This ambitious mathematical research project has to do with actions of Lie groups, and certain of their discrete subgroups, on manifolds. A very simple example of this situation is the group of rotations of three-dimensional space about axes passing through the origin, acting on a sphere centered at the origin. In higher dimensions, the examples become more exotic, and more importantly, as Professor Zimmer has discovered, the rules of the game that say what sorts of actions can occur on what sorts of manifolds change in striking and fundamental ways. A whole new subject, which might be called nonlinear representation theory, is growing up to take its place next to the older and much better understood theory of group representations on vector spaces. This research impinges directly on four core areas of mathematics: analysis, geometry, algebra, and topology. More specifically, Professor Zimmer will investigate the topology of manifolds admitting an action of a semisimple Lie group or discrete subgroup by describing features of the representation theory of the manifold's fundamental group. The arithmetic nature of actions of of these groups will be investigated. There will be continued study of the phenomena alluded to above that make the higher-dimensional case qualitatively different from the lower-dimensional.
