Ergodic-Theoretic Properties of Actions of Discrete Groups

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，只有很少的结果是关于轨道的统计性质的大不可调群已知。这个项目的一部分解决了作用于齐次空间的李群中格的禁分布问题。这些方法利用了单幂流的等分布特性（由于拉特纳和其他人），适用于各种各样的格的自然作用。由于基于拉特纳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来给出有效的误差项。该提议的预期结果对数论有几个潜在的应用。特别地，我们计划研究由线性和二次型组成的系统在整数点上的值分布。另一个研究方向是研究零流形的大群自同构的混合性质。本课题的主要目的是研究大型离散和连续运动群在不同空间上的轨道分布。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：遍历论、数论（如数学）。几何（如紧曲面上闭测地线的分布）、偏微分方程（如拉普拉斯算子特征值的分布）等。当一类物体表现出非常复杂的行为时，这类物体的统计特性提供了对其结构的重要见解。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希望的相互作用，同时丰富了这两个研究领域。这门学科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极好介绍，并可用于不同层次的听众的演讲。

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.