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

DMS-0245606 Vogan，大卫A.摘要标题：亚瑟的猜想，谱理论和高阶解析数论摘要。这个建议涉及两个问题的应用tosautomorphic形式。 第一个问题是在表示论：这是一个猜想，在当地的调和分析，动机是采取亚瑟的turtures一起的结果伯格，李和Sarnak。这个问题作为表示论中的一个问题很有趣，它也为亚瑟的理论提供了一个试验场，并提供了更好地理解自守谱的可能性。第二个问题是在高阶群的自守形式的背景下研究解析数论。 这个梦想的目标是更好地理解L-函数的高阶矩，但也有一些更容易和具体的问题，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“自守形式”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。这些联系使得非自守形式的工作在其他领域产生了重大影响。许多加密算法--通过互联网进行安全通信所必需的--都是基于素数的非常微妙的性质，而这些算法中的许多算法都是从解析数论和自守形式中得到的困难结果。自守形式的另一个应用是“拉马努金图”的构造--这些图具有显著的连通性，并已应用于通信网络和理论计算机科学。所考虑的问题将加深我们对自守形式的理解。除了刚才讨论的应用类型之外，这些问题位于不同数学领域的交叉点，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.