New analytic techniques in group theory

拟议的研究的主旨是引进和发展新的分析技术，在群论和群体行动的研究。主要项目包括：（a）利用粗微分的新概念研究准等距（在最近与Eskin和Whyte的联合工作中介绍）;（B）基于PDE的分析技术，特别是波前集理论，发展用于研究动力系统刚性的技术;（c）使用动力学和几何群论的思想对在其边界上光滑作用的双曲群进行分类;（d）在研究群作用的刚性方面发展新的上同调方法。所有的研究在于广泛的跨学科领域的刚性动力学，几何和拓扑结构。许多建议的项目涉及将分析思想和技术应用于传统上通过动力学，几何或拓扑方法研究的问题。在数学对象的研究中，对象的对称性通常起着关键作用-特别是当对象具有许多对称性时。PI研究表征，描述和研究在各种动力学，几何和拓扑设置中具有许多对称性的空间的方法。这些问题本质上是跨学科的，通常需要学习，适应和应用数学许多领域的思想和技术。PI的工作与数学的各个领域都有联系：从微分方程（使用波前集来研究方程解的精细分析性质）到理论计算机科学（扩展图，Kazhdan性质（T）和粗嵌入问题）。

The main thrust of the proposed research is to introduce and develop new analytic techniques in group theory and the study of group actions. Major projects include: (a) studying quasi-isometries using a new notion of coarse differentiation (introduced in recent joint work with Eskin and Whyte); (b) developing techniques for studying rigidity of dynamical systems based on analytic techniques from pde, particularly the theory of wavefront sets; (c) classifying hyperbolic groups which act smoothly on their boundaries using ideas from both dynamics and geometric group theory; (d) developing new cohomological methods in the study of rigidity of group actions. All research lies in the broad interdisciplinary area of rigidity in dynamics, geometry and topology. Many proposed projects involve applying analytic ideas and techniques to problems traditionally studied by dynamical, geometric or topological methods.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. The PI investigates ways of characterizing, describing and studying spaces with many symmetries in various dynamical, geometric and topological settings. These questions are interdisciplinary in nature and often require learning, adapting and applying ideas and techniques from many areas of mathematics. The PI's work has connections with diverse areas of mathematics: from differential equations (the use of waverfront sets to study fine analytic properties of solutions to equations) to theoretical computer science (expander graphs, Kazhdan's property (T) and coarse embedding problems).