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

DMS-0245606Vogan，David A.摘要标题：亚瑟猜想，特殊理论，和高等数论。该提案涉及应用于自同构形式的两个问题。第一个问题是在表示理论中：它是局部调和分析中的一个猜想，它是由Arthur猜想与Burger，Li和Sarnak的结果一起引起的。这个问题是表象理论中的一个有趣的问题；它也为阿瑟的猜想提供了一个试验场，并为更好地理解自同构谱提供了可能性。第二个问题是在高阶群的自同构型的背景下研究解析数理论。梦想的目标是更好地理解L函数的高阶矩，但还有一些更容易和具体的问题，如大筛不等式的发展，这些问题的解决也将对解析数论产生直接的影响。该项目涉及“自同构形式”领域的两个问题。在朗兰兹计划的指导下，这是一个相对较新的数学领域--它试图在某些(显然)相距甚远的数学领域之间建立联系。这些联系使得自形形式的工作在其他领域产生了重大影响。许多密码算法--在互联网上进行安全通信所必需的--都基于质数的非常微妙的性质，而这些算法中的许多都是解析数论和自形形式的困难结果。自同构图的另一个应用是构造“Ramanujan图”--这些图具有显著的连通性，在通信网络和理论计算机科学中都有应用。正在考虑的问题将加深我们对自同构形式的理解。除了刚才讨论的应用类型之外，这些问题还存在于不同数学领域的交叉，并将鼓励这些不同领域的专家之间的合作。

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.