Rigidity Phenomena of Higher Rank Actions

高阶动作的刚性现象

• 批准号: 1501095
• 负责人: Zhiren Wang
• 金额: $ 16.33万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-07-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1501095&HistoricalAwards=false
• 关键词: Rigidity; Phenomena; Higher; Rank; Actions

This project belongs to the field of dynamical systems, which aims to understand long-term behaviors of movements that take place over time in a geometric space and are governed by a fixed set of mathematical rules. The theory of dynamical systems has its roots in celestial mechanics and has become an important tool in many other scientific fields. The proposed research deals with a special class of dynamical systems, where the rules governing the movement form a sufficiently large group so that a point is allowed to move in one of several possible ways in each step. In such cases, there is often a small collection of possible structures that remain unchanged over time. The goal of this project is to determine such structures. The PI will conduct research on the topic, collaborate with other researchers, work with graduate and undergraduate students, and disseminate ideas developed in the project. This research will study dynamical systems of group actions. The principal objects are actions on tori, nilmanifolds and other homogeneous spaces by higher-rank commutative groups or by lattices in higher-rank semisimple Lie groups. These actions are often expected to have rigidity, a phenomenon characterized by the scarcity of invariant structures under the action. The main objective is to classify such group actions under suitable assumptions, as well as invariant structures under these actions. The method of the research will combine tools from ergodic theory, smooth dynamics and representation theory.

该项目属于动力系统领域，旨在了解在几何空间中随时间发生的运动的长期行为，并由一组固定的数学规则控制。动力系统理论起源于天体力学，并已成为许多其他科学领域的重要工具。所提出的研究涉及一类特殊的动力系统，其中控制运动的规则形成了一个足够大的组，使得一个点可以在每一步中以几种可能的方式之一移动。在这种情况下，通常有一小部分可能的结构随着时间的推移保持不变。本项目的目标是确定这种结构。PI将对该主题进行研究，与其他研究人员合作，与研究生和本科生合作，并传播项目中开发的想法。本研究将研究群体作用的动力系统。主要对象是高秩交换群或高秩半单李群中的格在环面、零流形和其他齐次空间上的作用。 这些动作通常被期望具有刚性，这种现象的特征在于动作下的不变结构的稀缺性。其主要目标是在适当的假设下对这种群体行为进行分类，以及在这些行为下的不变结构。研究方法将结合遍历理论、光滑动力学和表示理论等联合收割机工具。