modular functor and a weakly holomorphic genus-zero weakly conformal field theory. Putting two copies of a weakly holomorphic genus-zero weakly conformal field theory in a suitable way, we obtained a genus- zero conformal field theory. These constructions depends on two major technical results obtained by us: One is the solution of the central charge problem for vertex operator algebras. The other is the proof of the associativity of intertwining operators for a suitable vertex operator algebra.

5.

Jointly with Jim Lepowsky, we developed a tensor product theory for modules for a vertex operator algebra. We introduced the notion of vertex tensor category which is a sophisticated analogue, involving geometry and not just topology, of the notion of symmetric tensor category. We proved that the category of modules for a suitable vertex operator algebra is a vertex tensor category. We also proved that any vertex tensor category gives a braided tensor category. In particular, we proved that the category generated by the standard modules of a fixed positive integral level for an affine Lie algebra and certain categories of modules for the Virasoro algebra are braided tensor categories.

6.

We  studied topological vertex algebras and their cohomology. We gave a geometric formulation of topological vertex operator algebras in terms of holomorphic forms on a certain moduli space (parameter space) of spheres with punctures and local coordinates. As a consequence, we obtain a geometric and conceptual construction of the Gerstenhaber or Batalin-Vilkovisky algebra structure on the cohomology of a topological vertex algebra.

7.

We studied the basic axiomatic aspects of intertwining operator algebras in detail. We formulated fundamental associativity, commutativity and generalized rationality properties of intertwining operator algebras in terms of both complex and formal variables, and we formulated and proved basic implications among these axioms. In the theory of vertex operator algebras, the Jacobi identity is an analogue of the Lie algebra Jacobi identity and plays a particularly important role. For intertwining operator algebras, it is natural to ask whether there is also a Jacobi identity, and we found such a Jacobi identity for intertwining operator algebras. We also established that the category of intertwining operator algebras is isomorphic to the category of algebras over genus-zero modular functors satisfying a certain meromorphicity property.