Theta Functions, Intersection Theory and Representation Theory

Theta 函数、交集理论和表示论

• 批准号: 0901249
• 负责人: Prakash Belkale
• 金额: $ 23.93万
• 依托单位: University of North Carolina at Chapel Hill
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-06-01 至 2013-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901249&HistoricalAwards=false
• 关键词: Theta; Functions; Intersection; Theory; Representation

This project will focus on the interconnected areas of non-abelian theta functions, intersection theory and representation theory. The view point is that the theory of theta functions and invariant theory (with relations to the classical Schubert calculus) constitute a common generalization of the classical theory of theta functions. The spaces of theta functions correspond to physical states in the WZW model of conformal field theory, and this correspondence links the subject to modern mathematical physics.The Hitchin connection on the spaces of non-abelian theta functions lies at the heart of the physics view point on theta functions and several outstanding problems concerning its properties remain open (unitarity, whether it is ``motivic''). The aim of the project is to study the Hitchin connection, and to have techniques from conformal field theory bear upon open problems in invariant theory such as the saturation conjecture for the even orthogonal groups. A study of the Hitchin connection will be carried out with potential applications to strange duality questions (that is, relations between non-abelian theta functions for different groups) in their most general formulation. It is hoped that a study of the strange dualities will lead to insights on the Hitchin connection.Algebraic geometry studies the solutions to systems of polynomial equations. Representation theory (the theory of symmetry), combinatorics, and mathematical physics have fundamental links with algebraic geometry. Theta functions, originating in algebraic geometry, are some of the most important functions in mathematics. These functions appear frequently in mathematical physics. The project aims to deepen the links between theta functions and representation theory. It is hoped that the project will lead to better algorithms in various combinatorial questions. Possible applications outside of mathematics are via contributions to the theory of differential equations (the Riemann-Hilbert problem), to theoretical computer science and physics (string theory and conformal field theory).

该项目将重点关注非阿贝尔 theta 函数、交集理论和表示论的相互关联领域。该观点认为，theta 函数理论和不变理论（与经典舒伯特微积分有关）构成了经典 theta 函数理论的共同推广。 theta 函数的空间对应于共形场论 WZW 模型中的物理状态，这种对应关系将这一主题与现代数学物理联系起来。非阿贝尔 theta 函数空间上的希钦联系是物理学观点的核心关于 theta 函数的研究以及有关其性质的几个突出问题仍然悬而未决（幺正性，是否是“动机”）。 该项目的目的是研究希钦联系，并利用共形场论技术解决不变理论中的开放问题，例如偶正交群的饱和猜想。对希钦联系的研究将在其最一般的表述中潜在地应用于奇怪的对偶问题（即不同群的非阿贝尔 theta 函数之间的关系）。希望对奇怪的对偶性的研究能够带来对希钦联系的见解。代数几何研究多项式方程组的解。表示论（对称性理论）、组合学和数学物理与代数几何有着根本的联系。 Theta 函数起源于代数几何，是数学中最重要的函数之一。这些函数在数学物理中经常出现。该项目旨在加深 theta 函数和表示理论之间的联系。希望该项目能够在各种组合问题上带来更好的算法。数学之外的可能应用包括对微分方程理论（黎曼-希尔伯特问题）、理论计算机科学和物理学（弦理论和共形场论）的贡献。