Arthur's Conjecture, Spectural Theory, and Analytic Number Theory in Higher Rank

亚瑟猜想、谱论和高阶解析数论

• 批准号: 0245606
• 负责人: Akshay Venkatesh
• 金额: --
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-04-01 至 2008-02-29
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0245606&HistoricalAwards=false
• 关键词: Arthur; Conjecture; Spectural; Theory; Analytic

DMS-0245606Vogan, David A.AbstractTitle: Arthur's Conjecture, Spectural Theory, and Analytic Number Theory in Higher RankAbstract. The proposal concerns two problems with applications toautomorphic forms. The first problem is in representation theory:it is a conjecture in local harmonic analysis that ismotivated by taking Arthur's conjectures together with resultsof Burger, Li and Sarnak. This problem is of interest as aquestion in representation theory; it also offers a testingground for Arthur's conjectures and affords the possibilityof a better understanding of the automorphic spectrum. Thesecond problem is to study analytic number theory in the contextof automorphic forms on groups of higher rank. The dream goal ofthis is a better understanding of higher moments of L-functions,but there are a number of easier and concrete problems, such asthe development of large-sieve inequalities, whose solution wouldalso have immediate consequences for analytic number theory. The project concerns two questions in the field of``automorphic forms.'' This is a relatively new field of mathematics,guided by the Langlands program -- it seeks to establishconnections between certain (apparently) far-separatedareas of mathematics. These connections have allowed work inautomorphic forms to have a significant impact in other fields.Many cryptographic algorithms -- necessary for secure communicationover the Internet -- are based onvery subtle properties of prime numbers, and underlyingmany of these algorithms are difficult results from analytic numbertheory and automorphic forms. Another applicationof automorphic forms has been the construction of ``Ramanujan graphs''-- these are graphs with remarkable connectivity, and have hadapplication to communication networks and to theoretical computerscience. The questions under consideration will deepenour understanding of automorphic forms. In additionto the type of application just discussed,these questions lie at the intersection of different fields ofmathematics, and will encourage collaboration between expertsin these different fields.

DMS-0245606Vogan，David A.Abstracttitle：较高的Rankabstract中的Arthur的猜想，幽灵理论和分析数理论。该提案涉及应用的两个问题。 第一个问题是代表理论：它是局部谐波分析中的一个猜想，它通过将亚瑟的猜想与汉堡，李和萨尔纳克的结果一起吸引而激发。这个问题是代表理论中的水生的感兴趣。它还为亚瑟的猜想提供了一个测试地，并可以更好地理解对汽车频谱的可能性。第三个问题是在较高等级的群体上研究自动形式的上下文中的分析数理论。 这一梦想的目标是更好地理解L功能的更高时刻，但是存在许多更简单，具体的问题，例如大型宣告不平等的发展，其解决方案的解决方案也对分析数理论产生了直接的后果。该项目涉及“自动形态形式”领域的两个问题。这是一个相对较新的数学领域，在兰兰兹计划的指导下 - 它试图在某些（显然）远距离的数学学位之间建立连接。这些连接使工作不形式形式可以在其他领域产生重大影响。许多加密算法（对于安全通信互联网所必需的互联网所必需）是基于质量数字的基本微妙属性，这些算法的基础是分析性数字理论和自动形态形式的困难。自动形式的另一种应用是构造``Ramanujan graphs''的构建 - 这些图形具有显着连接性，并且对通信网络和理论计算机科学具有无关。所考虑的问题将使对自动形式的理解。除了刚刚讨论的应用程序的类型外，这些问题还位于不同领域的不同领域的交集，并将鼓励专家之间在这些不同领域之间的协作。