Special Meeting: Collaborative Research: Affine Hecke algebras, the Langlands Program, Conformal Field Theory and Matrix Models

特别会议：协作研究：仿射赫克代数、朗兰兹纲领、共形场论和矩阵模型

• 批准号: 0603329
• 负责人: Pavel Etingof
• 金额: $ 15万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-05-01 至 2008-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0603329&HistoricalAwards=false
• 关键词: Special; Meeting; Collaborative; Research; Affine

The conference ``Affine Hecke algebras, Langlands program, conformal field theory, and matrix models'' will bring together a diverse group of leading mathematicians and physicists and young researchers, to discuss striking recentdevelopments in these fields, and allow experts in one of them tolearn about the others. The conference will consist of three parts: 1) Affine Hecke algebras (1 week) 2) Langlands program (1 week)3) conformal field theory and matrix models (2 weeks).There are numerous and deep connections between these areas, some of whichwere found quite recently. To explore these connections is one ofthe main goals of the conference. Some of the topics to bediscussed in the conference are: representations of affine and double affine Hecke algebras; cyclotomic Hecke algebras; Macdonald theory; geometric representation theory; representations of double loop groups; representations of affine Lie algebras at the critical level; representations of p-adic groups; geometric Langlands conjecturesand their connections to string theory; Seiberg-Witten theory; AdS-CFTcorrespondence; relations between matrix models and string theory. The Langlands program, formulated by R. Langlands in 1967 in hisletter to A. Weil, is a far-reaching program deeply connectingrepresentation theory and number theory. It is crucial forunderstanding central number-theoretic objects called automorphicforms. Affine Hecke algebras are algebraic structures which playa crucial role in the Langlands program. Conformal field theoriesare physical models for the behavior of quantum particles, which are especially tractable mathematically, because of presence of a large amount of symmetry, called conformal symmetry. Such theories (in two spacetime dimensions) are also importantin formulating the basics of string theory. Matrix models is essentially a branch of probability theory, dealing withproperties of random matrices. However, it recently turned out that they have deep applications to string theory. In the last 10 years, there has been significant progress in allfour areas, and it became clear that they are intimatelyrelated. The conference is designed to help experts and youngresearchers (including many women and minorities) to explorethis progress and connections.

会议“仿射Hecke代数，朗兰兹计划，共形场论和矩阵模型”将汇集一个不同的领导数学家和物理学家和年轻的研究人员，讨论这些领域的惊人的最新发展，并允许其中一个领域的专家学习其他领域。会议将包括三个部分：1）仿射Hecke代数（1周）2）Langlands程序（1周）3）共形场论和矩阵模型（2周）这些领域之间有许多深刻的联系，其中一些是最近发现的。探索这些联系是会议的主要目标之一。会议讨论的题目有：仿射和双仿射Hecke代数的表示;分圆Hecke代数; Macdonald理论;几何表示理论;双环群的表示;仿射李代数在临界水平上的表示; p-adic群的表示;几何Langlands表示及其与弦理论的联系; Seiberg-Witten理论; AdS-CFT对应;矩阵模型与弦理论之间的关系Langlands纲领是由R.朗兰兹在1967年给A. Weil，是一个意义深远的程序，将表示论和数论紧密联系在一起。这对于理解称为自同构形式的中心数论对象至关重要。仿射Hecke代数是在Langlands纲领中起着重要作用的代数结构。共形场论是量子粒子行为的物理模型，由于存在大量的对称性，称为共形对称性，因此在数学上特别容易处理。这样的理论（在两个时空维度上）在阐述弦理论的基础上也很重要。矩阵模型本质上是概率论的一个分支，主要研究随机矩阵的性质。然而，最近发现它们在弦理论中有很深的应用。在过去10年中，所有四个领域都取得了重大进展，显然它们是密切相关的。这次会议旨在帮助专家和研究者（包括许多妇女和少数民族）探索这一进展和联系。