Classical and Quantum Geometric Langlands Correspondence

对于代数闭域k上的光滑完备代数曲线X和约化群G，存在一个基本对象，称为X上G-丛的模空间，记为Bun(G)。自同构层理论是经典自同构函数理论的几何化，它研究了Bun(G)上的各种类型的层。最基本的是，我们研究了拟凝聚层QCoh(Bun(G))的(派生)范畴，以及各种扭D-模范畴。几何朗兰兹对应的目的是研究这个范畴与另一个类似定义的范畴之间的显著关系，这个范畴被称为G的朗兰兹对偶。正如20世纪中叶发现的那样，人们可以给出自然界中出现的所有基本对称定律的列表--这些定律被动态金图分类，也被称为根系。我们的基本研究对象，自同构轮理论，将这些对称性定律与现代数学中的另一个基本对象，即代数曲线(也称为黎曼曲面)结合在一起。朗兰兹对应的发现表明，自同构层理论本身具有更多的对称性。目前的项目旨在从最基本的角度研究这些新的对称性。

For a smooth complete algebraic curve X over an algebraically closed field k, and a reductive group G, there is a fundamental object called the moduli space of G-bundles on X, and denoted Bun(G). The theory of automorphic sheaves, which is a geometrization of the classical theory of automorphic functions, studies various categories of sheaves on Bun(G). Most fundamentally, we study the (derived) category of quasi-coherent sheaves QCoh(Bun(G)), as well as various categories of twisted D-modules. The goal of Geometric Langlands Correspondence is to study the remarkable relation of this category to the similarly defined category for another reductive group, called the Langlands dual of G. As was discovered in the middle of the 20th century, one can give a list of all fundamental laws of symmetry that occur in nature--these are classified by Dynkin diagrams, also known as root systems. Our fundamental object of study, the theory of automorphic sheaves, couples these symmetry laws with another fundamental object in modern mathematics, namely algebraic curves (also known in their incarnation as Riemann surfaces). The discovery of Langlands correspondence indicates that the theory of automorphic sheaves admits additional symmetries of its own. The current project aims to study these new symmetries at their most fundamental.