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

马古利斯将继续研究齐次空间上子群作用理论在数论和丢番图近似中的不同应用。他将研究仿射变换的离散群，特别是仿射变换的紧致离散群的代数结构。他还将继续研究任意仿射变换离散群的线性部分的Zariski闭包。这个项目涉及遍历理论的研究。遍历理论一般关注理解系统的平均行为，其动力学过于复杂或混乱，无法在微观细节上进行跟踪。在“动力学”的标题下，可以放置关于抽象变换如何作用于光滑空间的现代理论。通过这种方式，遍历理论在寻求对均匀空间上的流动进行分类时与几何联系在一起。

Margulis will continue his work on different application of the theory of subgroup actions on homogeneous spaces to number theory and Diophantine approximations. He will study discrete groups of affine transformations, and in particular the algebraic structure of cocompact discrete groups of affine transformations. He will also continue his work on the Zariski closure of the linear part of an arbitrary discrete group of affine transformations. This project involves research in ergodic theory. Ergodic theory in general concerns understanding the average behavior of systems whose dynamics is too complicated or chaotic to be followed in microscopic detail. Under the heading "dynamics can be placed the modern theory of how groups of abstract transformations act on smooth spaces. In this way ergodic theory makes contact with geometry in its quest to classify flows on homogeneous spaces.