Thin Groups and Dynamics

薄群和动力学

• 批准号: 1361673
• 负责人: Hee Oh
• 金额: $ 60万
• 依托单位: Yale University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1361673&HistoricalAwards=false
• 关键词: Thin; Groups; Dynamics

A homogeneous space is a geometric object that manifests a continuous family of symmetries. Dynamics of flows on homogeneous spaces are related to many natural problems in number theory and geometry. Traditionally, the homogeneous spaces of interest in most studies have finite volume. However, it has recently been discovered that the dynamics on homogeneous spaces of infinite volume also arise in many natural problems such as Apollonian circle packings and in building efficient networks. This proposal will investigate this rich topic.The proposal deals with dynamics on infinite homogeneous spaces, which turns out to yield quite a rich theory and has several deep and striking applications. The proposed study involves new interactions between hyperbolic geometry, Kleinian groups, ergodic theory, and number theory. Investigating this new territory requires many new ideas both from geometry and dynamics, and opens up new interactions between these fields.

齐次空间是一个几何对象，它表现出连续的对称性。齐次空间上的流动力学与数论和几何中的许多自然问题有关。传统上，在大多数研究中感兴趣的齐性空间具有有限体积。然而，最近人们发现，无穷体积的齐次空间上的动力学也出现在许多自然问题中，如阿波罗圆填充和建立有效的网络。 这个建议将探讨这个丰富的主题。这个建议涉及无限齐次空间上的动力学，结果产生了相当丰富的理论，并有一些深刻而引人注目的应用。 拟议的研究涉及双曲几何，克莱因集团，遍历理论和数论之间的新的相互作用。 研究这一新领域需要几何学和动力学的许多新思想，并在这些领域之间开辟了新的相互作用。