Arithmetic and Analysis on Locally Symmetric Spaces and Applications

局部对称空间的计算与分析及应用

• 批准号: 0500191
• 负责人: Peter Sarnak
• 金额: --
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2010-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0500191&HistoricalAwards=false
• 关键词: Arithmetic; Analysis; Locally; Symmetric; Spaces

(mathematical) This proposal is concerned with the analytic theoryof automorphic forms. Specifically the study of the sizeof a general automorphic L-function on the critical lineand the size of an eigenfunction on an arithmetic locally symmetricspace .In both cases the main goal is to establish a subconvexestimate . In the first case the sharpest form of such an estimatewould follow from the Grand Riemann Hypothesis .However many of thedesired applications of such estimates only require subconvexity.The applications are varied ,one such being the recent subconvexestimate by Cogdell,Piatetsky-Shapiro and the proposer which allowsfor the resolution of Hilbert's 11th problem on the representationof integers by quadratic forms in a number field.Other applicationsare to problems in mathematical physics ,specifically the behaviorof quantum states in quantizations of classically chaotic systems. Thesecond problem of the size of an eigenfunction on such locallysymmetric spaces is closely associated with the first and the proposal is concerned with understanding this more difficult problem.It constitutes a generalization of the Ramanujan conjectures. (general): The proposal is concerned with the study of special types of geometric spaces which are defined via arithmetic and number theoretic constructions. This has been an active area of investigation for the last 60 or so years ,primarily since it carries some of the most powerful tools that we know in number theory (to diophantine problemsas well as ones associated with prime numbers). Specifically theseemingly technical issues that are being investigated have applicationsto resolve some simple well known problems in number theory. One such isone of Hilbert's problems from 1900 concerning which numbers are sums of3 integer squares in an extention of the ordinary whole numbers tointegers in a number field. Other applications of this theory are perhapsless traditional and are to Mathematical Physics,specifically QuantumChaos. These spaces provide one of the few (in fact the only one) classically chaotic Hamiltonian systems whose quantization can be satisfactorally studied mathematically. The techniques to do so are number theoretic and the results are often quite surprizing.

（数学）这个提议涉及自守形式的分析理论。具体来说，研究临界线上一般自同构 L 函数的大小和算术局部对称空间上本征函数的大小。在这两种情况下，主要目标都是建立次凸估计。在第一种情况下，这种估计的最尖锐形式将来自大黎曼假设。然而，这种估计的许多所需应用只需要次凸性。应用是多种多样的，其中一个是 Cogdell、Piatetsky-Shapiro 和提议者最近提出的次凸估计，它允许解决希尔伯特关于用数中的二次形式表示整数的第 11 个问题。 其他应用涉及数学物理问题，特别是经典混沌系统量子化中量子态的行为。这种局部对称空间上的特征函数的大小的第二个问题与第一个问题密切相关，该提案涉及理解这个更困难的问题。它构成了拉马努金猜想的推广。 （一般）：该提案涉及通过算术和数论构造定义的特殊类型几何空间的研究。在过去 60 年左右的时间里，这一直是一个活跃的研究领域，主要是因为它拥有我们所知道的数论中一些最强大的工具（针对丢番图问题以及与素数相关的问题）。 具体来说，正在研究的看似技术问题可用于解决数论中一些简单的众所周知的问题。其中一个问题是 1900 年提出的希尔伯特问题，该问题涉及数字是普通整数到数域中整数的扩展中的 3 个整数平方和。该理论的其他应用也许不那么传统，并且是数学物理，特别是量子混沌。这些空间提供了少数（实际上是唯一的）经典混沌哈密顿系统之一，其量化可以得到令人满意的数学研究。这样做的技术是数论的，结果往往令人惊讶。