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

要调查的问题是在该地区的理论李群及其离散子群。其中一个主要目标是继续建立一个齐次空间的方法作为一个强大的工具，数论，特别是在丢番图近似程序。特别注意的是齐次动力学中的有效密度和有效等分布问题。也将强调幂单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。本文还建议在两个方向上继续研究仿射变换离散群：（1）Auslander猜想及其分类问题;（2）Fuchsian群的真仿射变换集的边界问题。李群及其离散子群理论是数学的中心领域之一。在过去的几十年中，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的长期问题，这些问题无法通过其他方法解决。该提议与刚性理论有关，该理论研究当关于几何和数学对象的相当弱的数据完全或几乎完全确定这些对象的结构时的现象。该建议应建立新的联系理论李群及其离散子群，数论，几何，动力系统和遍历理论，概率论，理论计算机科学，数学物理，并在一般之间的离散和连续的数学。

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.