Spring, 2009

• Speaker Masood Aryapoor, Yale
  • Title The Penrose Transform in the Split Signature
  • Time/place Friday, 1/30/2009, 11:45 am in Hill 423
  • Abstract The Penrose Transform in Split Signature In this talk I will introduce a version of the Penrose transform in split signature. It relates the cohomological data (with compact support) on an appropriate open subset of the complex 3-projective space and kernel of di.erential operators on the (real) Grassmannian of 2-planes in R4 . As an application I will derive a cohomological interpretation of the so-called X-ray transform (Radon transform) and a cohomological realization of the minimal representation of SL(4, R). Possible generalizations and applications to Representation Theory will be discussed as well.

• Speaker Tom Robinson, Rutgers
  • Title A difference equation interpretation of equivalent axioms for vertex algebras
  • Time/place Friday, 2/6/2009, 11:45 am in Hill 423
  • Abstract We consider axioms for the notion of vertex algebra. The "Jacobi identity" is the main axiom. We use a certain well-known difference equation to show how, in the presence of certain minor axioms, a certain natural class of axioms, of which the Jacobi identity is a distinguished member, are equivalent. In particular, the axiom consisting of weak commutativity together with weak associativity is included in this class, as are certain axioms involving the recently-formulated axiom of weak skew-associativity. Each member of the class corresponds to specifying a particular boundary condition which completely determines the solution of the difference equation mentioned above. Time permitting, we shall also use this point of view to establish (modulo a certain calculation which we will sketch) the (known, and nontrivial) equivalence between the notions of representation of, and of module for, a vertex algebra. In this talk, we will not assume previous familiarity with vertex algebra theory.

• Speaker Shrawan Kumar, University of North Carolina at Chapel Hill
  • Title Cachazo-Douglas-Seiberg-Witten Conjecture for Simple Lie Algebras and its generalization for symmetric spaces
  • Time/place Friday, 2/13/2009, 11:45 am in Hill 423
  • Abstract Let g be a finite dimensional simple Lie algebra over the complex numbers. Consider the exterior algebra R on two copies of g. The diagonal adjoint action of g gives rise to a g-algebra structure on R compatible with the bigrading. There are three `standard' copies of the adjoint representation g in the total degree 2 component R^2. Let J be the (bigraded) ideal of R generated by these three copies and define the bigraded g-algebra A :=R/J. The Killing form gives rise to a g-invariant S in A^{1,1}.

    Motivated by supersymmetric gauge theory, Cachazo-Douglas-Seiberg-Witten made the following conjecture. Various cases of the conjecture were proved by Cachazo-Douglas-Seiberg-Witten, Witten, Etingof-Kac, Etingof.

    Conjecture (i) The subalgebra A^{g} of g-invariants in A is generated, as an algebra, by the element S.
    (ii) S^h = 0.
    (iii) S^{h-1} is not 0.
    The aim of this talk is to give a uniform proof of the above conjecture part (i). We will also give an analog of this result for symmetric spaces.

    The main ingredients in the proof are: Garland's result on the Lie algebra cohomology of \hat{u} := g\otimes tC[t]; Kostant's result on the `diagonal' cohomolgy of \hat{u} and its connection with abelian ideals in a Borel subalgebra of g; and a certain deformation of the singular cohomology of the innite Grassmannian introduced by Belkale-Kumar.

• Speaker Corina Calinescu, Ohio State University
  • Title On applications of vertex operator algebras in Lie theory and partition theory
  • Time/place Friday, 2/27/2009, 11:45 am in Hill 423
  • Abstract In this talk we survey recent developments in the study of principal subspaces of the standard modules for untwisted affine Lie algebras of types A, D, E. We discuss generators-and-relations results for these subspaces and recursions satisfied by the characters of the principal subspaces. Our methods are based on intertwining vertex operators and general facts about affine Lie algebras. This is joint work with Jim Lepowsky and Antun Milas. The talk is introductory.

• Speaker Siddhartha Sahi, Rutgers
  • Title Binomial formula in higher dimensions
  • Time/place Friday, 3/6/2009, 11:45 am in Hill 423
  • Abstract For each integer k>=0, define p_k(x)=x(x-1)...x(x-k+1)/k!, then the value of p_k at each integer n is precisely the binomial coefficient nCk. It turns out that this has a beautiful generalization to higher dimensions, where k (and n) are replaced by partitions and p_k is a symmetric polynomial. These polynomials have a deep connection with the representation theory of the symmetric group as well as the general linear group.

• Speaker Lisa Carbone, Rutgers
  • Title Hyperbolic Kac-Moody symmetry, arithmetic and applications
  • Time/place Friday, 3/13/2009, 11:45 am in Hill 423
  • Abstract Hyperbolic Kac-Moody groups and algebras appear to play a fundamental role in high-energy theoretical physics, particularly the proposed unification of string theories, yet our knowledge of their internal structure is sketchy and fragmented. We discuss some of the mathematical problems that arise in the study of hyperbolic Kac-Moody symmetry and its applications.

• Speaker Haisheng Li, Rutgers-Camden
  • Title Quantum vertex F((t))-algebras and quantum affine algebras
  • Time/place Friday, 4/3/2009, 11:45 am in Hill 423
  • Abstract In the general field of vertex algebras, a problem for a while has been to develop a suitable theory of quantum vertex algebras, so that quantum affine algebras can be associated to quantum vertex algebras in the same way that affine Lie algebras are associated to vertex algebras. In this talk, we shall discuss a recent development on this problem. More specifically, we shall discuss a new notion of (weak) quantum vertex F((t))-algebra and we show how to associate quantum affine algebras to weak quantum vertex F((t))-algebras. This talk will be introductory; knowledge of vertex algebras or quantum affine algebras is not assumed.

• Speaker Yi-Zhi Huang, Rutgers
  • Title Representations of vertex operator algebras and braided finite tensor categories
  • Time/place Friday, 4/17/2009, 11:45 am in Hill 423
  • Abstract I will discuss what has been achieved in the past twenty years on the construction and study of a braided finite tensor category structure on a suitable module category for a suitable vertex operator algebra. The category of finite direct sums of standard (integrable highest weight) modules of a fixed positive integral level for an affine Lie algebra will be discussed as an important example. The talk is based on a recent paper in the archive with the same title and the archive number arXiv:0903.4233.

• Speaker Minxian Zhu, Rutgers
  • Title Regular representations of quantum groups at roots of unity
  • Time/place Friday, 4/24/2009, 11:45 am in Hill 423
  • Abstract I will discuss how the quantum function algebra decomposes as a bimodule of the big quantum group at roots of unity. I will explain how it is related to the structure of a family of vertex algebras associated to algebraic groups in rational levels.

• Speaker Siddhartha Sahi, Rutgers
  • Title q-binomial formula in higher dimensions
  • Time/place Friday, 5/1/2009, 11:45 am in Hill 423
  • Abstract For each integer k>=0, define p_k(x)=x(x-1)...x(x-k+1)/k!, then the value of p_k at each integer n is precisely the binomial coefficient nCk. In my last talk I explained that this has a beautiful generalization to higher dimensions, where k (and n) are replaced by partitions and p_k is a symmetric polynomial. These polynomials have a deep connection with the representation theory of the symmetric group as well as the general liner group. In this talk I will describe the q-analog of this theory, which involves the q-exponential.