Arithmetic, Geometric and Ergodic Aspects of the Theory of Lie Groups and their discrete subgroups

所要研究的问题是李群及其离散子群的理论领域。主要目标之一是继续建立齐次空间方法作为数论的有力工具，特别是丢番图近似。本文将特别关注均匀动力学中的有效密度和有效均匀分布问题。也将着重于幂偶流的对数定律和齐次动力学在丢番图近似度量理论中的应用。并建议在两个方向上继续研究仿射变换离散群：(a) Auslander猜想及其相关的分类问题；(2) Fuchsian群的固有仿射变换集的边界。李群及其离散子群理论是数学研究的中心领域之一。在过去的几十年里，人们意识到该理论的某些方面可以应用于解决数论、量子混沌和相关主题中长期存在的问题，这些问题是其他方法无法解决的。该提议与刚性理论有关，刚性理论研究的是几何和数学对象的相当弱的数据完全或几乎完全决定这些对象的结构的现象。该提案应在李群及其离散子群理论、数论、几何、动力系统与遍历理论、概率论、理论计算机科学、数学物理之间建立新的联系，并在数学中的离散与连续之间建立新的联系。

The problems to be investigated are in the area of the theory of Lie groups and their discrete subgroups. One of the main objectives is to continue the program of establishing a homogeneous space approach as a powerful tool in number theory and, in particular in Diophantine approximation. Special attention will be given to the problem of the effective density and the effective equidistribution in homogeneous dynamics. There will also be emphasis on logarithm laws for unipotent flows and applications of homogeneous dynamics to metric theory of Diophantine approximation. It is also proposed to continue the work on discrete groups of affine transformations in two directions: (a) Auslander conjecture and classification problems related to it; (2) the boundary of the set of proper affine transformations of Fuchsian groups.The theory of Lie groups and their discrete subgroups is one if the central areas in mathematics. During the last few decades, it was realized that some aspects of the theory can be applied to solve longstanding problems in number theory, quantum chaos and related topics, which could not be tackled by other methods. The proposal is related to rigidity theory that studies phenomena when rather weak data about geometric and mathematical objects determines completely or almost completely the structure of those objects. The proposal should establish new connections between theory of Lie groups and their discrete subgroups, number theory, geometry, dynamical systems and ergodic theory, probability theory, theoretical computer science, mathematical physics, and in general between discrete and continuous in mathematics.