Rigidity Phenomena in Geometry and Dynamics

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AbstractAward: DMS-0906085Principal Investigator: Ralf SpatzierThe research proposed lies between dynamical systems andgeometry. Its principal goal is the investigation of thedynamical and geometric structures of "higher rank" systems inthese areas. These systems also appear naturally in seeminglyquite separate areas such as number theory. The investigator willstudy rigidity properties of actions of higher rank abelian andsemi-simple Lie groups and their lattices with the ultimate goalof classifying such systems under suitable geometric or dynamicalhypotheses. In particular, he will study higher rank hyperbolicabelian actions on tori and Cartan actions of rank 2. Thesespecial cases, important in their own right, will serve astesting ground for more general conjectures. Also, additionaltools for classification are available in these cases. Theinvestigator will also study actions by semi-simple groups andtheir lattices preserving affine and other geometric structures.In addition, he will analyze Riemannian manifolds (especiallyhigher rank ones) and their geodesic flows. Geometric, dynamicaland group theoretic tools will be used in this research.Dynamical systems and ergodic theory investigate the evolution ofa physical or mathematical system over time (e.g. turbulence in afluid flow). New ideas and concepts such as chaos and fractalshave changed our perception of the world fundamentally. Dynamicsand ergodic theory provide the mathematical tools and analysisfor these investigations. Dynamical systems have had a majorimpact on the sciences and engineering. Symbolic dynamics forinstance has been instrumental in developing efficient and safecodes for computer science. Tools and ideas from smooth dynamicsare used as far afield as cell biology and meteorology. Geometryis one of the oldest fields in mathematics, and generally studiescurves, surfaces and their higher dimensional analogues, theirshapes, shortest paths, and maps between such spaces.Differential geometry had its roots in cartography, and is nowstudied for its close ties with physics and other sciences andapplied areas (computer vision e.g.) as well as internalaesthetic reasons. Geometry and dynamics are closely related assome important dynamical systems originate from geometry, andgeometry also provides tools to study dynamical systems. One maingoal of this project studies when two dynamical systems commute,i.e. when one system is unaffected by the changes brought on bythe other. Important examples of such systems arise fromgeometry.