Group Actions and Rigidity

在数学对象的研究中，对象的对称性常常扮演着关键的角色，特别是当对象有许多对称性时。本研究探讨在各种动力学、几何学和拓扑学环境中表征、描述和研究具有许多对称性的空间的方法。这些问题往往需要学习，适应和应用数学的许多领域的想法和技术。这项工作与数学的各个领域都有联系：从微分方程（使用波阵面集来研究方程解的精细分析性质）到理论计算机科学（（超级）扩展器、Kazhdan性质（T）、Lafforgue强性质（T））和粗糙嵌入问题）。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的结构上取得了重大突破。 另一个主要的重点是双曲流形的结构，PI最近取得了另一个重大突破的问题McMullen和里德。其他主题包括研究由动力学结构产生的膨胀器和超膨胀器族，既要了解它们的粗糙几何，又要了解它们在计算机科学、动力学或算子代数中是否有新的应用。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。

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.