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

【摘要】本文研究的是李群及其离散子群的理论问题。共同的主题是将这一理论与数论、几何和遍历理论联系起来研究。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问题。特别是，首席研究员计划继续与其他数学家共同研究仿射变换的适当不连续群和标准代数作用的局部刚性。李群理论，约简理论，遍历理论和动力系统理论的技术将在整个拟议的工作中使用。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，这个理论主要是由它与几何、力学和微分方程理论的联系所推动的。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，这个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.