Vertex Operator Algebras and Mathematical Conformal Field Theory

Huang和Lepowsky打算解决一系列与顶点（算子）代数理论和共形场论有关的问题，以及它们与物理学和物理学各个领域的关系和应用。 他们打算使用新的和不断发展的代数，几何和分析的思想和方法。 黄建议继续发展代数，分析和几何理论的基础共形fieldtheory和应用所取得的成果解决问题的代数，几何和拓扑。 Lepowsky打算继续发展顶点算子代数理论及其与共形场论、无限维代数结构、数论和组合恒等式理论的关系。 黄和Lepowsky还建议解决有关的问题，可循环的想法称为“可怕的月光”，以geometricuniformizations和对数张量积理论。 本建议中讨论的各种主题实际上是相互密切联系的，其中某些问题的解决方案有望有助于其他问题的分析和解决。 Huang和Lepowsky的长期目标是为数学理论的深入发展做出贡献，这将增强对数学和理论物理许多分支之间许多已知和未知联系的概念理解。 黄和Lepowsky将继续非常积极地在他们的教学，写作，组织会议和研讨会，以及他们的其他活动，旨在增加和加强理解的数学社区非常丰富的领域，他们的工作。 他们还将继续努力传达给理论物理界的成员，通过写作和演讲，他们的热情，丰富和实用的方法，以顶点算子代数理论和数学共形场论。顶点算子代数理论自然出现在研究某些无限维对称和解决有关的基本问题的“怪物”，一个有限的对称群，顶点算子代数是共形场论的重要组成部分，共形场论是凝聚态物理学和弦论中产生的一种物理理论，而弦论则是试图统一宇宙中所有相互作用的物理理论。 共形场论正在迅速发展成为一个丰富而美丽的数学理论。 这种发展已经导致了新的想法，令人惊讶的结果和许多数学问题的解决方案，并预计将导致更多。 共形场论的数学发展也被应用于物理现象的研究，如无序系统，边界表面上的凝聚态物理和某些更高维的物理对象，在弦理论中称为D膜。

Huang and Lepowsky intend to solve a range of problems related tovertex (operator) algebra theory and conformal field theory, and theirrelations with and applications to a variety of areas of mathematicsand physics. They intend to use new and evolving algebraic, geometricand analytic ideas and methods. Huang proposes to continue to developan algebraic, analytic and geometric theory underlying conformal fieldtheory and to apply the results obtained to the solution of problemsin algebra, geometry and topology. Lepowsky intends to continue todevelop vertex operator algebra theory and its relations withconformal field theory, infinite-dimensional algebraic structures,number theory and the theory of combinatorial identities. Huang andLepowsky also propose to solve problems related to the remarkablecircle of ideas called ``monstrous moonshine,'' to geometricuniformizations and to logarithmic tensor product theory. The variedtopics discussed in this proposal are in fact deeply connected withone another, and the solutions of certain of these problems areexpected to be useful in the analysis and solution of others. Along-term goal of Huang and Lepowsky is to contribute to the deeperdevelopment of a mathematical theory that will enhance the conceptualunderstanding of many known and still-unknown connections among manybranches of mathematics and theoretical physics. Huang and Lepowskywill continue to be very active in their teaching, writing, organizingof conferences and seminars, and their other activities designed toincrease and enhance the understanding in the mathematical communityof the very rich field in which they work. They will also continue totry to convey to members of the theoretical physics community, throughwriting and their speaking, their enthusiasm for the richness andutility of their approach to vertex operator algebra theory andmathematical conformal field theory.The theory of vertex operator algebras arose naturally in the study ofcertain infinite-dimensional symmetries and in the solution offundamental problems relating the ``Monster,'' a certain finitesymmetry group, with number theory, and this theory is fundamental toa wide range of problems in both mathematics and theoretical physics.Vertex operator algebras are crucial ingredients in the mathematicalstudy of conformal field theory, a physical theory that arose in bothcondensed matter physics and string theory, which in turn is aphysical theory attempting to unify all the interactions in theuniverse. Conformal field theory is in the process of being rapidlydeveloped into a rich and beautiful mathematical theory. Thisdevelopment has already led to new ideas, surprising results and thesolutions of many mathematical problems, and it is expected to lead tomany more. The mathematical development of conformal field theory hasalso been applied to the study of physical phenomena such as disordersytems, condensed matter physics on surfaces with boundaries andcertain higher-dimensional physical objects called D-branes in stringtheory.