Vertex Operator Algebras

Lepowsky计划继续他对顶点算子代数理论和共形场理论的代数和几何特征的研究，并将这项工作应用于一系列问题。他将继续与黄宜智合作，研究他们的顶点张量范畴理论，并将其应用于代数、拓扑和几何问题，包括与被称为“怪异月光”的非凡现象相关的问题。他还将探索顶点算子代数理论与代数几何和数论之间的联系。近年来，理论物理中与弦理论相关的一系列数学理论取得了很大的进展。本课题主要研究顶点算子代数理论，它为共形量子场论提供了代数基础，是弦理论的基础。数学理论有自己的动机和观点，有时与物理结构直接相关，有时与之无关。在项目中，将解决顶点算子代数理论的代数和几何的基本问题，并将结果应用于代数，几何，数论和有限群论的问题。

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.