CAREER: Rigidity of Group Actions on Manifolds

职业：流形上群体行动的刚性

• 批准号: 2020013
• 负责人: Aaron Brown
• 金额: $ 35.52万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-09-01 至 2025-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2020013&HistoricalAwards=false
• 关键词: CAREER; Rigidity; Group; Actions; Manifolds

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.

动力系统理论描述了随时间变化的系统，例如太阳系中行星的运动、行星天气或股票市场；该理论提供了描述系统预期长期行为、量化系统复杂性并提供系统稳定性和不稳定性概念的工具。 群体行动自然地出现在数学的许多领域。例如，群动作描述了正多边形的可能对称性，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的工具来研究更一般的空间群体行为。 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的对象都具有某种原型形式。 在该项目期间，首席研究员将为高中生提供有关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子作用的项目（包括Zimmer纲领中的一些猜想）以及由几何结构产生的群作用，例如对字符变异的作用。 在这些项目中，首席研究员打算使用平滑动力系统理论中的许多工具，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 NSF 的法定使命，并已被通过使用基金会的智力优点和更广泛的影响审查标准进行评估，认为值得支持。