Rigidity Phenomena of Higher Rank Actions

该项目属于动力系统领域，旨在了解运动在几何空间中随时间发生的长期行为，并受一套固定的数学规则支配。动力系统理论源于天体力学，并已成为许多其他科学领域的重要工具。这项拟议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，以便允许一个点在每一步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。这个项目的目标是确定这样的结构。PI将就这一主题进行研究，与其他研究人员合作，与研究生和本科生合作，并传播在该项目中形成的想法。本研究将研究群体行动的动力系统。主要对象是高阶交换群或高阶半单李群中的格在环面、零流形和其他齐次空间上的作用。这些动作通常被认为具有刚性，这种现象的特征是在动作下缺乏不变的结构。其主要目的是在适当的假设下对这类群体行为进行分类，以及在这些行为下的不变结构。研究方法将遍历理论、光滑动力学和表征理论的工具结合起来。

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.