New analytic techniques in group theory

群论中的新分析技术

• 批准号: 1308291
• 负责人: David Fisher
• 金额: $ 23.76万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2018-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1308291&HistoricalAwards=false
• 关键词: New; analytic; techniques; group; theory

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).

拟议研究的主要目的是介绍和开发小组理论和小组行动研究中的新分析技术。主要项目包括：（a）使用新的粗分化概念研究准静态（在与Eskin和Whyte的最新联合工作中引入）； （b）开发基于PDE的分析技术的动态系统刚性的技术，尤其是波前集合的理论； （c）使用动力学和几何组理论的思想对双曲线组进行分类； （d）在研究小组行动的刚性研究中开发新的共同体学方法。所有研究都在动态，几何学和拓扑结构中的僵化跨学科领域。许多提出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。在研究数学对象的研究中，对象的对称性通常会扮演关键角色，尤其是当对象具有许多对称性时。 PI研究了在各种动力学，几何和拓扑设置中具有许多对称性的空间表征，描述和研究空间的方法。这些问题本质上是跨学科的，通常需要从许多数学领域学习，适应和应用思想和技术。 PI的工作与数学的不同领域有联系：从微分方程（使用Waverfront集来研究方程式与方程式的良好分析特性）到理论计算机科学（Expander Graphs，Kazhdan的属性（T）和粗嵌入问题）。