Geometric Langlands Correspondence

9800511 Gaitsgory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，其基本特征是采用现代代数几何的方法（例如，反常层理论、代数栈、D-模、共形场论），用于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关于朗兰兹对偶群的主丛的模空间上的反常层联系起来。在这个建议中，提出了三个特定的研究方向：1）使用Whittaker模型的几何模拟构造GL（n）的自守层; 2）研究对应于从最大环面导出的局部系统的自守层，即Eisenstein级数的情况; 3）构造对应于2维分歧局部系统的GL（2）的自守层。在上述每个方向上的进展将有助于更好地理解朗兰兹对应的一般结构以及相关的数论和表示论问题。这个建议在数学中被称为朗兰兹纲领的部分。朗兰兹纲领是数论的一部分。数论是研究整数的性质，是数学最古老的分支。从一开始，数论中的问题就为在数学的各个领域创造新的思想提供了动力。朗格兰纲领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。这个建议的目的是探讨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.