Langlands Duality and Moduli Spaces of G-Bundles

Braverman DMS-0300271获奖摘要本提案由3部分组成。第一部分是联合R.Bezrukavnikov和致力于研究类似的几何朗兰兹猜想（如制定的贝林森和德林费尔德）的情况下，基场有积极的特点。 第二部分是与M.Finkelberg和D. Gaitsgory联合编写的。有PI提出研究交叉上同调的所谓Uhlebeck空间的G-丛上的投影平面。这个答案对于用涅克拉索夫最近提出的方法研究超对称规范理论应该是有用的。 在第三部分中，PI提出了应用形式弧的所谓的“非线性变形”（最近由Drinfeld和Grinberg-Kazhdan定义）来构造局部正特征域上约化群的局部L-函数的问题。这个建议在数学中被称为Langlands程序。 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这门学科的其他不同部分创造新的数学提供了动力。朗兰兹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。这个建议的一个方面是探讨几何技术在朗兰兹程序中的应用。

Abstract for award of Braverman DMS-0300271This proposal consists of 3 parts. The first part is joint with R.Bezrukavnikov and is devoted to the study of an analog of the geometric Langlands conjecture (as formulated by Beilinson and Drinfeld) in the case when the base field has positive characteristic. The second part is joint with M.Finkelberg and D.Gaitsgory. There the PI proposes to study the intersection cohomology of the so called Uhlebeck spaces of G-bundles on the projective plane. The answer should be useful for studying super-symmetric gauge theories in the way suggested recently by Nekrasov. In the third part the PI proposes to apply the so called "versal deformations" of formal arcs (defined recently by Drinfeld and Grinberg-Kazhdan) to the problem of construction of local L-functions for reductive groups over a local field of positive characteristic.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 whole numbers and is the oldest branch of mathematics. From the beginning problems in number theory have furnished a driving force in creating new mathematics in other diverse parts of the discipline. The Langlands program is a general philosophy that connects number theory with calculus; it embodies the modern approach to the study of whole numbers. One aspect of this proposal is to explore the applications of geometric techniques within the Langlands program.