Fall, 2013

• Speaker Matthew Tai, University of Pennsylvania
  • Title Introduction to Family Algebras
  • Time/place October 4, Friday, 12:00 in Hill 705
  • Abstract For a complex simple Lie algebra g, all of the interesting g-module structure of U(g) is captured by the harmonic polynomials H(g), but H(g) has infinitely many isotypic components, making studying its structure difficult. Family algebras are a set of algebras over I(g) that each have only finitely many isotypic components and whose algebraic structure is linked to how those isotypic components fit into H(g). I will present some results for general family algebras, as well as examples of specific computations done using family algebras.

• Speaker Brian Paljug, Temple University
  • Title An Introduction to Algebraic Operads and Homotopy Algebras
  • Time/place October 18, Friday, 12:00 in Hill 705
  • Abstract Operads are combinatorial tools developed to model complex algebraic information, and have since found use in the study of deformation quantization, homological algebra, and mathematical physics. In this introductory talk I will show how certain common algebraic structures - associative algebras, Lie algebras, perhaps others - are governed by certain operads. I will then shift to talking about homotopy algebras, which are defined by complex infinite systems of operations and relations, and indicate how operads provide a powerful and convenient framework for their study. I will then describe an interesting and challenging problem in homotopy algebras, how to simultaneously modify homotopy algebras and a morphism between them, and explain how operads may be used in its solution.

• Speaker Chris Sadowski, Rutgers University
  • Title Presentations of the principal subspaces of the standard sl(3)^-modules
  • Time/place October 25, Friday, 12:00 in Hill 705
  • Abstract Principal subspaces of standard modules have been a topic of recent interest due to their connection to the Rogers-Ramanujan and Gordon-Andrews identities. In this talk, we give an overview of some known results about principal subspaces. In particular, following the work of Calinescu, Capparelli, Lepowsky, and Milas, we review certain presentations of principal subspaces and the exact sequences among principal subspaces obtained once these presentations are known. These exact sequences lead to recursions (namely, the Rogers-Ramanujan and Rogers-Selberg recursions) whose solutions yield the graded dimensions of principal subspaces. These graded dimensions, with certain specializations of variables, give the sum sides of the Rogers-Ramanujan and Gordon-Andrews identities in the sl(2)^ case. We give new results proving the presentations of the principal subspaces of the standard sl(3)^-modules and discuss the proof. We also give a conjecture for the presentations of the principal subspaces of the higher level standard sl(n+1)^-modules.

• Speaker Robert McRae, Rutgers University
  • Title Linear automorphisms of vertex operator algebras associated with formal changes of variable and Bernoulli numbers
  • Time/place November 1, Friday, 12:00 in Hill 705
  • Abstract I will explain, following Zhu and Huang, how a formal change of variable induces a linear automorphism of any vertex operator algebra V and thus gives rise to a new vertex operator algebra structure on V. When the formal change of variable is taken to be e^x-1, the new vertex operator on V involves Bernoulli numbers, Bernoulli polynomial values, and related rational numbers. I will show how relations among vertex operators can be used to gain information about these numbers, and I will also give another construction of the linear automorphism on V induced by e^x-1 that gives rise to a series of rational numbers that shares some characteristics of the Bernoulli numbers.

• Speaker Dongwen Liu, University of Connecticut
  • Title Archimedean zeta integrals on U(2,1)
  • Time/place November 8, Friday, 12:00 in Hill 705
  • Abstract In the work of Harris and Li the theory of theta correspondence was applied to show that, for a dual pair of unitary groups with equal rank, zeta integrals arising from Rallis inner product formula give the central values of certain automorphic L-functions, which are of arithmetic interest. In this talk I will describe the explicit calculation of Archimedean zeta integrals of this type for U(2,1). In particular we compute the matrix coefficients of Weil representations using joint harmonics in the Fock model, and those of discrete series using Schmid operators.

• Speaker Yusra Naqvi, Rutgers University
  • Title A Product Formula for Coefficients of Jack Polynomials
  • Time/place November 8, Friday, 2:00 in Hill 705 (note the special time)
  • Abstract In this talk, we will discuss Jack symmetric functions, which form a basis for the ring of symmetric functions over the field of rational functions in one parameter. They also generalize several classical families of symmetric functions, which can be obtained by specializing the parameter. In 1989, Richard Stanley conjectured that certain coefficients which appear when a product of Jack polynomials is expressed in this basis can be described combinatorially in terms of weighted hooks of Young diagrams. I will outline a proof of Stanley's conjecture for a sub-family of Jack polynomials.

• Speaker Jinwei Yang, Rutgers University
  • Title Twisted generating functions for singular vectors in Verma modules for sl(2)
  • Time/place November 15, Friday, 12:00 in Hill 705
  • Abstract We construct certain "twisted" generating functions incorporating the well-known classical singular vectors in the Verma modules for sl(2), and we then characterize the singular vectors by deriving partial differential equations for the generating functions. We also construct certain differential equations for generating functions over singular vectors in suitably localized Verma modules for sl(2). These ideas suggest generalizations to further cases. This is joint work with J. Lepowsky.

• Speaker Edinah K. Gnang, IAS
  • Title Hypermatrix Algebra and their spectral decomposition
  • Time/place December 6, Friday, 12:00 in Hill 705
  • Abstract In this talk we will present an overview of the hypermatrix generalization of matrix algebra proposed by Mesner and Bhattacharya in 1990. We will discuss a spectral theorem for hypermatrices deduced from this algebra as well as connections with other tensor spectral decompositions. Finally if time permits we will discuss some applications and related open problems. Joint work with Vladimir Retakh and Ahmed Elgammal.