Abstracts

Bojko Bakalov (North Carolina State University)

Title: $\mathcal W$-constraints for the total descendant potential of a
simple singularity

Abstract: Simple singularities are classified by Dynkin diagrams of type ADE.
Let $\mathfrak g$ be the corresponding finite-dimensional Lie algebra,
and $W$ its Weyl group. The set of $\mathfrak g$-invariants in the
basic representation of the affine Kac--Moody algebra $\hat{\mathfrak
g}$ is known as a $\mathcal W$-algebra and is a subalgebra of the
Heisenberg vertex algebra $\mathcal F$. Using period integrals, we
construct an analytic continuation of the twisted representation of
$\mathcal F$. Our construction yields a global object, which may be
called a $W$-twisted representation of $\mathcal F$. Our main result
is that the total descendant potential of the singularity, introduced
by Givental, is a highest weight vector for the $\mathcal W$-algebra.
(Joint work with T.\ Milanov.)

Title: Twisted modules for vertex operator superalgebras

Abstract: We will discuss twisted modules for vertex operator superalgebras that naturally arise in superconformal field theory---in particular, twisted modules that arise from automorphisms of the super extensions of the Virasoro algebra such as the N=1 and N=2 Neveu-Schwarz algebras in the case of supersymmetric vertex operator superalgebras. These include Ramond twisted sectors, mirror twisted sectors and the realization of the phenomenon of spectral flow via twisted sectors. We will present some of our recent constructions of such twisted modules, and some current problems that arise in constructing new twisted modules.

Maarten Bergvelt (University of Illinois at Urbana-Champaign)

Title: Integrable Systems and Cluster Algebras

Abstract: Using the Toda Latice as basic example we will discuss how
in cluster algebras integrable systems appear. No prior exposure to
integrable systems or cluster algebras will be assumed.

 Vyjayanthi Chari (University of California, Riverside)

Title: BGG reciprocity for current algebras.

Abstract: We discuss the relationship between representations of quantum affine algebras, Demazure modules in the  level one representations of affine Lie algebras and Macdonald polynomials.  We then develop a framework in which one can formulate and prove a version of BGG reciprocity for the current algebra (a maximal parabolic in the affine Lie algebra) associated to sl(n+1).

  Kyu-Hwan Lee  (University of Conneticut)

Title: Weyl group multiple Dirichlet series for symmetrizable Kac-Moody root systems

Abstract: Weyl group multiple Dirichlet series, introduced by Brubaker, Bump, Chinta, Friedberg and Hoffstein, are expected to be Whittaker coeffcients of Eisenstein series on metaplectic groups. Chinta and Gunnells constructed these multiple Dirichlet series for all the finite root systems using the method of averaging a Weyl group action on the field of rational functions. In this paper, we generalize Chinta and Gunnells' work and construct Weyl group multiple Dirichlet series for the root systems associated with symmetrizable Kac-Moody algebras, and establish their functional equations and meromorphic continuation.

Geoff Mason (University of California, Santa Cruz)

Title. Some cohomology classes attached to the Moonshine Module and the Monster simple group.

Abstract. We discuss various cohomology classes canonically
associated with the Monster $M$, the Moonshine VOA $V^{\natural}$,
and the $g$-twisted sectors for $g\in M$. In particular, we show that the
natural class in the Hochschild cohomolgy group $HH^3(\mathbb{Z}M)$
defined by $V^{\natural}$ has order exactly $24$. Time permitting, we
discuss some topological and $K$-theoretic consequences of this result.

Title. On the structure of $\mathbb{N}$-graded vertex operator algebras.

Abstract. We discuss the algebraic structure of an $\mathbb{N}$-graded
vertex operator algebra $V$, focusing in particular on the algebraic
structure on $V_0\oplus V_1$. We describe a general conjecture for the
case that $V$ is regular, and prove it when $V$ is a shifted VOA.
(This is joint work with Gail Yamskulna.)

Swarnava Mukhopadhyay (University of North Carolina, Chapel Hill, NC)

Title: Rank-level duality for conformal blocks of type so(2m+1)

Abstract: Classical invariants of tensor products of representations of one Lie group can often be related to invariants of some other Lie group. Physics suggests that the right objects to consider for these questions are certain refinements of these invariants known as conformal blocks. Conformal blocks appear in algebraic geometry as spaces of global sections of line bundles on the moduli stack of parabolic bundles on a smooth curve. Rank-level duality connects a conformal block associated to one Lie algebra to a conformal block for a different Lie algebra. In this talk we will discuss a formulation of rank-level duality using conformal embeddings of Lie algebras. We will also give an outline of our proof of the rank-level duality for type so(2m+1) conjectured by T. Nakanishi and A. Tsuchiya.

Daniel Orr (University of North Carolina, Chapel Hill, NC)

Title: Double affine Hecke algebras and difference Whittaker functions

Abstract: Difference Whittaker functions are eigenfunctions of the finite-difference Toda Hamiltonian H. In its original form due to Ruijsenaars, H can be viewed as an operator acting on functions on the maximal torus of diagonal matrices in GL(N). By work of Etingof and Sevostyanov, there are natural analogues of H for any complex simple (or reductive) group G in place of GL(N). In this general setting, difference Whittaker functions appear in surprisingly diverse contexts, from their origins in mathematical physics to the representation theory of quantum groups to the quantum K-theory of flag varieties. In this talk, I will discuss an explicit construction of certain difference Whittaker functions W with remarkable properties. The construction relies on the representation theory of double affine Hecke algebras. I will focus on recent work on "nonsymmetric" variants of W and the corresponding Dunkl-type operators for the finite-difference Toda Hamiltonians. These nonsymmetric functions are naturally expressed in terms of the level-one Demazure characters for the associated (twisted) affine Kac-Moody algebra.

 Martin Schlichenmaier (University of Luxembourg) 

Title: An elementary proof of the formal rigidity of the Witt and Virasoro algebra

Abstract:
A sketch of a proof that the Witt and the Virasoro algebra are
infinitesimally and formally rigid is given.
This is done by elementary and direct calculations showing that
the 2nd Lie algebra cohomology of these algebras with values
in the adjoint module is vanishing. The relation between deformations
and Lie algebra cohomology is explained.
I like to point out that this result does not imply that families
over non-formal bases are locally trivial.
As the speaker showed in joint work with Alice Fialowski in some earlier
work, there exists geometrically induced families of Krichever-Novikov vector field algebras containing the Witt algebra as
element over the special point 0, and all other algebras are non-isomorphic to it.

  Shabanskaya, Anastasia (Eastern Connecticut State University)

Title: Six-dimensional Lie algebras with a five-dimensional nilradical and solvable extensions of a special class of nilpotent Lie algebras

Abstract: I will talk about all the difficulties to classify solvable Lie algebras on the example of the paper "Six-dimensional Lie algebras with a five-dimensional nilradical " which attempts to correct and simplify an old paper by G. M. Mubarakzyanov which classifies the six-dimensional solvable indecomposable Lie algebras for which the nilradical is five-dimensional. Also I will introduce another approach to classify solvable Lie algebras in an arbitrary finite dimension, where you start with a certain nilpotent Lie algebra and find nilpotent extensions of it to an arbitrary finite dimension. Further working with a such nilpotent Lie algebra you could find all possible solvable indecomposable extensions of this algebra to an arbitrary finite dimension. It was done for two different nilpotent Lie algebras A_{6,11} and A_{6,19} using the notation of the paper Invariants of real low dimension Lie algebras, J.Math. Phys. 17(6) (1976), 986-994. The construction continues Winternitz' and colleagues' program of investigating solvable Lie algebras using special properties rather than trying to extend one dimension at a time.

Oleg Sheinman  (Steklov Mathematics Institute and Independent University of Moscow)

Title: Lax operator algebras: unexpected outcome, and a new tool of the theory
of integrable systems.

Abstract:  In 2001 I.Krichever invented a quite general notion of Lax operators with the spectral
parameter on a Riemann surface. This concept has emerged as a result
of a long term comprehending the role of vector bundles in the
theory of integrable systems and thus heavily uses the notion of Tyurin
parameters. It provides a general approach to the Hamiltonian theory
of a broad class of finite-dimensional integrable systems including Hitchin systems,
Calogero-Moser systems, gyroscopes, integrable cases of flow around a solid.

In 2006-2007 I.Krichever and the author have observed that
these Lax operators form an associative algebra, and obtained their
orthogonal and symplectic generalizations constituting Lie algebras.
The latter were given the name of Lax operator algebras. Every Lax operator
algebra is given by a Riemann surface with several marked points, a fixed
Tyurin data on it, and a complex finite-dimensional simple or reductive Lie
algebra. In this context the Lax operators initially defined by I.Krichever
correspond to the full linear algebra. Lax operator algebras possess
an almost-graded structure and central extensions. They are a natural
generalization of both affine Kac-Moody and Krichever-Novikov algebras.

It is quite intriguing that the same relations in terms of Tyurin parameters
imply closure of the Lax operator algebras with respect to their brackets,
and the Hamiltonian theory for the corresponding integrable systems as well.

Lax operator algebras provide a natural tool to obtain a Dirac-type prequantization
of Lax integrable systems in question. Every such system can be canonically related
with a kind of Conformal Field Theory, and the corresponding Knizhnik-Zamolodchikov-type
operators give a projective unitary representation of the Lie algebra of Hamiltonian vector
fields of the integrable system we are quantizing.

In the talk I will define the Lax operator algebras and outline the above listed
applications to the Hamiltonian theory and quantization of Lax integrable systems.