9626173 Spatzier The proposed research lies at the interface of differential geometry and dynamical systems. A principal goal is to investigate the dynamical and geometric structures of "higher rank" systems. In particular, the investigator intends to classify hyperbolic actions of higher rank abelian and semisimple Lie groups or actions which preserve geometric structures, and also establish other rigidity properties. The investigator also plans to study Riemannian manifolds (especially higher rank ones) and their geodesic flows. Both geometric, dynamical and group theoretic tools are used in this research. Geometry is one of the oldest fields in mathematics, and generally studies curves, surfaces and their higher dimensional analogues, their shapes, shortest paths, and maps between such spaces. Differential geometry had its roots in cartography, and is now studied for aesthetic reasons and its close ties with physics and other sciences and applied areas (computer vision e.g.). Dynamical systems is a relatively new field that investigates the evolution of a physical or mathematical system over time (e.g. turbulence in a fluid flow). New ideas and concepts from dynamics (e.g. chaos, fractals) have changed our perception of the world fundamentally. Dynamical systems have had a major impact on the sciences and engineering. Symbolic dynamics for instance has been instrumental for developing efficient and safe codes for computer science. Tools and ideas from smooth dynamics are used as far afield as cell biology and meteorology.
