Abstract 9705712 Robert J. Zimmer The goal of this project is to study a variety mf questions within the general program of investigating actions of semisimple Lie groups and their discrete subgroups on manifolds. This includes an investigation of the topological, geometric, and analytic features of the spaces on which these groups act, and the corresponding features of the actions themselves. Applications of these questions to ergodic theory and differential geometry will be studied as well. The techniques of algebraic ergodic theory, i.e., the interaction of the ergodic theory with algebraic groups, in conjunction with differential geometry, will be used throughout the project. The study of symmetry has played a fundamental role in mathematics and its applications. Within mathematics it has become understood as one of the central themes of most disciplines, and it has had applications as diverse as particle physics, design of efficient networks, coding, and radar. This project aims to contribute to this general development by the study of situations in which there is a large symmetry group, in many cases exactly the situation in which applications are most present.
