Arithmetic and Analysis on Locally Symmetric Spaces and Applications

这个建议是关于自同构形式的解析理论的。具体地研究了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特征函数的大小。在这两种情况下，主要目标都是建立一个次凸估计。在第一种情况下，这种估计的最尖锐形式将遵循大黎曼假设。然而，这种估计的许多期望应用只需要次凸性。其应用是多种多样的，其中一个是最近由Cogdell，Piatetsky-Shapiro和提议者提出的次凸估计，它允许解决希尔伯特的第11个问题，即整数在数字域中的二次型表示。其他应用是数学物理问题，特别是经典混沌系统量子化中的量子态行为。在这种局部对称空间上的特征函数大小的第二个问题与第一个问题密切相关，本文的建议是关于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结构定义的特殊类型的几何空间的研究。在过去60年左右的时间里，这一直是一个活跃的研究领域，主要是因为它包含了我们所知道的数论中一些最强大的工具（用于丢芬图问题以及与素数相关的问题）。具体来说，正在研究的这些看似技术性的问题可以应用于解决数论中一些简单的众所周知的问题。1900年希尔伯特的一个问题就是这样的：在一个数字域中，数字是普通整数到整数的扩展中3个整数平方的和。这一理论的其他应用可能不那么传统，而是应用于数学物理，特别是量子混沌。这些空间提供了为数不多的（事实上是唯一的）经典混沌哈密顿系统之一，其量化可以在数学上得到满意的研究。这样做的技术是数论的，结果往往是相当惊人的。

(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.