Ergodic theory of large groups, counting, and equidistribution

大群遍历理论、计数和均匀分布

• 批准号: 0654413
• 负责人: Alexander Gorodnik
• 金额: $ 8.01万
• 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: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0654413&HistoricalAwards=false
• 关键词: Ergodic; theory; large; groups; counting

AbstractGorodnikIn 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作用。注意，关于大型非顺从群的轨道的统计性质的结果很少，本项目的一部分解决了作用在齐次空间上的李群中格的轨道分布问题。这些方法利用了幂单流的等分布性质（由于Ratner等人），适用于各种各样的格的自然作用。由于基于Ratner理论的方法不能提供有效的收敛速度估计，因此打算开发一种不同的方法，给出有效的误差项。该建议的预期结果在数论中有几个潜在的应用。特别地，我们计划研究由线性和二次型组成的系统的整数点处的值的分布。另一个研究方向是研究nilmanifold的大群自同构的混合性质。这个项目的主要目标是研究各种空间上的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许多不同的数学领域：遍历理论，数论（例如，素数的分布），几何形状（例如，紧密表面上的闭合测地线的分布），PDE（例如，拉普拉斯算子的本征值的分布）等。当一个对象族表现出非常复杂的行为时，这个对象族的统计特性提供了对其结构的重要观察。在这个项目中，我们使用动力系统理论的方法来推导数论中关于整数点分布的结果。我们的研究揭示了遍历理论和数论之间有前途的相互作用，同时丰富了这两个领域的研究。这一主题及其与许多其他分支的联系，为研究生和本科生提供了一个很好的介绍，以积极的研究，并可用于演示访问不同层次的观众。