Geometry of Langlands Duality

该项目广泛地指向几何，有望成为数学（朗兰兹程序）和物理（规范理论）中各种对偶现象的基础。一个目标是从统计力学的角度重构环格拉斯曼和Mirkovic-Vilonen环，并将其应用于二维环格拉斯曼的构造。另一个课题是建立仿射Steinberg变异上等变相干束范畴的朗兰兹对偶性。一个更技术性的项目是对Feigin-Frenkel临界中心描述的几何澄清最后，量子场论的项目是重新表述Costello？在流动方面的微扰QFT的数学表述和对QFT与Grothendiecks函数集字典关系的推测。最雄心勃勃的项目试图通过“随机集合理论”为数论的算术案例创建几何。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，它是对数论经典学科的现代观点。随着时间的推移，该计划纳入了一些数学的中心学科，从表示理论，代数几何和目前的同伦理论开始。与物理的关系是当前数学和物理之间的障碍融化的一部分，这种障碍是在两个学科分开发展的时期产生的。在过去的25年里，对数学的核心影响是量子场论（特别是弦理论）思想的引入，量子场论是研究基本粒子的物理学的一部分。这项提议的工作试图在两个方向上工作，将物理学的思想应用于数学，并将数学结构应用于物理学。它也旨在更深入地理解量子场论和数论之间的关系。提案还包含了表征理论和朗兰兹程序中更多的标准主题。

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.