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，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，可以有效地简化为有关齐次空间上的点或测度的均匀分布的问题。这些有关均等分配的问题可以从多个角度来探讨。已被证明特别适合研究此类问题的两种理论是自守形式的谱理论，它与 L 函数理论密切相关，以及动力系统理论，特别是对单能和更一般的流的研究这些同质的空间。最近，在一些工具方面取得了相当大的进展，例如 L 函数的特殊值公式，以及在更高阶环面作用下不变的测量的（部分）分类结果。特别令人兴奋的是结合这些技术的可能性，在某些情况下已经实现。拟议的 FRG 的目的是进一步研究这一思想圈，我们相信它有可能影响与上述相关的许多其他问题。这些研究的结果将是对齐次空间上群作用的动力学、自同构形式的分析理论以及对算术性质问题的（有时是意想不到的）应用的更深入的理解。本项目涉及两个经典领域之间令人惊讶的联系起源完全不同的数学：数论和动力学。数论的研究始于数千年前，很大程度上是受到有关素数的问题的推动。 另一方面，遍历理论和动力学是最近起源的数学领域，它们产生于对复杂确定性过程（例如行星运动）的长期演化的研究。一个引人注目的事实（直到最近才开始被大量利用）是，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。 该项目将加深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效的学科的知识结合起来。