Mathematical Sciences: Ergodic Theory, Groups, and Geometry

Zimmer将研究流形上半单李群及其离散子群的作用。这将包括这些群体活动的空间的拓扑、几何和分析特征，以及活动本身的相应特征。他还将研究这些问题在遍历理论和微分几何中的应用。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。＊＊＊

9321385 Zimmer Zimmer will investigate several questions concerning the actions of semisimple Lie groups and their discrete subgroups on manifolds. This will include the topological, geometric, and analytic features of the spaces on which these groups act, and the corresponding features of the actions themselves. He will also study application of these questions to ergodic theory and differential geometry. 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. ***