Rigidity in Dynamics and Geometry

动力学和几何中的刚性

• 批准号: 2246556
• 负责人: David Fisher
• 金额: $ 43.84万
• 依托单位: William Marsh Rice 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: 2022-10-01 至 2025-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2246556&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.

在数学对象的研究中，对象的对称性常常起着关键的作用，特别是当对象具有许多对称性时。本研究探讨如何表征，描述和研究在各种动力学，几何和拓扑设置的空间与许多对称性。这些问题通常需要学习，适应和应用数学许多领域的思想和技术。这项工作与数学的不同领域有联系：从微分方程到理论计算机科学，再到描述性集合论，再到数论。研究生资助将用于培养新一代的专家。该项目的主旨是利用广泛领域之间的联系，进一步了解李群中与格相关的基本结构。一个主要的重点是研究流形上的群作用，PI最近在Zimmer的Astructures上取得了重大进展。另一个主要的重点是双曲流形的结构，PI最近取得了突破性的问题McMullen和里德。其他主题包括研究具有许多隐藏对称性的流形（稠密流形）如何与几何、拓扑和群论中的几个经典问题相关。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Zimmer's conjecture: Subexponential growth, measure rigidity, and strong property (T)

齐默猜想：次指数增长、测度刚性、强性质（T）

• DOI: 10.4007/annals.2022.196.3.1
• 发表时间: 2022
• 期刊: Annals of Mathematics
• 影响因子: 4.9
• 作者: Brown, Aaron;Fisher, David;Hurtado, Sebastian
• 通讯作者: Hurtado, Sebastian

A new proof of finiteness of maximal arithmetic reflection groups

最大算术反射群有限性的新证明

• DOI: 10.5802/ahl.162
• 发表时间: 2023
• 期刊: Annales Henri Lebesgue
• 作者: Fisher, David;Hurtado, Sebastian
• 通讯作者: Hurtado, Sebastian

Arithmeticity, superrigidity and totally geodesic submanifolds of complex hyperbolic manifolds

复双曲流形的算术性、超刚性和全测地线子流形

• DOI: 10.1007/s00222-023-01186-5
• 发表时间: 2023
• 期刊: Inventiones mathematicae
• 影响因子: 3.1
• 作者: Bader, Uri;Fisher, David;Miller, Nicholas;Stover, Matthew
• 通讯作者: Stover, Matthew