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

摘要Margulis在这个研究计划中解决的问题是在李群及其离散子群的理论领域。 共同的主题是研究这个理论与数论，几何和遍历理论。 其中一个主要目标是继续建立一个强大的工具，在数论，特别是丢番图逼近理论的空间方法的程序。 本文还提出研究李群及其离散子群在齐性空间、流形和一般度量空间上的作用的各种问题。 特别是首席研究员计划继续他的联合工作与othermathematicians适当不连续群体的仿射变换和对当地刚性的标准代数行动. 李群理论、约化理论、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的中心领域之一。 从历史上看，这一理论主要是由其与几何，力学和微分方程理论的联系。在过去的15-20年里，人们意识到该理论的某些方面可以应用于解决数论和相关主题中的某些问题，这些问题几十年来无法用其他方法解决。 建议在建议中继续这一发展，以获得新的应用在数论，动力系统和几何。 总的来说，这个建议应该在数学中的离散和连续之间建立新的联系。

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.