The theory of vertex operator algebras and their representations is a very important subject of study, which has many applications in mathematics and physics. The first area of this project is the study of the representation theory of vertex operator algebras. This study will lead to the construction of irreducible twisted modules which are always assumed to exist in the physics literature. The second area of the project is the study of the orbifold theory associated with a finite cyclic group G for a holomorphic vertex algebra V. The last area of the project is to study the moonshine module vertex operator algebra and to prove the Frenkel-Lepowsky-Meurman conjecture on the moonshine module vertex operator algebra. Conformal field theory is an important physical theory describing both two-dimensional critical phenomena in condensed matter physics and classical motions of strings in string theory. Besides its importance in physics, the beautiful and rich mathematical structure of conformal field theory also has interested many mathematicians. New relations between different branches of mathematics, such as representations of infinite-dimensional Lie algebras and groups, Riemann surfaces and algebraic curves, the Monster sporadic group, modular functions and modular forms, elliptic genera, and knot theory, is revealed in the study of conformal field theory. It is believed that the study of the mathematics involved in conformal field theory will lead to new mathematical structures which will be important both in mathematics and theoretical physics.
