Langlands Duality and Moduli Spaces of G-Bundles

布雷弗曼DMS-0300271获奖摘要本建议书由三部分组成。第一部分与R.Bezrukavnikov共同研究了基场具有正特征时几何朗兰兹猜想(Beilinson And Drinfeld)的一个类比。第二部分是与M.Finkelberg和D.Gaitsgory的合作。在那里，PI建议研究射影平面上G-丛的所谓Uhlebeck空间的交上同调。这个答案对于用奈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建议将形式弧(最近由Drinfeld和Grinberg-Kazhdan定义)的所谓“垂直变形”应用于构造具有正特征的局部域上的约化群的局部L函数的问题。朗兰兹计划是数论的一部分。数论是研究全体数的性质的学科，是数学中最古老的分支。从一开始，数论中的问题就为在该学科的其他不同部分创造新的数学提供了动力。朗兰兹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。这项提议的一个方面是探索几何技术在朗兰兹计划中的应用。

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.