Vertex Operator Algebras

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ABSTRACT, Lepowsky Lepowsky plans to continue his investigations of the algebraic and geometric features of vertex operator algebra theory and conformal field theory, and to apply this work to a range of problems. He will continue to collaborate with Yi-Zhi Huang on their theory of vertex tensor categories, with applications to algebraic, topological and geometric questions, including questions associated with the remarkable phenomena known as ``monstrous moonshine.'' He will also pursue connections between vertex operator algebra theory and both algebraic geometry and number theory. In recent years, there has been a great deal of progress on a range of mathematical theories related to string theory in theoretical physics. This project focuses on vertex operator algebra theory, which provides the algebraic basis of conformal quantum field theory, a foundation of string theory. The mathematical theory has its own motivations and points of view, sometimes directly related to physical structures and sometimes not. In the project, basic problems on the algebra and geometry of vertex operator algebra theory will be addressed, and the results will be applied to problems in algebra, geometry, number theory and finite group theory.