Group Actions and Rigidity

在数学对象的研究中，对象的对称性往往起着关键作用——特别是当对象具有许多对称性时。本研究探讨了在各种动态、几何和拓扑环境中，表征、描述和研究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关：从微分方程（使用波前集来研究方程解的精细解析性质）到理论计算机科学(（超级）展开器、Kazhdan性质（T)、Lafforgue强性质(T)和粗嵌入问题）。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突破。另一个主要焦点是双曲流形的结构，PI最近在McMullen和Reid的问题上取得了另一个重大突破。其他主题包括研究由动力结构产生的膨胀子和超膨胀子族，既要理解它们的粗糙几何，又要看看它们是否在计算机科学、动力学或算子代数中有新的应用。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 (the use of wavefront sets to study fine analytic properties of solutions to equations) to theoretical computer science ((super)expanders, Kazhdan's property (T), Lafforgue's strong property (T) and coarse embedding problems). 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 breakthroughs on conjectures of Zimmer's. Another major focus is on the structure of hyperbolic manifolds where the PI recently made another major breakthrough on a question of McMullen and Reid. Other topics include studying families of expanders and superexpanders arising from dynamical constructions, both to understand their coarse geometry and to see if they have novel applications to computer science, dynamics or operator algebras.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.