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

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。本提案的总体主题是寻找最对称空间和动力系统的特性，这些特性是这些空间的独特特征。PI提出了与此目标相关的项目，重点是：a)理解非紧型局部对称空间的有界上同调群，b)寻找描述紧型对称空间的测大地的性质，以及c)Zimmer关于闭流形上高阶非紧单李群格子群的分类作用的规划。对称在自然界中随处可见，比如动物、水晶和雪花。数学家在他们的工作中经常利用对称性，因为对称的空间和系统通常是最容易从理论上理解的。此外，对这些空间和系统的详细理解通常会导致对不对称对象的更好理解。PI将研究空间中测地线（或点之间的最短路径）的性质与该空间所表现出的对称性之间的相互作用。例如，由于地球表面不是完美的圆形，它的测地线应该与理想球体的测地线不同。

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.