Spring, 2003

• Speaker Masahiko Miyamoto, University of Tsukuba, Japan
  • Title An introduction to a new stage of vertex operator algebra theory
  • Time/place Tuesday, 1/28/2003 3:00 pm in Hill 525
  • Note Joint Geometry/Topology/Quantum Mathematics Seminar/Colloquium
  • Abstract I will show a new relationship of vertex operator algebra theory with formal series. In particular, I will introduce a new concept of trace function of logarithmic form and show a modular invariance property with the help of finite dimensional ring theory.

• Speaker Masahiko Miyamoto, University of Tsukuba, Japan
  • Title Interlocked modules and pseudo-trace functions
  • Time/place Tuesday, 1/28/2003 4:30 pm (see note below) in Hill 705
  • Note Different day and time this week only due to Jean Taylor Symposium.
  • Abstract I will explain the proof of my recent result.

• Speaker Kiyokazu Nagatomo, Osaka University, Japan
  • Title Conformal field theory over the projective line
  • Time/place Friday, 2/21/2003 3:00 pm in Hill 705
  • Abstract I will explain a result recently obtained on conformal field theories over the projective line (joint work with A. Tsuchiya, math.QA/0206223). Let V be a vertex operator algebra satisfying the C_2-finiteness condition and certain regularity properties. We define spaces of conformal blocks associated to V using the geometry of the projective line, and prove the finiteness of conformal blocks and a factorization property. I start with the notion of vertex operator algebra and I will explain the definition of conformal blocks. No special knowledge of vertex operator algebras and conformal field theory is required; part of the talk is intended to be a comprehensive introduction to both fields.

• Speaker Yucai Su, Shanghai Jiaotong University and Harvard University
  • Title Lie algebras associated with derivation-simple algebras
  • Time/place Friday, 3/7/2003 3:00 pm in Hill 705
  • Abstract We first classify the pairs (A,D), where A is a commutative associative algebra with an identity element over an algebraically closed field F of characteristic zero, D is a finite dimensional F-vector space consisting of commuting locally finite derivations of A. Then using these pairs, we construct some in general not finitely graded Lie algebras of generalized Cartan type, and study their structure theory and representation theory.

• Speaker Chengming Bai, Nankai University and Rutgers University
  • Title Novikov algebras and vertex (operator) algebras
  • Time/place Friday, 3/14/2003 3:00 pm in Hill 705
  • Abstract Novikov algebras were introduced first by I.M. Gelfand and I.Y. Dorfman in connection with Hamiltonian operators in the formal variational calculus. Later they were also introduced to study the Poisson brackets of hydrodynamic type by S.P. Novikov et al. The finite-dimensional Novikov algebras can induce an interesting class of the infinite-dimensional Virasoro type Lie algebras. Hence they correspond to a class of vertex (operator) algebras. I will give a brief survey of the study of finite-dimensional Novikov algebras and the relations between them and their corresponding vertex (operator) algebras.

• Speaker David Radnell, Rutgers University
  • Title Schiffer Variation in Teichm\"uller Space and Determinant Line Bundles
  • Time/place Friday, 3/28/2003 3:00 pm in Hill 705
  • Abstract A construction of conformal field theories, in the sense of Segal and Kontsevich, from vertex operator algebras is essentially complete in genus zero and one. However, in higher genus, some basic analytic and geometric problems must be solved in order to even formulate some fundamental structures, such as modular functors and holomorphic weakly conformal field theories in the sense of Segal.
          The basic geometric objects are Riemann surfaces with analytically parametrized boundaries and their associated determinant lines. The formulations of modular functors and holomorphic weakly conformal field theories are based on the highly nontrivial assumptions that the moduli space of such surfaces is an infinite-dimensional complex manifold, the determinant lines form a holomorphic line bundle over this moduli space and that the sewing operation is holomorphic. I will outline a proof of these results using deep classical results from Teichmüller theory and Schiffer variation.
          No background in geometry will be assumed.

• Speaker Lin Zhang, Rutgers University
  • Title Tensor category theory for modules for a vertex operator algebra -- introduction and generalization
  • Time/place Friday, 4/11/2003 3:00 pm in Hill 705
  • Abstract There is a well-developed tensor category theory for certain modules of a fixed positive integral level for an affine Lie algebra, important in the study of conformal field theory. Two mathematical constructions are available: one by Finkelberg following Kazhdan-Lusztig's work on a related category, using algebro-geometric methods, and the other by Huang and Lepowsky using vertex operator algebra theory, which works for a very large class of categories of modules for rational vertex operator algebras. In this talk I will give an introduction to the vertex algebraic approach, and explain how, by using logarithmic intertwining operators, this can be applied to the category considered by Kazhdan and Lusztig. I will also explain the relation of the two constructions of this category. No tehnical background will be assumed in this talk.

• Speaker Michael Roitman, University of Michigan
  • Title Affinization of commutative algebras
  • Time/place Friday, 4/25/2003 3:00 pm in Hill 705
  • Abstract If V = V_0 + V_1 + V_2 + ... is a vertex algebra graded so that dim V_0 = 1 and V_1 = 0, then V_2 has a structure of commutative (but non-associative in general) algebra with an invariant bilinear form. We show that for any commutatve algebra A with a non-degenerate invariant bilinear form there is a vertex algebra V, graded as above, such that A = V_2. Moreover, if A has a unit e, then V can be chosen so that 2e is a Virasoro element.

• Speaker Frederick Gardiner, City University of New York
  • Title The pure mapping class group of a Cantor set (joint work with Nikola Lakic)
  • Time/place Friday, 5/2/2003 1:30 pm in Hill 705
  • Note Special time. Joint Quantum Mathematics/Topology/Geometry Seminar
  • Abstract It is shown that the pure mapping class group of the complement in the complex plane of the standard middle-thirds Cantor set acts discretely on the Teichmuller space of Cantor sets of bounded geometric type.