Rigidity in Dynamics and Geometry

在数学对象的研究中，对象的对称性常常起着关键作用——特别是当对象具有许多对称性时。本研究探讨了表征、描述和研究在各种动态、几何和拓扑设置中具有许多对称性的空间的方法。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。这项工作与数学的各个领域有联系：从微分方程到理论计算机科学，从描述性集合论到数论。研究生基金将用于培养新一代专家。该项目的主要目的是利用广泛区域之间的联系，以进一步了解李群中与晶格相关的基本结构。一个主要的焦点是流形上的群体行为的研究，PI最近在Zimmer的猜想上取得了重大进展。另一个主要焦点是双曲流形的结构，PI最近在McMullen和Reid的问题上取得了突破。其他主题包括研究具有许多隐藏对称性的流形（密集通约数）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。

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.