Fall, 2001

• Speaker Alexander Kirillov, Jr., SUNY Stony Brook
  • Title On a q-analog of the McKay correspondence
  • Time/place Friday, 9/14/2001 3:00 pm in Hill 705
  • Abstract It is well known that finite subgroups in SU(2) are classified by simply-laced affine Dynkin diagrams, i.e., affine ADE diagrams. This calssification, known as McKay correspondence, is one of the many related ADE-type classifications (e.g., it is related with ADE classification in singularity theorey). In this talk, we give an analogue of this result for the quantum group U_q sl(2) with q beig a root of unity. This turns out to be related with the classification of modular invariants in Conformal field theory based on integrable representations of affine sl(2).

• Speaker Yi-Zhi Huang, Rutgers University
  • Title Vertex operator algebras and conformal field theories
  • Time/place Friday, 10/12/2001 3:00 pm in Hill 705
  • Abstract Conformal field theories were defined mathematically around 1987 by Kontsevich and Segal in terms of properties of path integrals. A construction of such a theory can be viewed in a certain sense as a construction of certain path integrals. However, up to now, there is still no complete published construction of examples of conformal field theories satisfying this definition. <\br>       On the other hand, around 1986, a notion of vertex operator algebra was introduced and studied in connection with the representation theory of infinite-dimensional Lie algebras and the Monster by Borcherds and Frenkel-Lepowsky-Meurman. Since then, the theory of vertex operator algebras has been developed rapidly and has found applications in a number of branches of mathematics. In this series of talks, I will explain a research program to construct conformal field theories in the sense of Kontsevich and Segal. Both the existing results and unsolved problems will be discussed.

• Speaker Yan Soibelman Kansas State Univerisity
  • Title Elliptic curves and quantum tori
  • Time/place Friday, 10/26/2001 3:00 pm in Hill 705
  • Abstract I plan to discuss the program of non-commutative compactifications I suggested two years ago. The main example will be non-commutative degenerations of elliptic curves. I will explain why quantum tori appear on the boundary of the moduli space of elliptic curves. I will discuss the relations to algebraic and symplectic geometry, q-difference equations, etc. The talk will consist largely of conjectures and speculations.

• Speaker Yi-Zhi Huang, Rutgers University
  • Title Vertex operator algebras and conformal field theories II
  • Time/place Friday, 11/3/2001 3:00 pm in Hill 705
  • Abstract This talk is a continuation of my talk on October 12. I will explain how to construct genus-zero conformal field theories from vertex operator algebras and why we need modules and intertwining operators when we want to construct maps associated to genus-one surfaces. I will also discuss weakly-conformal field theories and consequences of the existence of such theories, including the Verlinde formula.

• Speaker Deepak Parashar, MPI Leipzig
  • Title Some biparametric examples of quantum groups
  • Time/place Friday, 11/9/2001 3:00 pm in Hill 705
  • Abstract I will present simple examples of the standard and nonstandard (or Jordanian) quantum groups as well as their biparametric versions. The scheme is then extended in the wider context of the corresponding `coloured' counterparts. Within the framework of the R-matrix approach, I will also discuss some basic algebraic and geometric results from the theory of coloured quantum groups and outline possible physical and mathematical applications.

• Speaker Yi-Zhi Huang, Rutgers University
  • Title Vertex operator algebras and conformal field theories III
  • Time/place Friday, 11/16/2001 3:00 pm in Hill 705
  • Abstract I will continue the discussions on modular functors and weakly conformal field theories and on the consequences of the existence of such theories, including the Verlinde formula. I will also briefly explain Segal's idea on how to obtain real conformal field theories from rational weakly conformal field theories. At the end, the existing results and open problems will be discussed.

• Speaker Hai-Sheng Li, Rutgers University at Camden
  • Title Certain noncommutative analogues of vertex algebras
  • Time/place Friday, 11/30/2001 3:00 pm in Hill 705
  • Abstract We define and study a certain noncommutative analogue of the notion of vertex algebra. We show how to construct such algebras by using a set of compatible weak vertex operators.