Results Obtained

1.

Jointly with Chongming Xu, we derived an analog of the Ernst equation in general relativity when there is a cosmological term in the Einstein equation.

2.

We found all types of Backlund transformations in the three dimensional Minkowski space. There are three types: one tran forms a space-like surface of negative constant curvature to a surface of same kind. Another transforms a time-like surface of negative constant curvature to a surface of same kind. The third transforms a time-like surface of positive constant curvature to a space-like surface of positive constant curvature and vice versa. We also discovered that the singularities of  transformations of the third type occurs only when the images of the transformations go to the null infinity of the Minkowski

3.

Jointly with Frenkel and Lepowsky, we developed the axiomatic theory of vertex operator algebras and modules. We introduced basic notions, e.g., contragredient modules and intertwining operators, and proved basic theorems about vertex operator algebras, modules and intertwining operators.

4.

We completely solved the problem of constructing genus-zero conformal field theories (conformal field theories defined on spheres, corresponding to ``tree diagrams'' in quantum field theory) from suitable vertex operator algebras and their repreentations. In the process of constructing such theories, we constructed many other important structures and obtained many useful results. We constructed structures called intertwining operator algebras, genus-zero modular functors and genus-zero weakly conformal field theories. Intertwining operator algebras (a notion that we introduced) are natural nonmeromorphic generalizations of vertex operator algebras. (``Nonmeromorphic'' refers to multivaluedness of the underlying formal or complex functions; vertex operator algebras themselves are meromorphic in this sense.) In the study of conformal field theories and related mathematical problems, even for a problem whose statement involes only vertex operator algebras, the solution often involves nonmeromorphic operator algebras. So intertwining operator algebras are not simply generalizations; they are part of the important structure whenever there are vertex operator algebras. We showed that for a suitable vertex operator algebra, the direct sum of all irreducible modules for the vertex operator algebra is an intertwining operator algebra, and starting from an intertwining operator algebra, we constructed a genus-zero