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亚伯theta函数，交叉理论和表示理论的相互联系。观点是，theta函数和不变理论的理论（与经典的舒伯特微积分有关）构成了theta函数经典理论的共同概括。 theta功能的空间对应于WZW的wzw综合场理论模型中的物理状态，并且这种对应关系将主题与现代数学物理学联系起来。非亚伯利亚theta函数空间上的Hitchin连接在于物理学核心的核心，theta的核心点上的theta函数函数和几个出色的问题仍然是开放的（它的构成其开放式）（是否有动力，它是否有动力）。 该项目的目的是研究Hitchin连接，并具有共形野外理论的技术对不变理论中的开放性问题，例如均匀的群体的饱和猜想。对Hitchin连接的研究将在其最一般的表述中对奇怪的二元性问题的潜在应用（即非亚伯theta函数之间的关系）进行。希望对奇怪二元性的研究能够对希钦连接有所了解。代数几何研究对多项式方程系统的解决方案。表示理论（对称理论），组合学和数学物理学与代数几何形状具有基本联系。源自代数几何形状的theta函数是数学中最重要的功能。这些功能在数学物理学中经常出现。该项目旨在加深theta函数与表示理论之间的联系。希望该项目能够在各种组合问题中提供更好的算法。数学之外的可能应用是通过对微分方程理论（Riemann-Hilbert问题）的贡献，对理论计算机科学和物理学（弦理论和保形场理论）。