Mathematical Science: RUI: Actions of Discrete Subgroups of Lie Groups

普通欧氏空间的经典结晶群是李群的离散子群的基本例子。在非常严格的几何结构中出现的高阶局部对称空间中的结晶学群已被Marguis分类。Witte将研究这些重要的群是否可以实现为一维空间的对称性。Witte还将研究在均匀空间中出现的结晶学基团，而不是通常的对称空间。这个项目涉及对遍历理论的研究。一般说来，遍历理论涉及理解系统的平均行为，这些系统的动力学太复杂或太混乱，以至于无法在微观细节上进行跟踪。在标题“动力学可以被放在现代理论如何抽象变换如何作用于光滑空间。这样，遍历理论与几何联系起来，寻求在均匀空间上对流动进行分类。

The classical crystallographic groups of ordinary Euclidean space are basic examples of discrete subgroups of Lie groups. The crystallographic groups arising in the very rigid geometric structures known as locally symmetric spaces of higher rank have been classified by Margulis. Witte will investigate whether these important groups can be realized as symmetries of one-dimensional spaces. Witte will also investigate the crystallographic groups that arise in homogeneous spaces other than the usual symmetric spaces. 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.