Rigidity Phenomena in Geometry and Dynamics

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ABSTRACT DMS - 0203735.The research proposed lies at the interface of dynamical systems and differential geometry. Its principal goal is the investigation of the dynamical and geometric structures of "higher rank" systems. Such systems naturally appear both in dynamical systems and geometry. In particular, the principal investigator will study rigidity properties of actions of higher rank abelian and semisimple Lie groups and their lattices with the ultimate goal of classifying such systems under suitable geometric or dynamical hypotheses. In particular, he will study higherrank hyperbolic abelian actions, and actions by semisimple groups and their lattices preserving affine and geometric structures. He will also investigate invariant measures of algebraic higher rank abelian actions. The investigator will also investigate Riemannian manifolds (especially higher rank ones)and their geodesic flows. Geometric, dynamical and group theoretic tools will be used in this research.Dynamical systems and ergodic theory are relatively new fields that investigate 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. Dynamics and ergodic theory provide the mathematical tools and analysis for these investigations. Dynamical systems have had a major impact on the sciences and engineering. Symbolic dynamics for instance has been instrumental in developing efficient and safe codes for computer science. Tools and ideas from smooth dynamics are used as far afield as cell biology and meteorology. 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 its close ties with physics and other sciences and applied areas (computer vision e.g.)as well as internal aesthetic reasons. Geometry and dynamics are closely related as some important dynamical systems originate from geometry, and geometry also provides tools to study dynamical systems. One main goal of this project studies when two dynamical systems commute, i.e. when one system is unaffected by the changes brought on by the other. Important examples of such systems arise from geometry.