CAREER: Rigidity of Group Actions on Manifolds

动力系统理论描述了随时间变化的系统，如太阳系中行星的运动、行星天气或股票市场；该理论提供了工具来描述系统的预期长期行为，量化系统的复杂性，并提供系统中稳定性和不稳定性的概念。群体行为在数学的许多领域中自然出现。例如，群体行为描述了正多边形可能的对称性，或者更一般地说，描述了将多边形切割成碎片并重新组装成原始多边形的方法。该研究项目将应用动力系统理论中的工具来研究空间上更一般的群作用。这些项目的目标是建立一定的刚性结果，表明所有动作或不变对象都具有一定的原型形式。在项目期间，首席研究员将为高中生开设动力系统的暑期课程，并完成一些关于群体行为刚性的最新发展的说明性著作。具体的研究项目包括一些研究高阶李群中格的作用（包括Zimmer程序中的一些猜想）的项目，以及由几何结构引起的群作用，如对特征变异的作用。在这些项目中，首席研究员打算使用光滑动力系统理论中的一些工具，包括非均匀双曲性、熵理论和范式理论，来研究更一般的群体行为。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Dynamical systems theory describes systems changing over time such as the motion of the planets in the solar system, the planetary weather, or the stock market; the theory provides tools to describe the expected long-term behavior of a system, quantify the complexity of a system, and provide notions of stability and instability in a system. Group actions arise naturally in many areas of mathematics. For instance, group actions describe the possible symmetries of a regular polygon or, more generally, the ways to cut a polygon into pieces and reassemble into the original polygon. The research project will apply tools from the theory of dynamical systems in order to study more general group actions on spaces. The goal of these projects is to establish certain rigidity results showing that all actions or invariant objects are of a certain prototypical form. During the project, the principal investigator will give summer courses on dynamical systems to high-school students and complete a number of expository writings on recent developments in rigidity of groups actions. The specific research projects include a number of projects studying actions of lattices in higher-rank Lie groups (including a number of conjectures in the Zimmer program) as well as group actions arising from geometric constructions such as actions on character varieties. In these projects, the principal investigator intends to use a number of tools from the theory of smooth dynamical systems including nonuniform hyperbolicity, entropy theory, and the theory of normal forms to study more general group actions.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.