Effective and sparse equidistribution problems on homogeneous spaces and linear dynamics of semisimple groups

PI(Nimish A.Shah)建议与他的研究生一起研究李群齐次空间上的广泛的极限分布问题，目的是应用于数论和算术几何中的问题。主要集中在以下三个方面：(1)酉流的轨道的有效密度和有效等分布；(2)齐次空间上子流形的伸缩平移的极限分布的刻画；(3)在部分双曲流作用下不稳定叶上相对稀疏点集的等分布。这项研究将结合不同数学领域的技术和结果：半单李群理论及其有限和无限维表示，遍历理论和动力系统，概率论，数论和自同构形式，代数几何等。这项工作将进一步发展这些领域之间的桥梁，并产生有趣的新结果和技术，具有动力学、数论、几何和组合性质。这项研究连接了几个重要的数学领域：我们应用遍历理论和动力系统(物理学中有起源的领域)的概念来解决数论中的一类问题，使用李群、代数和微分几何以及概率论的技术。所有这些领域都与物理、天文学、统计学和计算机科学等其他科学有着深刻的联系。拟议项目的广泛目的是调查一系列想法和创造有可能影响所有这些领域的许多感兴趣的问题的技术。该项目的一个重要方面将是培训研究生和博士后研究员进行研究，将一个领域的想法应用于解决另一个领域的问题。

The PI (Nimish A. Shah) proposes to investigate, together with his graduate students, a wide range of limit distribution problems on homogenous spaces of Lie groups with an aim of applications to questions in number theory and arithmetic geometry. The following three areas will be focused on: (1) effective density and effective equidistribution of orbits of unipotent flows, (2) description of limit distributions of stretching translates of sub-manifolds on homogeneous spaces, and (3) equidistribution of relatively sparse set of points on unstable leaves under the action of a partially hyperbolic flow. This study will combine techniques and results from diverse areas of mathematics: theory of semi-simple Lie groups and their finite and infinite dimensional representations, ergodic theory and dynamical systems, probability theory, number theory and automorphic forms, algebraic geometry, etc. The work will further the development of bridges between these areas, and yield interesting new results and techniques, of dynamical, number theoretical, geometrical and combinatorial nature. The proposed research links several important areas of mathematics: we apply concepts of ergodic theory and dynamical systems (the area with origin in physics) to solve a class of problems in number theory, using techniques of Lie groups, algebraic and differential geometry, and probability theory. All of these areas have deep connections with other sciences such as physics, astronomy, statistics, and computer science. The broad purpose of the proposed project is to investigate a range of ideas and create techniques that have the potential to impact many problems of interest in all of these fields. An important aspect of this project will be training of graduate students and post doctoral fellows to conduct research where ideas of one filed are applied to solve problem in another field.