New Analytic Techniques in Group Theory

奖项：DMS 1607041，首席研究员：David M. fisher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。提出的研究项目探讨表征、描述和研究各种动态、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（展开图、Kazhdan性质(T)和粗嵌入问题）。提出的研究的主要推力是在群论和群体行为研究中引入和发展新的分析技术。主要项目包括：(a)使用粗微分的新概念（在最近与Eskin和Whyte的联合工作中引入）和涉及非标准分析和Loeb测度的新思想研究准等距；(b)发展基于偏微分方程解析技术的动力系统刚性研究技术，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平滑运动的双曲群进行分类；(d)在群作用的刚性研究中发展了新的上同调方法。所有的研究都是在动力学、几何和拓扑的刚性的广泛跨学科领域。许多被提议的项目涉及应用分析思想和技术来解决传统上用动力学、几何或拓扑方法研究的问题。

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.