Geometric Langlands Transform and Dualities

主要研究人员将进行几何朗兰兹程序的研究。给定一个连接光滑投影代数曲线X /零和一个还原G组的特点,这个项目的目标是构建一个规范DG类别之间的等价D-modules G-bundles X和一个堆栈的栈上的某些DG o模块类别相关的本地系统的Langlands双X对G的主要调查人员将致力于建设这个等效为G = GL (n)。利用非交换几何的方法，他们将研究d模的DG范畴上对应于o模的Serre对偶的对偶性。他们还将研究这种对偶的经典自同构对应物，即自同构形式空间上的非标准标量积。朗兰兹程序是在1967年作为数论和自同构形式理论中的一系列猜想而形成的。后来，几何朗兰兹程序在代数闭场的代数几何中被公式化，并在量子场论中被解释为物理上的对偶。本课题的目的是在一个重要的特殊情况下证明几何朗兰兹程序的主要猜想。它结合了几何和经典自同构形式理论以及非交换几何的思想。

The principal investigators will conduct research in the geometric Langlands program. Given a connected smooth projective algebraic curve X over a field of characteristic zero and a reductive group G, the goal of this program is to construct a canonical equivalence between the DG category of D-modules on the stack of G-bundles on X and a certain DG category related to O-modules on the stack of local systems on X with respect to the Langlands dual of G. The principal investigators will work on the construction of this equivalence for G=GL(n). Using methods of non-commutative geometry, they will study the duality on the DG category of D-modules that corresponds to Serre duality for O-modules. They will also study the classical automorphic counterpart of this duality, which is a non-standard scalar product on the space of automorphic forms.The Langlands program was formulated in 1967 as a series of conjectures in number theory and the theory of automorphic forms. Later the geometric Langlands program was formulated in terms of algebraic geometry over algebraically closed fields and interpreted in physical terms as a duality in quantum field theory. This project is aimed at proving the main conjecture of the geometric Langlands program in an important particular case. It combines ideas from the geometric and classical theory of automorphic forms as well as non-commutative geometry.