Spring, 2008

Spring, 2008

  • Speaker Antun Milas, SUNY-Albany
    • Title W-algebras, quantum groups and combinatorial identities
    • Time/place Tuesday, 2/5/2008, 2:15 pm in Hill 124
    • Note Special time and place
    • Abstract I will discuss a conjectural relationship between certain quantum W-algebras (vertex algebras) and finite-dimensional quantum groups associated to $sl_2$ (Hopf algebras). In the process we shall encounter interesting multisum identities.

  • Speaker Kevin McGerty, Chicago and IAS
    • Title Hall algebras and the Quantum Frobenius
    • Time/place Friday, 2/15/2008, 11:45am in Hill 423
    • Abstract See the preprint math.QA/0601150.

  • Speaker Olivier Schiffman, Paris and IAS
    • Title Macdonald polynomials and Eisenstein series on elliptic curves
    • Time/place Friday, 2/29/2008, 11:45am in Hill 423
    • Abstract

  • Speaker Tom Robinson, Rutgers
    • Title The automorphism property, differential representations and classical combinatorial identities
    • Time/place Friday, 3/7/2008, 11:45 am in Hill 423
    • Abstract We shall first recall the automorphism property of exponentiated derivations, and then discuss representing two variable derivations (formal partial differential operators) as one variable derivations. Then we shall show how one may use these two algebraic ingredients to compute simple classical identities involving special hyperbinomial numbers like the Stirling numbers.

  • Speaker Tom Robinson, Rutgers
    • Title Formal differential representations, Faa di Bruno and the Riordan Group
    • Time/place Friday, 3/14/2008, 11:55 am in Hill 425
    • Note Adjusted time and room
    • Abstract First I will show explicitly how a calculation in Frenkel-Lepowsky-Meurman's book on vertex operator algebras, which I will in its essentials redo, can be viewed as an application of a formal representation of exponentiated derivations. The outcome of the calculation is Faa di Bruno's formula for the higher derivatives of a composite function. Then building on this result I will show how another application of an easy class of formal differential representation leads to the Riordan Group. No prerequisites necessary.

  • Speaker David Ben-Zvi, Univ. of Texas, Austin and IAS
    • Title Real Groups and Topological Field Theory
    • Time/place Friday, 3/28/2008, 11:45 am in Hill 423
    • Abstract I will explain current joint work with David Nadler, in which the representation theory of real reductive Lie groups is examined through the lens of topological field theory and the geometric Langlands program. Our main results show how to recover the representation theory of real forms of a complex group G from the representation theory of G, and how to deduce a Langlands dual description of the representation theory (a form of Soergel's conjecture, generalizing results of Vogan and Langlands).

  • Speaker Liang Kong, Max Planck Institute for Mathematics at Bonn
    • Title A tensor-categorical study of open-closed rational conformal field theory
    • Time/place Friday, 4/4/2008, 11:45 am in Hill 423
    • Abstract I propose a reformulation of open-closed rational conformal field theory in terms of certain algebras in modular tensor categories. I will explain where it comes from. Then I will give a somewhat dual formulation. I will also discuss the so-called open-closed duality in this framework.

  • Speaker Hadi Salmasian, Alberta
    • Title On the structure and geometry of infinite-dimensional classical Lie groups
    • Time/place Friday, 4/11/2008, 11:45 am in Hill 423
    • Abstract Although structure theory and representations of finite-dimensional classical Lie groups/Lie algebras have been studied for about a hundred years, only recently have infinite-dimensional classical Lie groups/Lie algebras been investigated systematically. In this talk I begin by surveying recent progress on classification of simple locally finite Lie algebras, their Cartan and Borel subalgebras, and the geometry of their associated flag manifolds. Next I will focus on certain B-stable subvarieties of these flag manifolds which probably deserve to be called infinite-dimensional Schubert varieties. The talk will conclude with a result on finiteness of weight multiplicities of modules canonically associated to line bundles of these varieties.

  • Speaker Robert Wilson, Rutgers
    • Title Andrews' multisum identities and affine Lie algebras
    • Time/place Friday, 4/18/2008, 11:45 am in Hill 423
    • Abstract Certain important power series identities (in particular, the Rogers-Ramanujan identities and their combinatorial generalizations by Andrews, Gordon and Bressoud) have been explained in terms of filtrations of standard modules for affine Lie algebras (in particular, A_1^{(1)}). Andrews has given another generalization of the Rogers-Ramanujan identities in which the sum side is replaced by a multisum. We give a Lie theoretic interpretation of this expression using the operators X^{(i)} introduced by Meurman and Primc on a standard A_1^{(1)}-module.

  • Speaker Daniel Sage, LSU and NYU
    • Title The Springer correspondence, unipotent characters, and perverse coherent sheaves
    • Time/place Friday, 4/25/2008, 11:45 am in Hill 423
    • Abstract In this talk, I will discuss a combinatorial coincidence in representation theory as well as some hints at its geometric interpretation. Let G be a connected reductive algebraic group with Weyl group W. The Springer correspondence assigns to each irreducible representation of W an irreducible equivariant local system on a unipotent class of G. A major ingredient of Lusztig's parametrization of unipotent characters of finite reductive groups (or of unipotent character sheaves) has a similar flavor, with each irreducible representation of W associated to an element of a certain finite set which may be thought of as irreducible equivariant local systems on certain finite groups. In this talk, I will describe a remarkable compatibility between these two assignments of local systems to irreducible representations of W. This result is related to a conjecture of Lusztig on the geometry of special pieces in the unipotent variety. I will explain how an extension of the Deligne-Bezrukavnikov theory of perverse coherent sheaves can be used to address this conjecture. This work is joint with Pramod Achar.