Georgia Benkart (University of Wisconsin, Madison, USA)

A Talk on the Weyl Side

Abstract.
The Weyl algebra arises in many different contexts in mathematics
and physics and plays an essential role in the representation theory of
nilpotent Lie algebras. This talk will focus on a family of Weyllike
algebras that includes the Weyl algebra itself, various algebras that
arise in noncommutative geometry, and many others. The aim is to
give a uniform approach to the structure and representations
of these algebras.

Alexandre Bouayad (Université Paris VII, France)

Generalized quantum enveloping algebras and Langlands interpolating quantum groups

Abstract.
We will define and study deformations along several parameters of the enveloping algebra of
a semisimple finite dimensional Lie algebra, called the Generalized Quantum Enveloping (GQE)
algebras.
We will see how GQE algebras can be used to define Langlands Interpolating Quantum (LIQ) groups
and solve conjectures motivated by the geometric Langlands program. These conjectures have been
suggested by E. Frenkel and D. Hernandez in an original work on different LIQ groups.

Corina Calinescu (Yale University, USA)

Vertexalgebraic structure of representations of affine Lie
algebras

Abstract.
This talk is an overview of the known results and open problems
about vertexalgebraic structure of the principal subspaces of standard
modules for certain untwisted and twisted affine Lie algebras. We discuss
presentations of these subspaces and their fermionic characters. The talk
is based on joint work with James Lepowsky and Antun Milas.

Jiarui Fei (UC Riverside, USA)

Hall algebra and counting

Abstract.
In this talk, we will use Hall Algebras to count rational points of moduli spaces and flag varieties of quiver representations. One of our main tools is the counting character (first used by M. Reineke) from Hall algebras to various quantum tori. This character can play with the full
Hopf structure of Hall algebras of quivers to generate interesting results. When playing with
comultiplication, the counting character can be specialized to Qin and Rupel's quantum cluster
character, which leads to several easy proofs of hard theorems in the cluster algebra.

Toshiki Nakashima (Sophia University, Tokyo, Japan)

Elementary characters and monomial realizations of
crystal bases

Abstract.
We shall present the explicit forms of
certain elementary characters for the classical groups and
describe some relations between their explicit forms and
the monomial realizations of crystal bases.
We also mention the refined polyhedral realizations
of crystal bases.

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

KazhdanLusztig cells and CalogeroMoser spaces

Abstract.
KazhdanLusztig cells of Weyl groups are related to primitive
ideals of complex semisimple Lie algebras. We will explain a
conjectural description of left cells in terms of Galois theory for
CalogeroMoser spaces, which are deformations of symplectic
singularities constructed by Etingof and Ginzburg from rational
Cherednik algebras (joint work with Cedric Bonnafé).

Edward Richmond (University of British Columbia, Canada)

LittelwoodRichardson coefficients

Abstract.
A tight fusion frame is a sequence of orthogonal projection
matrices which sum to a scalar multiple of the identity. To any such
sequence, we can associate a weakly decreasing sequence of positive
integers given by the ranks of these projections.
The question we address is the following: For which sequences of positive
integers do tight fusion frames exist?
In this talk, I will discuss joint work with K. Luoto and M. Bownik where
we explore this problem. In particular, we give a combinatorial
characterization in terms of nonvanishing LittlewoodRichardson
coefficients. Classically, LittlewoodRichardson coefficients appear in the
representation theory of $GL_n$ as the structure constants when decomposing a
tensor product of irreducible representations. This connection between
frame theory and algebraic combinatorics yields several interesting results
in both fields of mathematics.

Dylan Rupel (University of Oregon, Eugene, USA)

Quantum Cluster Characters

Abstract.
In this talk I will describe the noninitial cluster
variables in a quantum cluster algebra in terms of the representation
theory of a corresponding quiver. Time permitting, I will explain why
quantum cluster characters should have ever been discovered by realizing
them as the image under a certain algebra homomorphism from the
RingelHall algebra of the quiver to the initial quantum torus.

Daisuke Sagaki (University of Tsukuba, Japan)

Maximal dimension of tensor products and Schur positivity
for classical Lie algebras

Abstract.
In my talk, we would like to discuss the following questions:
Let $\mathfrak{g}$ be a finitedimensional simple Lie algebra,
and fix a dominant integral weight $\lambda$.
Assume that $\mu$ runs over all dominant integral weights
such that $\lambda\mu$ are also dominant.
Question 1.
Determine the condition for $\mu$ so that
the dimension of $V(\lambda) \otimes V(\lambda\mu)$
is maximal. Here, $V(\lambda)$ and $V(\lambda\mu)$
are irreducible $\mathfrak{g}$modules of highest weight
$\lambda$ and $\lambda\mu$, respectively.
Question 2.
Determine the condition for $\mu$ so that
the character of $V(\lambda) \otimes V(\lambda\mu)$ is
"maximal" in the sense that the multiplicity of $V(\nu)$
in $V(\lambda) \otimes V(\lambda\mu)$ is maximal
for every dominant integral weight $\nu$.

Alistair Savage (University of Ottawa, Canada)

Equivariant map superalgebras

Abstract.
Suppose a finite group acts on a scheme (or algebraic variety) $X$ and a
"target'' Lie superalgebra $\mathfrak g$. Then the space of equivariant algebraic maps
from $X$ to $\mathfrak g$ is a Lie superalgebra under pointwise multiplication. We call
this an "equivariant map superalgebra". An important class of examples are
the (twisted) loop superalgebras, where the variety $X$ is the
onedimensional torus.
In this talk we will present a classification of the irreducible
finitedimensional representations of an equivariant map superalgebra where
the target is a basic classical Lie superalgebra and the group in question
acts freely on $X$. It turns out that all irreducible finitedimensional
representations are generalized evaluation representations. In the case
that the even part of $\mathfrak g$ is semisimple, they are in fact all evaluation
representations. As a corollary of our general result, we obtain the first
classification of the twisted loop superalgebras.

Peter Tingley (MIT, USA)

Various constructions of MirkovicVilonen polytopes

Abstract.
Consider a symmetrizable KacMoody algebra. In finite type,
MirkovicVilonen (MV) polytopes give nice combinatorial realizations
of Kashiwara's crystals. These polytopes were originally defined using
the geometry of the affine grassmannian, but they also arise naturally
in several other contexts. For instance, they can be understood in
terms of quiver varieties, or in terms of KhovanovLaudaRouquier
(KLR) algebras. Both of these points of view have the advantage that
they make sense beyond finite type. I will explain as much of this
story as I can, focusing on recent work with Ben Webster developing
the KLR algebra construction. In particular, we obtain MV polytopes in
all affine types which can be described combinatorially and which have
many properties analogous to the finitetype situation.

R. Venkatesh (Institute of Mathematical Sciences, Chennai, India)

Unique factorization of tensor products for KacMoody algebras

Abstract.
Suppose $V$ is a representation of a complex simple Lie
algebra that can be written as a tensor product of irreducible
representations. A theorem of C.S. Rajan states that the irreducible
factors that occur are uniquely determined, up to reordering, by the
isomorphism class of $V$. I will present an elementary proof of Rajan's
theorem, which generalizes with no extra effort to the infinite
dimensional (KacMoody) Lie algebras. This is joint work with S. Viswanath.
