Ergodic-Theoretic Properties of Actions of Discrete Groups

离散群作用的遍历理论性质

• 批准号: 0527082
• 负责人: Alexander Gorodnik
• 金额: $ 1.77万
• 依托单位: California Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-09-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0527082&HistoricalAwards=false
• 关键词: Ergodic; Theoretic; Properties; Actions; Discrete

In this project, the PI will study the actions of Lie groups and their discretesubgroups from ergodic-theoretic point of view. Note that only few resultsabout statistical properties of orbits of large nonamenable groups are known.A part of this project addresses the question about the distribution oforbits of lattices in Lie groups acting on homogeneous spaces. The methods,which utilize the equidistribution properties of unipotent flows (due to Ratnerand others), apply to a large variety of natural actions of lattices. Sincethe methods based on the Ratner theory do not provide effective estimateson the rate of convergence, it is intended to develop a different approachthat gives effective error terms. The expected results of the proposal haveseveral potential applications to number theory. In particular, we plan toinvestigate distribution of values at integer points of systems consistingof linear and quadratic forms. Another direction of research is the studyof mixing properties of large groups of automorphisms of nilmanifolds.The main objective of this project is to investigate distribution of orbitsof large discrete and continuous groups of motions on various spaces.Questions about asymptotic distribution of families of mathematical objectsappear in many different areas of mathematics: ergodic theory, number theory(e.g., distribution of prime numbers), geometry (e.g., distribution of closedgeodesics on a compact surface), PDE (e.g., distribution of eigenvalues of theLaplace operator), and others. When a family of the objects exhibits verycomplicated behavior, statistical properties of this family provide importantinsights into its structure. In this project, we use methods of the theory ofdynamical systems to derive results in number theory on the distribution ofinteger points. Our study uncovers promising interplay between ergodic theoryand number theory simultaneously enriching both of these areas of research.This subject and its connections with many other branches of mathematicsprovide an excellent introduction for graduate as well as undergraduatestudents to active research and can be used for presentations accessible toaudiences of different levels.

在这个项目中，PI将从遍历理论的角度研究李群及其离散子群的行为。请注意，关于大型不可命名群的轨道统计特性的结果很少。该项目的一部分解决了作用于齐次空间的李群中晶格轨道的分布问题。这些方法利用单能流的等分布性质（由拉特纳等人提出），适用于晶格的多种自然作用。由于基于拉特纳理论的方法不能提供对收敛速度的有效估计，因此旨在开发一种给出有效误差项的不同方法。该提案的预期结果对数论有几个潜在的应用。特别是，我们计划研究由线性和二次形式组成的系统整数点处的值分布。另一个研究方向是研究尼尔曼流形大群自同构的混合性质。该项目的主要目标是研究各种空间上大的离散和连续运动群的轨道分布。关于数学对象族的渐近分布的问题出现在数学的许多不同领域：遍历理论、数论（例如素数的分布）、几何（例如素数的分布） 紧凑表面上的闭合测地线）、偏微分方程（例如，拉普拉斯算子的特征值分布）等。当一个对象族表现出非常复杂的行为时，该族的统计特性为其结构提供了重要的见解。在这个项目中，我们使用动力系统理论的方法来推导出整数点分布的数论结果。我们的研究揭示了遍历理论和数论之间有希望的相互作用，同时丰富了这两个研究领域。该主题及其与许多其他数学分支的联系为研究生和本科生提供了积极研究的精彩介绍，并且可用于向不同水平的观众进行演示。