Arithmetic, Geometric, and Ergodic Aspects of the Theory of Lie Groups and Their Discrete Subgroups

本文所涉及的问题是李群及其离散子群的理论领域。共同的主题是结合数论、几何和遍历理论来研究这一理论。主要目标之一是继续建立齐次空间方法作为数论中的一个强大工具的计划，特别是在丢番图近似理论中。研究了李群及其离散子群在齐次空间、流形和一般度量空间上的作用等问题。特别是，首席研究员计划继续他与其他数学家在仿射变换的适当不连续群和标准代数作用的局部刚性方面的联合工作。李群理论、归约理论、遍历理论和动力系统理论将在整个工作中得到应用。李群及其离散子群理论是数学中的中心领域之一。从历史上看，这一理论主要是因为它与几何学、力学和微分方程式理论的联系。在过去的15-20年里，人们认识到该理论的某些方面可以用来解决数论及相关问题中的某些问题，而这些问题是几十年来其他方法无法解决的。建议继续这一发展，以便在数论、动力系统和几何中获得新的应用。总的来说，这项提议应该在数学上建立离散和连续之间的新联系。

AbstractMargulisThe problems addressed in this research proposal are in the area of the theoryof Lie groups and their discrete subgroups. The common theme is to study thistheory in connection with number theory, geometry and ergodic theory. One ofthe main objectives is to continue the program of establishing a homogeneousspace approach as a powerful tool in number theory and, in particular, in thetheory of Diophantine approximations. It is also proposed to study a variety ofquestions about actions of Lie groups and their discrete subgroups onhomogeneous spaces, manifolds and general metric spaces. In particular theprincipal investigator plans to continue his joint work with othermathematicians on properly discontinuous groups of affine transformations and onlocal rigidity for standard algebraic actions. The technique of Lie grouptheory, reduction theory, ergodic theory and the theory of dynamical systemswill be used throughout the proposed work.The theory of Lie groups and their discrete subgroups is one of the centralfields in mathematics. Historically this theory was mostly motivated by itsconnections with geometry, mechanics and the theory of differential equations. During the last 15-20 years it was realized that some aspects of the theory canbe applied to solve certain problems in number theory and related topics, whichcould not be tackled by other methods for several decades. It is suggested inthe proposal to continue this development in order to obtain new applications innumber theory, dynamical systems and geometry. In general, the proposal shouldestablish new connections between discrete and continuous in mathematics.