Geometry of Langlands Duality

朗兰兹对偶的几何

• 批准号: 1303434
• 负责人: Ivan Mirkovic
• 金额: $ 14.93万
• 依托单位: University of Massachusetts Amherst
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-09-01 至 2016-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1303434&HistoricalAwards=false
• 关键词: Geometry; Langlands; Duality

The project is broadly directed towards geometry that is expected to underlie variousduality phenomena in mathematics (Langlands program) and physics (Gauge Theory).One goal is to reconstruct Loop Grassmannians and Mirkovic-Vilonen cycles from thepoint of view of Statistical Mechanics, and apply this to constructing two dimensionalloop Grassmannians. Another project is a to establish Langlands duality for categoriesof equivariant coherent sheaves on affine Steinberg varieties. A more technical projectis a geometric clarification of the Feigin-Frenkel description of the critical center Finally,the projects on Quantum Field Theory are on reformulating Costello?s mathematicalformulation of perturbative QFT in terms of flows and a speculation on the relation ofQFT to Grothendiecks function-sheaf dictionary. The most ambitious project attemptsto create geometric for the arithmetic case of Number Theory through a 'stochastic settheory'.The project aims towards relating and unifying developments in mathematics and theo-retical particle physics. The mathematical origin of these developments is the so calledLanglands program which is the modern view on a classical discipline of Number Theory. In time this program incorporated a number of central disciplines of mathematics, starting with Representation Theory, Algebraic Geometry and currently the Homotopy Theory. The relation to physics is a part of the current melting of barriers between mathematics and physics which arose in a period when the two subjects developed separately. The central impact on mathematics in the last quarter century was the import of the ideas from Quantum Field Theory (in particular String Theory), which is the part of physics that studies elementary particles. The proposed work attempts to work in both directions by applying ideas from physics to mathematics and mathematical constructions to physics. It also aims towards deeper understanding of the relation between Quantum Field Theory and Number Theory. The proposal also contains more standard topics within Representation Theory and Langlands Program.

该项目的主要目标是研究数学（朗兰兹计划）和物理学（规范理论）中各种对偶现象的几何基础，其中一个目标是从统计力学的角度重建Loop Grassmannian和Mirkovic-Vilonen循环，并将其应用于构造二维Loop Grassmannian。另一个项目是建立仿射Steinberg簇上等变凝聚层范畴的Langlands对偶。一个更技术性的项目是一个几何澄清费金-弗伦克尔描述的临界中心最后，量子场论的项目是重新制定科斯特洛？给出了微扰QFT的流形式，并对QFT与Grothen微分函数层字典的关系作了一个推测。最雄心勃勃的项目是通过“随机集合论”为数论的算术情况创建几何。该项目旨在联系和统一数学和理论粒子物理学的发展。这些发展的数学起源是所谓的朗兰兹纲领，它是数论经典学科的现代观点。随着时间的推移，该计划纳入了一些中心学科， 数学，从表示论，代数几何开始，现在是同伦理论。与物理学的关系是当前数学和物理学之间的障碍融化的一部分，这些障碍是在这两个学科分别发展的时期出现的。在过去的25世纪对数学的主要影响是引入了量子场论（特别是弦论）的思想，这是研究基本粒子的物理学的一部分。拟议的工作试图在两个方向的工作，从物理应用到数学和数学结构的物理思想。它还旨在更深入地理解量子场论和数论之间的关系。该提案还包含表示论和朗兰兹纲领中的更多标准主题。