Fall, 2017

Fall, 2017

  • Speaker Siddhartha Sahi, Rutgers University
    • Title The Capelli eigenvalue problem for Lie superalgebras
    • Time/place 9/8/2017, Friday, 12:00 in Hill 705
    • Abstract The Tits-Kantor-Koecher (TKK) construction attaches a simple Lie algebra to a simple Jordan algebra. In this setting one has a Jordan "norm" that generalizes the determinant, and a family of invariant differential operators generalizing the Capelli operators of classical invariant theory. In the early 1990s Bert Kostant and I studied the eigenvalues of these generalized Capelli operators, and a few years later Friedrich Knop and I discovered a surprising connection to Macdonald polynomials.

      It turns out that these ideas have analogs for Lie superalgebras, although there are several subtle issues and new phenomena. I will describe a number of recent results in this direction, which have been obtained in joint work with Hadi Salmasian, Alexander Alldridge, and Vera Serganova.

  • Speaker Sven Moeller, Rutgers University
    • Title Dimension Formulae in Genus Zero and Uniqueness of Vertex Operator Algebras
    • Time/place 9/15/2017, Friday, 12:00 in Hill 705
    • Abstract We prove a dimension formula for orbifold vertex operator algebras of central charge 24 by automorphisms of order $n$ such that $\Gamma_0(n)$ is a genus zero group. We then use this formula together with the inverse orbifold construction for automorphisms of orders 2, 4, 5, 6 and 8 to establish that each of the following fifteen Lie algebras is the weight-one space $V_1$ of exactly one holomorphic, $C_2$-cofinite vertex operator algebra $V$ of CFT-type of central charge 24: $A_5C_5E_{6,2}$, $A_3A_{7,2}C_3^2$, $A_{8,2}F_{4,2}$, $B_8E_{8,2}$, $A_2^2A_{5,2}^2B_2$, $C_8F_4^2$, $A_{4,2}^2C_{4,2}$, $A_{2,2}^4D_{4,4}$, $B_5E_{7,2}F_4$, $B_4C_6^2$, $A_{4,5}^2$, $A_4A_{9,2}B_3$, $B_6C_{10}$, $A_1C_{5,3}G_{2,2}$ and $A_{1,2}A_{3,4}^3$.

      This is joint work with Nils Scheithauer (Darmstadt) and Jethro van Ekeren (IMPA, Rio de Janeiro).

  • Speaker Bin Gui, Vanderbilt University
    • Title A unitary tensor product theory for unitary representations of unitary vertex operator algebras
    • Time/place 9/22/2017, Friday, 12:00 in Hill 705
    • Abstract A formal definition of unitary vertex operator algebras was introduced by Dong, Lin. For many examples of unitary VOAs (unitary minimal models, affine Lie algebras at non-negative integer levels), all representations are unitarizable. It is natural to ask whether their tensor product theories are unitary. In this talk, we try to answer this question. Let V be a unitary vertex operator algebra. We define a sesquilinear form on the tensor product of two unitary V-modules. We show that, when these sesquilinear forms are positive definite (i.e., when they are inner products), the modular tensor category for V is unitary. The positive definiteness of these sesquilinear forms, especially the positivity, is much harder to prove. We explain the main idea of the proof if time permitted.

  • Speaker Fei Qi, Rutgers University
    • Title A cohomological criterion for the reductivity for modules for vertex algebras
    • Time/place 10/6/2017, Friday, 12:00 in Hill 705
    • Abstract We use the cohomology theory of meromorphic open-string vertex algebras (MOSVA) to obtain a sufficient condition for the reductivity of left modules for such an algebra. In particular, this result gives a sufficient condition for vertex algebras, which are special MOSVAs. In this talk I will start from the theory of MOSVAs and its representations, discuss the cohomology theory of MOSVAs and present our main theorem. Many technical issues arise in the proof of this main theorem. I will address some of such issues.

      This is a joint work with Y.-Z. Huang.

  • Speaker Lisa Carbone, Rutgers University
    • Title Groups for Borcherds algebras
    • Time/place 10/13/2017, Friday, 12:00 in Hill 705
    • Abstract Borcherds algebras are generalizations of Kac-Moody algebras and have wide applications in physical theories and the study of automorphic forms. We discuss the problem of associating the analog of a Lie a group to a Borcherds algebra and we present some examples, including the Monster Lie algebra.

  • Speaker Jinwei Yang, Yale University
    • Title Braided tensor categories of admissible modules for affine Lie algebras
    • Time/place 10/20/2017, Friday, 12:00 in Hill 705
    • Abstract We construct a braided tensor category structure with a twist on a semisimple category of modules for an affine Lie algebra at an admissible level. We also prove the rigidity and modularity of this tensor category in the case of sl_2^. This is a joint work with T. Creutzig and Y.-Z. Huang.

  • Speaker Alina Vdovina, University of Bath and Hunter College
    • Title Buildings, surfaces and quaternions
    • Time/place 12/1/2017, Friday, 12:00 in Hill 705
    • Abstract Buildings are exciting objects which have geometric, algebraic and number theoretical aspects. We will give elementary constructions of several classes of buildings as universal covers of finite complexes. Then (as a geometric application) we will show a connection of our results with Gromov's surface subgroup question and (as an arithmetic application) we will give explicit examples of quaternionic lattices.

  • Speaker Joseph Bernstein, Tel Aviv University and Institute for Advanced Study
    • Title How to modify the Langlands' dual group
    • Time/place 12/8/2017, Friday, 12:00 in Hill 705
    • Abstract pdf file