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，关于大型不可服从群的轨道的统计性质的结果很少，这个项目的一部分解决了作用在齐次空间上的李群中的格比特的分布问题。这些方法利用了单位元流的等分布性质(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收敛速度的有效估计，因此它打算发展一种不同的方法来给出有效误差项。该方案的预期结果在数论中有几个潜在的应用。特别是，我们计划研究由线性和二次型组成的系统在整点上的值的分布。另一个研究方向是研究尼氏流形的大群自同构群的混合性质。这个项目的主要目的是研究不同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许多不同的数学领域：遍历理论、数论(例如素数分布)、几何(例如紧曲面上闭测地线的分布)、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，这个族的统计性质为了解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关于整点分布的数论结果。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。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一个很好的介绍积极研究的机会，并可用于不同水平的观众所能接触到的演讲。

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.