Geometric Langlands Correspondence

这个项目致力于几何朗兰兹程序的研究。这是一个相对较新的领域，它的基本特征是使用现代代数几何的方法(例如，倒槽理论、代数堆栈理论、D-模、保形场论)来研究表示论和数论中的各种问题。几何朗兰兹程序的目的是将代数曲线上关于某个约化群的局部系统与该曲线上主丛的模空间上的关于朗兰兹对偶群的倒叶联系起来。本文提出了三个具体的研究方向：1)利用Whittaker模型的几何模拟构造GL(N)的自同构层；2)研究由极大环面诱导的局部系统对应的自同构层，即Eisenstein级数的情况；3)构造GL(2)的对应于二维分支局部系统的自同构层。上述每个方向的进展都将有助于更好地理解朗兰兹通信的一般结构以及相关的数论和表象理论问题。这一建议是在被称为朗兰兹程序的数学部分提出的。朗兰兹计划是数论的一部分。数论是研究整数性质的学科，是数学中最古老的分支。从一开始，数论中的问题就提供了在所有数学领域创造新思想的驱动力。朗兰德的计划是一种将数论与微积分联系起来的一般哲学；它体现了研究整数的现代方法。这项提议的目的是探索几何技术在朗兰兹计划中的应用。

9800511GaitsgoryThis project is devoted to the study of the geometric Langlands program. This is a relatively new field whose basic feature is the employment of methods of modern algebraic geometry (e.g., the theory of perverse sheaves, algebraic stacks, D-modules, conformal field theory) for investigation of various problems in representation theory and in number theory. The purpose of the geometric Langlands program is to relate local systems on an algebraic curve with respect to some reductive group to perverse sheaves on the moduli space of principle bundles on that curve with respect to the Langlands dual group. In this proposal three particular directions of study are suggested: 1) Construction of automorphic sheaves for GL(n) using a geometric analog of the Whittaker model; 2) The study of automorphic sheaves that correspond to local systems induced from the maximal torus, i.e. the case of Eisenstein series; 3) Construction of automorphic sheaves for GL(2) that correspond to 2-dimensional ramified local systems. Progress in each of the above directions will contribute to a better understanding of the general structure of the Langlands correspondence and of the related number-theoretic and representation-theoretic problems.This proposal is in the part of mathematics known as the Langlands program. The Langlands program is part of number theory. Number theory is the study of the properties of the integral numbers and is the oldest branch of mathematics. From the outset, problems in number theory have furnished a driving force in creating new ideas in all areas of mathematics. The Langland's program is a general philosophy that connects number theory with calculus; it embodies the modern approach to the study of integral numbers. The object of this proposal is to explore the applications of geometric techniques within the Langlands program.