FRG: Collaborative Research: Arithmetic and equidistribution on homogeneous spaces

FRG：协作研究：齐次空间上的算术和等分布

• 批准号: 0554365
• 负责人: Akshay Venkatesh
• 金额: $ 28.5万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-06-01 至 2008-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0554365&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Arithmetic; equidistribution

In recent years, it has become clear that many interesting problems,in particular problems in arithmetic, quantum chaos and the theory ofL-functions, may be profitably reduced to questions concerningequidistribution of points or measures on homogeneous spaces. These questions regarding equidistribution can be approached from manyangles. Two theories which have proved to be particularly well-suitedto study such questions are the spectral theory of automorphic forms,which is closely related to the theory of L-functions, and the theory of dynamical systems, particularly the study of unipotent and moregeneral flows on these homogeneous spaces. Recently there has beenconsiderable progress involving tools such as special value formulaefor L-functions, and (partial) classification results for measuresinvariant under higher rank torus actions. Particularly exciting is the possibility, already realized in some instances, of combiningthese techniques. The purpose of the proposed FRG is to investigatefurther this circle of ideas, which we believe has the potential toimpact many other problems related to the above. The result of theseinvestigations will be a deeper understanding of the dynamics of groupactions on homogeneous spaces, of the analytic theory of automorphic forms, and the (sometimes unexpected) applications to problems ofarithmetic nature.The present project is concerned with a surprising link between twoclassical fields of mathematics of quite disparate origin: numbertheory and dynamics. The study of number theory began thousands of years ago, motivated, in significant part, by questions about primenumbers. On the other hand, ergodic theory and dynamics aremathematical fields of more recent provenance, which arose fromstudying the long-term evolution of complicated deterministicprocesses -- such as planetary motion. It is a striking fact (which has only recently begun to be heavily exploited) that, in certaincontexts, ideas from ergodic theory interact very deeply with classicalproblems in number theory. This project will enhance ourunderstanding of this inter-relation and how we can combine knowledgefrom both of these fruitful disciplines effectively.

近年来，人们已经清楚地认识到，许多有趣的问题，特别是算术、量子混沌和L-函数理论中的问题，可以有利地归结为关于齐次空间上的点或测度的均匀分布的问题。这些关于平均分配的问题可以从多个角度来探讨。两个理论已被证明是特别适合研究这些问题的自守形式的谱理论，这是密切相关的理论的L-功能，和理论的动力系统，特别是研究的单能和更一般的流动这些齐次空间。最近有相当大的进展，涉及的工具，如特殊的值formulaefor L-函数，和（部分）分类结果的测度不变下的高秩环面行动。特别令人兴奋的是，在某些情况下已经实现了将这些技术结合起来的可能性。拟议的联邦德国的目的是进一步调查这一圈的想法，我们相信有可能影响许多其他问题与上述有关。这些研究的结果将是对齐次空间上群作用的动力学、自守形式的分析理论以及对算术性质问题的应用（有时是意想不到的）有更深的理解。本项目关注的是两个起源完全不同的经典数学领域之间令人惊讶的联系：数论和动力学。对数论的研究始于几千年前，在很大程度上是出于对素数的质疑。 另一方面，遍历理论和动力学是最近起源的数学领域，它们起源于研究复杂的确定性过程的长期演化--比如行星运动。这是一个惊人的事实（直到最近才开始被大量利用），在某些情况下，遍历理论的思想与数论中的经典问题相互作用非常深刻。 这个项目将加强我们对这种相互关系的理解，以及我们如何有效地将这两个富有成果的学科的知识联合收割机结合起来。