Geometry of Langlands Duality

这个项目是针对搜索几何，预计将成为朗兰兹几何的基础。一个方面是解释朗兰兹对偶的约化群的同伦理论（在框架内导出代数几何）。另一个方面是继续发展一种特殊的几何Koszul对偶（"线性Koszul对偶"），并将其应用于表示论。在物理学方面，该项目设想了特胡夫特算子的数学形式，并在四维中构建了某种扩展的拓扑场论，并应用于纽结理论。该项目还包含了对数论和物理学关系的思考。在这里，数论的几何化有望通过与量子场论的联系而产生。该项目旨在联系和统一数学和理论粒子物理学的发展。数学中这些发展的起源是所谓的朗兰兹纲领，这是对数论经典学科的现代看法。随着时间的推移，这一计划纳入了一些中心学科的数学，开始与代表性理论，然后代数几何和目前的同伦理论。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。在上个世纪，对数学的主要影响是量子场论（特别是弦论）思想的引入，而量子场论是研究基本粒子的物理学的一部分。最近，维滕、卡普斯廷和古科夫在朗兰兹纲领和量子场论（特别是规范场论和弦理论）之间建立了一座长期寻求的桥梁。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与数论之间的关系。该提案还包含了表示论和朗兰兹纲领中的更多标准主题。

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.