The research proposed lies between dynamical systems, group theory and geometry. Principally, the investigator plans to study dynamical and geometric structures of "higher rank" systems in these areas. These appear naturally in seemingly quite separate areas, for example in number theory or in studying the spectrum of the Laplacian. The investigator will work on rigidity properties of actions of higher rank abelian and semi-simple Lie groups and their lattices striving to classify such systems under suitable geometric or dynamical hypotheses. In particular, he will study higher rank hyperbolic abelian actions and their cocycles on tori and homogeneous spaces as well as general Cartan actions of rank 2. These special cases provide tests for more general conjectures. The investigator will also study actions by semi-simple groups and their lattices preserving projective, affine and other geometric structures. In addition, the investigator will analyze Riemannian manifolds (especially those of higher spherical rank) and more general singular spaces and their geodesic flows. Geometric, dynamical and group theoretic tools will be used in this research.
Dynamical systems and ergodic theory investigate the evolution of a physical or mathematical system over time, such as turbulence in a fluid flow. New ideas and concepts such as chaos and fractals have changed our understanding of the world. Dynamics and ergodic theory provide excellent mathematical tools, and have a strong 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 a highly developed and ancient field in mathematics of amazing vigor. It studies curves, surfaces and their higher dimensional analogues, their shapes, shortest paths, and maps between such spaces. Differential geometry had its roots in cartography, starting with Gauss in the nineteenth century. It is closely linked with physics and other sciences and applied areas such as computer vision. Geometry and dynamics are closely related. Indeed, important dynamical systems come from geometry, and vice versa geometry 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 when the space contains many flat subspaces. Group theory finally enters both dynamics and geometry by studying the group of symmetries of a geometry or dynamical situation, or by investigating the dynamical and geometric behavior of the group of symmetries acting on a space.
