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

非正曲率、测地线和大群行为动力学中的对称性极值。

• 批准号: 0905906
• 负责人: Benjamin Schmidt
• 金额: $ 13.2万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-08-01 至 2010-03-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0905906&HistoricalAwards=false
• 关键词: Extremality; symmetry; nonpositive; curvature; geodesics

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.

该奖项是根据2009年《美国复苏和再投资法案》(公法111-5)提供资金的。这一建议的总主题是寻找最对称空间和唯一表征这些空间的动力系统的性质。PI提出了与这一目标相关的项目，重点是：a)了解非紧型局部对称空间的有界上同调群，b)寻找刻画紧型对称空间的测地线的性质，以及c)Zimmer关于闭流形上高秩非紧单李群的格子群的分类作用的程序。对称性贯穿于自然界中的各个形态，如动物、水晶和雪花。数学家在他们的工作中经常使用对称性，因为对称空间和系统通常是理论上最容易理解的。此外，对这些空间和系统的详细了解往往有助于更好地理解不太对称的物体。PI将研究一个空间中测地线(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，它的测地线应该不同于理想球体的测地线。