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. 摘要标题：亚瑟猜想、谱论和高阶解析数论摘要。该提案涉及自守形式应用的两个问题。 第一个问题是表示论中的问题：它是局部调和分析中的一个猜想，是由Arthur的猜想与Burger、Li和Sarnak的结果结合起来提出的。这个问题作为表示论中的一个问题很有趣。它还为亚瑟猜想提供了一个试验场，并提供了更好地理解自同构谱的可能性。第二个问题是在高阶群的自同构形式的背景下研究解析数论。 其梦想目标是更好地理解 L 函数的高阶矩，但是存在许多更简单和具体的问题，例如大筛不等式的发展，其解决方案也将对解析数论产生直接影响。该项目涉及“自同构形式”领域的两个问题。这是一个相对较新的数学领域，由朗兰兹纲领指导——它寻求在某些（显然）相距甚远的数学领域之间建立联系。这些联系使得自同构形式的工作能够在其他领域产生重大影响。许多密码算法（对于互联网上的安全通信所必需的）都是基于素数的非常微妙的属性，并且这些算法中的许多算法都是从解析数论和数学中得出的困难结果。自守形式。自守形式的另一个应用是构建“拉马努金图”——这些图具有显着的连通性，并且已应用于通信网络和理论计算机科学。正在考虑的问题将加深我们对自守形式的理解。除了刚刚讨论的应用类型之外，这些问题还处于数学不同领域的交叉点，并将鼓励这些不同领域的专家之间的合作。