Geometry of Langlands duality

米尔科维奇的建议是针对几何基础的朗兰兹计划。一个主要的和不断扩大的影响是来自量子场论的思想。"临界量子化“计划旨在发展一种形式主义，将在临界水平上统一模表示理论和仿射李代数表示的方法，并且应该具有广泛的研究，包括对相干层的导出类别的研究。”二次Koszul对偶“项目试图对出现在整个表示理论中的对偶族给出年龄测量方法，最后是CherednikFourier变换。 最后，我们提出了类场论和朗兰兹纲领的“完全代数几何”方法，该项目旨在联系和统一数学（表示论、代数几何、数论）和理论粒子物理（量子场论，特别是弦论）的发展。在过去的十年里，量子力学的主要发展是来自量子场论的预测的影响。这已经导致了一些进展，我希望方向与数学基本上有助于理解粒子物理学和“大统一”的项目，给出一个一致的描述已知的物理力。该项目的主要目标是采取一定的步骤，在这个方向上通过数学解释弦理论和量子场论的某些元素一般。

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.