Mathematical Sciences: Ergodic Theory, Groups and Geometry

数学科学：遍历理论、群和几何

• 批准号: 8805813
• 负责人: Robert Zimmer
• 金额: $ 20.34万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1988
• 资助国家: 美国
• 起止时间: 1988-07-01 至 1991-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8805813&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Ergodic; Theory; Groups

This ambitious mathematical research project has to do with actions of Lie groups, and certain of their discrete subgroups, on manifolds. A very simple example of this situation is the group of rotations of three-dimensional space about axes passing through the origin, acting on a sphere centered at the origin. In higher dimensions, the examples become more exotic, and more importantly, as Professor Zimmer has discovered, the rules of the game that say what sorts of actions can occur on what sorts of manifolds change in striking and fundamental ways. A whole new subject, which might be called nonlinear representation theory, is growing up to take its place next to the older and much better understood theory of group representations on vector spaces. This research impinges directly on four core areas of mathematics: analysis, geometry, algebra, and topology. More specifically, Professor Zimmer will investigate the topology of manifolds admitting an action of a semisimple Lie group or discrete subgroup by describing features of the representation theory of the manifold's fundamental group. The arithmetic nature of actions of of these groups will be investigated. There will be continued study of the phenomena alluded to above that make the higher-dimensional case qualitatively different from the lower-dimensional.

这个雄心勃勃的数学研究项目与李群及其离散子群在流形上的作用有关。这种情况的一个非常简单的例子是三维空间中围绕经过原点的轴的一组旋转，作用于以原点为中心的球体。在更高的维度中，例子变得更加奇特，更重要的是，正如齐默教授所发现的，游戏规则说明了什么样的行为可以发生在什么样的流形上，以惊人的方式发生根本性的变化。一个全新的学科，可能被称为非线性表示理论，正在崛起，以取代更古老的、更容易理解的向量空间群表示理论。这项研究直接影响到数学的四个核心领域：分析、几何、代数和拓扑。更具体地说，Zimmer教授将通过描述流形基本群的表示理论的特征来研究允许半单李群或离散子群作用的流形的拓扑结构。本文将研究这些基团作用的算术性质。将继续研究上面提到的现象，使高维情况与低维情况在质量上有所不同。