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，很明显，许多有趣的问题，尤其是算术，量子混乱和功能理论的问题，可能会降低到有关点的问题或均匀空间的措施的问题。这些有关等分分配的问题可以从多个旋转中解决。事实证明，这两种理论特别适合研究，这些问题是自动形式的频谱理论，它们与L功能理论密切相关，动力学系统的理论，尤其是对这些同质空间的单位和一般流量的研究。最近，涉及诸如特殊价值公式的工具以及（部分）分类结果的工具方面取得了巨大的进步，用于较高级别的圆环动作下的测量结果。特别令人兴奋的是，在某些情况下已经意识到了结合这些技术的可能性。拟议的FRG的目的是调查这一思想循环，我们认为这可能会影响与上述有关的许多其他问题。这些评估的结果将更深入地了解均匀空间的群体动态，对自动形式的分析理论以及对Arithmmetexthy问题的（有时意外的）应用（有时意外的）应用。目前的项目与相当不同的差异原点和动力学的数学数学数学和动力学的数学数学和数量理论：数字理论和数量理论学：数量理论和数字。数量理论的研究始于数千年前，在很大程度上是由于有关临时数学的问题而动机。 另一方面，近期出处的奇异理论和动力学浮膜领域，这是从研究复杂的确定性源源物（例如行星运动）的长期演变中产生的。这是一个惊人的事实（直到最近才开始被大量利用），在某些文章中，来自ergodic理论的思想与数量理论的经典问题非常深入。 该项目将增强我们对这一相互关系的理解，以及我们如何有效地结合这两个富有成果的学科的知识。