Limit distributions on homogeneous spaces: Interplay of dynamics and number theory

齐次空间上的极限分布：动力学与数论的相互作用

• 批准号: 1001654
• 负责人: Nimish Shah
• 金额: $ 13.4万
• 依托单位: Ohio State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2013-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1001654&HistoricalAwards=false
• 关键词: Limit; distributions; homogeneous; spaces; Interplay

In a wide variety of questions in number theory one often recognizes symmetries and invariance under group actions; and deep insights can be gained into such problems by understanding dynamics of subgroup actions on homogeneous spaces of Lie groups. In several such instances it is necessary to describe limiting distributions of sequences of translates of smooth measures on submanifolds of homogeneous spaces. Building up further from his earlier work on some typical cases, the principal investigator seeks in this project to develop new techniques in homogeneous dynamics to provide very general natural conditions on the manifolds and translating elements that lead to equidistribution of the limit measures. The work should lead to resolutions of some questions on metric properties of Diophantine approximation, counting integral and rational points on group varieties, and counting points on orbits of geometrically finite groups associated with certain geometric configurations. From a broader perspective, the research will apply the concepts and methods of ergodic theory and dynamical systems about the long-time behavior of a generic particle in a flow, which is originally a physical problem, to answer questions about deep properties of whole numbers. This conceptual bridge between two very different disciplines uses and influences other areas like representation theory, analysis, and differential geometry, not to mention physics.

在数论的各种问题中，人们经常认识到群作用下的对称性和不变性;通过理解李群齐次空间上子群作用的动力学，可以深入了解这些问题。在几个这样的情况下，有必要描述极限分布序列的平移光滑措施的子流形的齐次空间。进一步建立从他早期的工作在一些典型的情况下，主要研究人员在这个项目中寻求开发新技术在齐次动力学提供非常普遍的自然条件的流形和翻译元素，导致equidistribution的限制措施。这项工作应导致决议的一些问题的度量性质的丢番图逼近，计数积分和合理的点群品种，计数点轨道的几何有限群与某些几何配置。 从更广泛的角度来看，该研究将应用遍历理论和动力系统的概念和方法，关于流中一般粒子的长时间行为，这本来是一个物理问题，以回答有关整数的深层性质的问题。这两个非常不同的学科之间的概念桥梁使用和影响其他领域，如表示论，分析和微分几何，更不用说物理学。