Rigidity in Dynamics and Geometry

动力学和几何中的刚性

• 批准号: 2208430
• 负责人: David Fisher
• 金额: $ 43.84万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-09-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2208430&HistoricalAwards=false
• 关键词: Rigidity; Dynamics; Geometry

In the study of mathematical objects, a key role is often played by the symmetries of the object—particularly when the object has many symmetries. This research investigates ways of characterizing, describing, and studying spaces with many symmetries in various dynamical, geometric, and topological settings. These questions often require learning, adapting, and applying ideas and techniques from many areas of mathematics. This work has connections with diverse areas of mathematics: from differential equations to theoretical computer science to descriptive set theory to number theory. Graduate student funding will be used to train a new generation of experts.The main thrust of the project is to exploit connections between a wide set of areas to further understand fundamental structures related to lattices in Lie groups. A major focus is the study of group actions on manifolds where the PI recently made major advances on conjectures of Zimmer's. Another major focus is on the structure of hyperbolic manifolds where the PI recently made a breakthrough on a question of McMullen and Reid. Other topics include studying how manifolds with many hidden symmetries (dense commensurators) relate to several classical questions in geometry, topology, and group theory.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.

在对数学对象的研究中，对象的对称性通常会扮演关键角色，尤其是当对象具有许多对称性时。这项研究调查了在各种动态，几何和拓扑设置中具有许多对称性的空间表征，描述和研究空间的方法。这些问题通常需要学习，适应和应用许多数学领域的思想和技术。这项工作与数学的潜水领域有联系：从微分方程到理论计算机科学再到描述性集理论再到数字理论。研究生资金将用于培训新一代专家。该项目的主要目的是利用各个区域之间的连接，以进一步了解与Lie群体中与晶格有关的基本结构。主要重点是对PI最近对Zimmer's的猜想取得重大进步的群体行动的研究。另一个主要重点是PI最近在McMullen和Reid的问题上取得了突破的双曲线歧管结构。其他主题包括研究具有许多隐藏对称性（密集的同源器）的流形如何与几何，拓扑和群体理论中的几个经典问题相关联。该奖项反映了NSF的法定任务，并被认为是通过基金会的知识分子优点和更广泛的审查标准来通过评估来获得支持的。