Geometry of Langlands Duality

该项目旨在寻找朗兰兹猜想的几何基础。一方面是用同伦理论（在派生代数几何的框架下）解释约化群的朗兰兹对偶性。另一个方面是继续发展特定的几何Koszul对偶（“线性Koszul对偶”）并将其应用于表示理论。在物理方面，该项目设想了t'Hooft算子的数学形式和四维扩展拓扑场理论的构建，并应用于结理论。该项目还包含了对数论和物理关系的推测。在这里，数论的几何化有望通过与量子场论的联系而出现。该项目旨在将数学和理论粒子物理学的发展联系起来并统一起来。这些数学发展的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，这个项目纳入了一些数学的中心学科，从表示理论开始，然后是代数几何，现在是同伦理论。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。最近，Witten、Kapustin和Gukov在朗兰兹计划和量子场论（特别是规范理论和弦理论）之间建立了一座长期寻求的桥梁。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与数论之间的关系。提案还包含了表征理论和朗兰兹程序中更多的标准主题。

This project is directed towards search for geometry that is expected to underlie the Langlands conjectures. One aspect is the interpretation of Langlands duality of reductive groups in terms of homotopy theory (in the framework of Derived Algebraic Geometry). Another aspect is a continuation of the work on developing a particular geometric Koszul duality (``Linear Koszul Duality'') and applying it to representation theory. In relation to physics, the project envisions a mathematical formalism of t'Hooft operators and a construction of a certain extended topological field theory in four dimensions, with applications to knot theory. The project also contains a speculation on the relation of number theory and physics. Here, a geometrization of number theory is expected to arise through a connection to quantum field theory. The project aims towards relating and unifying developments in mathematics and theoretical particle physics. The origin of these developments in mathematics is the so called Langlands 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, then 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. Recently, Witten, Kapustin and Gukov established a long sought bridge between Langlands program and quantum field theory (in particular gauge theory and string theory). 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.