Extremality of symmetry in nonpositive curvature, geodesics, and the dynamics of large group actions.

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The general theme of this proposal is to find properties of the most symmetric spaces and dynamical systems that uniquely characterize these spaces. The PI proposes projects related to this goal with an emphasis on the following:a) understanding the bounded cohomology groups of locally symmetric spaces of noncompact type,b) finding properties of geodesics that characterize symmetric spaces of compact type, and c)Zimmer's program of classifying actions of lattice subgroups of higher rank noncompact simple Lie groups on closed manifolds. Symmetry appears throughout forms in nature such as animals, crystals, and snowflakes. Mathematicians exploit symmetry on a regular basis in their work since symmetric spaces and systems are usually the easiest to understand theoretically. Moreover, a detailed understanding of these spaces and systems frequently leads to a better understanding of less symmetric objects. The PI will investigate the interplay between properties of geodesics (or shortest paths between points) in a space and symmetry exhibited by that space. For example, since the surface of the Earth is not perfectly round, its geodesics should differ from those of an idealized sphere.