Geometry of Langlands duality

[摘要]米尔科维奇的建议是直接针对的几何基础的兰克兰计划。一个主要的和不断扩大的影响是量子场论的思想。“临界量化”计划旨在发展一种形式主义，将模表示理论和仿射李代数在临界水平上的表示方法统一起来，并应广泛包括对相干束的派生范畴的研究。“二次Koszul对偶性”项目试图给出几何方法来处理整个表示理论中出现的对偶性，最终以CherednikFourier变换告终。最后，我们提出了一种“完全代数-几何”的方法来研究类场论和朗兰兹程序。该项目旨在联系和统一数学（表示理论，代数几何，数论）和理论粒子物理（量子场论，特别是弦理论）的发展。过去十年数学的主要发展是量子场论预言的影响。这导致了一些进展，我希望方向能够逆转，因为数学本质上有助于理解粒子物理学和“大统一”项目，即对已知物理力给出一致的描述。该项目的主要目标是通过在数学上解释弦理论和量子场论的某些元素，在这个方向上采取某些步骤。

AbstractMirkovicThe proposal is directed towards the geometry underlying theLanglands program. A major and expanding influence are the ideas from quantum field theory. The ``Critical Quantization'' program aims to develop a formalism, that would unify approaches to modular representation theoriesand to representations of affine Lie algebras at a critical level, andshould have wide including the study of derived categories of coherent sheaves.The ``Quadratic Koszul Duality'' project attempts to give ageometric approach to a family of dualities thatappear throughout representation theory, culminating with the CherednikFourier transform. Finally, we proposecertain ``completely algebro-geometric'' approach to Class Field Theory and Langlands program.The project aims towards relating and unifying developments in mathematics(representation theory, algebraic geometry, number theory)and theoretical particle physics (quantum field theory and in particular string theory). The main development in mathematicsin the last decade was the influence of predictions derived from Quantum Field Theory. This has resulted in some inroads and I expect the direction to reverse with mathematics contributing essentially to the understanding of particle physics and the ``Grand Unification'' project of giving a consistent description of known physical forces.The main goal of the project is to take certain steps in this directionby explaining mathematically certain elements of String Theory and Quantum Field Theory in general.