New Analytic Techniques in Group Theory

群论中的新分析技术

• 批准号: 1607041
• 负责人: David Fisher
• 金额: $ 22.3万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-09-01 至 2020-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1607041&HistoricalAwards=false
• 关键词: New; Analytic; Techniques; Group; Theory

Award: DMS 1607041, Principal Investigator: David M. FisherIn 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 proposed research projects investigate 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 (expander graphs, Kazhdan's property (T) and coarse embedding problems).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) and newer ideas involving non-standard analysis and Loeb measures; (b) developing techniques for studying rigidity of dynamical systems based on analytic techniques from partial differential equations, 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.

奖项：DMS 1607041，主要研究者：大卫M.在数学对象的研究中，对象的对称性经常扮演着关键的角色-特别是当对象有许多对称性时。拟议的研究项目调查的表征，描述和研究空间的各种动力学，几何和拓扑设置具有许多对称性的方法。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。这项工作与数学的不同领域有联系：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）。拟议研究的主要目标是在群论和群作用的研究中引入和发展新的分析技术。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）和涉及非标准分析和Loeb测度的较新思想;（B）发展基于偏微分方程的分析技术的用于研究动力系统刚性的技术，特别是波前集理论;（c）利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）在群作用的刚性研究中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。许多拟议的项目涉及应用分析思想和技术的问题，传统上研究的动力学，几何或拓扑方法。