Geometry of Langlands Duality

该项目广泛地针对几何，这些几何被认为是数学(朗兰兹计划)和物理(规范理论)中各种二元性现象的基础。一个目标是从统计力学的角度重建环格拉斯曼循环和米尔科维奇-维洛宁循环，并将其应用于构造二维同种格拉斯曼循环。另一个项目是建立仿射Steinberg簇上等变凝聚层范畴的朗兰兹对偶。一个更技术性的项目是对临界中心的费金-弗伦克尔描述的几何澄清。最后，量子场论的项目是重新表述Costello？S关于流的微扰QFT的数学公式，以及关于QFT与Grothendiecks函数层词典的关系的推测。最雄心勃勃的项目试图通过‘随机集合论’为数论的算术案例创造几何。该项目旨在将数学和理论粒子物理的发展联系起来并统一起来。这些发展的数学起源是所谓的朗兰兹程序，它是对数论一个经典学科的现代观点。随着时间的推移，这个程序结合了许多数学的中心学科，从表示论、代数几何开始，现在是同伦论。与物理的关系是目前数学和物理之间障碍融化的一部分，这种障碍是在这两个学科单独发展的时期产生的。在过去的四分之一个世纪里，对数学的主要影响是从量子场论(特别是弦理论)中引进了思想，量子场论是物理学中研究基本粒子的部分。这项拟议的工作试图通过将物理学的思想应用于数学，将数学构造应用于物理学，来进行双向的工作。旨在加深对量子场论与数论之间关系的理解。该提案还在表征理论和朗兰兹计划中包含了更多的标准主题。

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.