Fall, 2012

Fall, 2012

  • Speaker Walter Freyn, Technische Universitat Darmstadt
    • Title The combinatorial geometry of Chevalley groups
    • Time/place 9/21/2012, Friday, 12:00 in Hill 705
    • Abstract

  • Speaker Dongwen Liu, University of Connecticut
    • Title Eisenstein series on loop groups
    • Time/place 9/28/2012, Friday, 12:00 in Hill 705
    • Abstract Based on Garland's work, we construct the Eisenstein series on the adelic loop groups over a number field, induced from either a cusp form or a quasi-character which is assumed to be unramified. We compute the constant terms and Fourier coefficients, prove their absolute and uniform convergence under the affine analog of Godement's criterion. For the case of quasi-characters the resulting formula is an affine Gindikin-Karpelevich formula. Then we prove the convergence of Eisenstein series themselves in certain analogs of Siegel subsets.

  • Speaker Tom Robinson, Rutgers University
    • Title A heuristic but easily rigorized construction of a non-trivial example of a vertex operator algebra
    • Time/place 10/5/2012, Friday, 12:00 in Hill 705
    • Abstract Essentially following the original approach of Frenkel, Lepowsky and Meurman, I will construct a non-trivial example of a vertex operator algebra. The main difference is that I will use a recently rigorized heuristic logarithm. (The heuristic was remarked upon in Frenkel, Lepowsky and Meurman's original work.)

  • Speaker Drazen Adamovic, University of Zagreb
    • Title On the representation theory of the triplet vertex operator algebra W(p,p')
    • Time/place 10/12/2012, Friday, 12:00 in Hill 705
    • Abstract In this talk we shall review our recent results on the triplet vertex operator algebra W(p,p') important in logarithmic field theory. We present our results on classification of irreducible representations and the structure of Zhu's algebra obtained in the case p'=2. We will also present an explicit realization of logarithmic W(p,p')-modules that have L(0) nilpotent rank three. This talk is based on a joint work with Antun Milas.

  • Speaker Roe Goodman, Rutgers University
    • Title Restricted roots and restricted form of Weyl dimension formula for spherical varieties (joint work with Simon Gindikin)
    • Time/place 10/19/2012, Friday, 12:00 in Hill 705
    • Abstract One can define the notion of restricted roots for a class of spherical homogeneous spaces of semisimple groups which includes simply connected symmetric spaces. For these spaces we give a detailed description (case by case) of the set of roots of the group associated with each restricted root of the space (which we call the "nest" of the restricted root). As an application, we obtain a refinement of the Weyl dimension formula in the case of spherical representations, expressing the dimension as a product over the set of indivisible positive restricted roots.

  • Speaker Ben Balsam, Fordham University
    • Title State sums, knot invariants and TQFTs
    • Time/place 10/26/2012, Friday, 12:00 in Hill 705
    • Abstract In recent years, the application of quantum groups to the study of low-dimensional topology has become an active topic of research. In three-dimensions, these yield the well-known Reshetihkin-Turaev (RT) invariants, which are a mathematical formulation of Chern-Simons theory and Turaev-Viro (TV) theory, which is a convergent form of the Ponzano-Regge state sum formula from Quantum Gravity. Both RT and TV are more than invariants; they have a far richer structure known as a Topological Quantum Field Theory.

      In this talk, we resolve a conjecture by Turaev. We demonstrate that Turaev-Viro theory based on spherical fusion category C is equivalent to Reshetikhin-Turaev theory based on Z(C) (The Drinfeld Center of C).

  • Speaker Anton Zeitlin, Columbia University
    • Title Continuous series of affine sl(2,R) and its close friends
    • Time/place 11/9/2012, Friday, 12:00 in Hill 705
    • Abstract I will talk about the construction of the continuous series of affine sl(2,R), based on the representation theory of loop ax+b group. I will also discuss possible relations with modular double representations of U_q(sl(2,R)).

  • Speaker Eugene Gorsky, Stony Brook University
    • Title Cherednik algebras and Khovanov-Rozansky homology
    • Time/place 11/16/2012, Friday, 12:00 in Hill 705
    • Abstract I will describe a model for the HOMFLY homology of the (m,n) torus knot using the finite-dimensional irreducible representation L_m/n of the rational Cherednik algebra with parameter m/n. The m-n symmetry of this construction and connections to the q,t-Catalan numbers of A. Garsia and M. Haiman will be also discussed. The talk is based on a joint work with A. Oblomkov, J. Rasmussen and V. Shende.

  • Speaker Jinwei Yang, Rutgers University
    • Title Tensor products of strongly graded vertex algebras and their modules
    • Time/place 11/30/2012, Friday, 12:00 in Hill 705
    • Abstract In this talk, I will define tensor products of strongly graded vertex algebras and tensor products of their strongly graded modules. Then I will classify certain irreducible strongly graded modules for a tensor product of strongly graded vertex algebras. As a consequence, I will classify certain irreducible strongly graded modules for the tensor product of the moonshine module vertex operator algebra and the vertex algebra associated with the self-dual Lorentzian lattice of rank 2 used by R. E. Borcherds to construct the "monster Lie algebra."

  • Speaker Robert McRae, Rutgers University
    • Title The Moonshine Module and Conway's Uniqueness Proof for the Leech Lattice
    • Time/place 12/7/2012, Friday, 12:00 in Hill 705
    • Abstract The Leech lattice is the unique even unimodular lattice of rank 24 that has no vectors of squared length 2. Similarly, the moonshine module is conjectured to be the unique vertex operator algebra of central charge 24 having no (nonzero) vectors of conformal weight 1 and such that every module is isomorphic to a direct sum of copies of the algebra. This talk will be an expository discussion of Conway's beautiful proof of the uniqueness of the Leech lattice and how one might obtain an analogous proof of the uniqueness of the moonshine module.