by partitions described recursively.
In this talk I will present a combinatorial approach to this family of partitions, providing a nonrecursive description using inequalities and a bijection with a family of binary tees enumerated by the Catalan numbers.
