Spring, 2014

• Speaker Yi-Zhi Huang, Rutgers University
  • Title Modular invariance in conformal field theory
  • Time/place 3/14/2014, Friday, 12:00 in Hill 705
  • Abstract In this talk I will survey the modular invariance results and conjectures in conformal field theory.

    In 1988, in a pioneering work of Moore and Seiberg, a modular invariance conjecture for chiral vertex operators (intertwining operators) in rational conformal field theories was formulated and was used to derive the important Moore-Seiberg polynomial equations, which in turn were used to show that the Verlinde conjecture follows from some basic conjectures on rational conformal field theories, including, in particular, the modular invariance conjecture. In 1990, in an important work of Zhu, a partial result on the modular invariance conjecture of Moore and Seiberg was obtained. Later, by adapting Zhu's method, Dong-Li-Mason and Miyamoto generalized Zhu's partial result to partial results on modular invariance for certain orbifold theories and on modular invariance involving one intertwining operator, respectively. However, Zhu's method cannot be used or adapted to prove the full modular invariance conjecture of Moore and Seiberg. In 2003, using a method involving modular invariant differential equations and operator product expansions, I proved this modular invariance conjecture. In 2006, in a joint work with Kong, I proved further the modular invariance of full rational conformal field theories. In the case of logarithmic conformal field theories, Miyamoto in 2002 generalized Zhu's partial result to a corresponding partial result with an additional assumption on irreducible modules. I also formulated several years ago a full modular invariance conjecture for logarithmic intertwining operators. In the thesis work of Fiordalisi, substantial progress has been made on this modular invariance conjecture.

    I will discuss the historical development and mathematical significance of the modular invariance results and conjectures mentioned above.

  • Slides of the talk Modular invariance in conformal field theory.

• Speaker Jinwei Yang, Rutgers University
  • Title Some results in the representation theory of strongly graded vertex algebras
  • Time/place 4/3/2014, Thursday, 11:50 am in Hill 525 (Note special time and room)
  • Abstract Ph.D. thesis defense.

• Speaker Robert McRae, Rutgers University
  • Title Integral forms for certain classes of vertex operator algebras and their modules
  • Time/place 4/3/2014, Thursday, 12:55 pm in Hill 525 (Note special time and room)
  • Abstract Ph.D. thesis defense.

• Speaker Chris Sadowski, Rutgers University
  • Title On the structure of principal subspaces of standard modules for affine Lie algebras of type A
  • Time/place 4/3/2014, Thursday, 2:00 pm in Hill 525 (Note special time and room)
  • Abstract Ph.D. thesis defense.

• Speaker Yusra Naqvi, Rutgers University
  • Title A product formula for certain Littlewood-Richardson coefficients for Jack and Macdonald polynomials
  • Time/place 4/7/2014, Monday, 2:00 pm in Hill 705 (Note special time and room)
  • Abstract Ph.D. thesis defense.

• Speaker Siddhartha Sahi, Rutgers University
  • Title A Coxeter-type presentation for double affine Hecke algebras and Artin groups
  • Time/place 4/11/2014, Friday, 12:00 in Hill 705
  • Abstract

• Speaker Siddhartha Sahi, Rutgers University
  • Title A Coxeter-type presentation for twisted double affine Hecke algebras and Artin groups
  • Time/place 4/18/2014, Friday, 12:00 in Hill 705
  • Abstract

• Speaker Eric Schippers, University of Manitoba
  • Title Conformal welding and the sewing equations
  • Time/place 4/25/2014, Friday, 12:00 in Hill 705
  • Abstract Consider a pair of spheres with simply-connected domains excised, and parametrizing maps from the circle to the resulting boundary curves. The spheres can be sewn together by identifying points under the parametrizations, and the resulting Riemann surface can then be uniformized to a sphere. The resulting relations between the uniformizing map and the parametrizations are referred to in conformal field theory as the sewing equations.

    A similar construction plays a central role in quasiconformal Teichmuller theory, where it is referred to as the conformal welding theorem. In this talk I will discuss the connection between the conformal welding theorem and the sewing equations, in light of a correspondence between quasiconformal Teichmuller theory and conformal field theory discovered with David Radnell. I will not assume any background in Teichmuller theory.

    Joint work with David Radnell and Wolfgang Staubach.

• Speaker Simon Wood, The Australian National University
  • Title Symmetric polynomials in free field VOAs
  • Time/place 5/30/2014, Friday, 12:00 in Hill 705
  • Abstract So called singular vectors are of crucial importance to VOA theory. They can be used to check if a VOA satisfies Zhu's c_2 cofiniteness property, classify modules over the VOA and to derive differential equations for conformal blocks. In this talk I will focus on the classification of modules in the example of the well known minimal models. In general singular vectors are notoriously hard to determine explicitly, however, if the VOA one wishes to study can be realised as a sub- or quotient VOA of some free field VOA, then these singular vectors can be expressed as images of screening operators. Screening operators have interesting expansions in symmetric polynomials and therefore there is an interesting and elegant connection between a VOA's representation theory, its singular vectors and symmetric polynomials.

• Speaker Genkai Zhang, Chalmers University of Technology and Gothenburg University, Sweden
  • Title Capelli identities and the Radon transform
  • Time/place 6/18/2014, Wednesday, 3:30 pm in Hill 705
  • Abstract We construct Capelli-type invariant differential operators on Grassmannian manifolds U/K and on their non-compact counterparts the symmetric domains G/K, and find their eigenvalues by using holomorphic extensions of representations of U/K as sections of line bundles. We give applications to inversion formulas for the Radon transform.