Ergodic Theory, Groups, and Geometry

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AbstractZimmerThis research will investigate actions of Lie groups and related questionsof dynamics, topology, and geometry. A focus will be on the actions ofnon-compact semisimple Lie groups on manifolds, and the relationshipbetween the group, geometric structures invariant under the action, and thetopology of the manifold, in particular the fundamental group. Use ofergodic theory, i.e., the measure theoretic structure and behavior of groupactions, will be extensive. In particular, the structure of stationarymeasures and invariant measures for these actions, and the extent to whichsuch general measures are similar to standard arithmetic or projectivemodels, will play a major role.The study of symmetry has played a major role in science and mathematics,and is a fundamental, often indispensable, tool for understanding manyphenomena. The connections to science and technology span the gamut fromelementary particle physics, to crystal structure of materials, to networkdesign, to dynamical systems. The underlying mathematics for studyingsymmetry is group theory. This research will focus on fundamentalmathematical questions concerning large symmetry groups, i.e. situations inwhich the structure under consideration has many symmetries. It isanticipated that this work will contribute to the fabric of fundamentalunderstanding of symmetry, thereby increasing the power of its potentialapplication across science.