Collaborative Research: Arithmetic and Equidistribution on Homogeneous Spaces

近年来，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，可以有益地归结为关于齐次空间上的点或度量的均匀分布的问题。关于均匀分布的问题可以从多个角度来探讨。已被证明特别适合于研究这类问题的两个理论是与L函数理论密切相关的自同构型谱理论和动力系统理论，特别是研究这些齐次空间上的酉幂等流和更一般的流。最近，关于L函数的特值公式和高阶环面作用下的度量不变量的(部分)分类结果等工具已经取得了相当大的进展。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在一起的可能性。拟议的FRG的目的是进一步调查这一思想圈，我们认为这可能会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的群作用的动力学、自同构形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。本项目涉及两个起源完全不同的经典数学领域之间的惊人联系：数论和动力学。数论的研究始于几千年前，在很大程度上是由关于原数的问题推动的。另一方面，遍历理论和动力学是较新起源的数学领域，起源于研究复杂的确定性过程的长期演化--例如行星运动。这是一个引人注目的事实(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。这个项目将加深我们对这种相互关系的理解，以及我们如何将这两个富有成效的学科的知识有效地结合在一起。

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. Thesequestions 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 theoryof 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 isthe 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 automorphicforms, 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 ofyears 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 (whichhas 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.