CAREER: New Analytic Techniques in Group Theory

DMS 0643546PI：David M.Fisher建议的研究的主要目的是在群论和群行动的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Ekin和Whyte的联合工作中引入)研究准等距；(B)开发刚性理论中的调和映射技术，特别是在无限维目标空间的背景下；以及(C)基于新的广义和变种的硬函数定理，开发群行动的范式理论。所有研究都位于动力学、几何和拓扑学中刚性的广泛交叉学科领域。这笔赠款还将支持致力于刚性理论新发展的暑期学校。这一领域很广泛，但定义并不完全明确，新的发展经常涉及到数学的其他领域的想法。这些暑期学校将为学生和年轻科学家提供快速了解新发展的机会，并建立与未来新发展保持同步所需的专业网络。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。PII研究在不同的动力学、几何和拓扑环境中刻画、描述和研究具有多种对称性的空间的方法。这些问题本质上是跨学科的，往往需要学习、适应和应用许多数学领域的思想和技术。PI的工作涉及数学的各个领域：从天体力学(KAM理论和太阳系的稳定性)到理论计算机科学(膨胀图、Kazhdan性质(T)和粗略嵌入问题)。许多提出的项目涉及将分析思想和技术应用于传统上用动力学、几何或拓扑方法研究的问题。

AbstractDMS 0643546PI: David M. FisherThe main thrust of the proposed research is to introduceand develop new analytic techniques in group theory andthe study of group actions. Major projects include:(a) studying quasi-isometries using a new notion ofcoarse differentiation (introduced in recent joint workwith Eskin and Whyte)(b) developing harmonic map techniques in rigiditytheory, particularly in the context of infinitedimensional target spaces and(c) developing a normal form theory for group actionsbased on new generalizations and variants of hardimplicit function theorems.All research lies in the broad interdisciplinary area ofrigidity in dynamics, geometry and topology. The grantwill also support summer schools devoted to newdevelopments in rigidity theory. The field is broad andnot entirely well defined, new developments frequentlyinvolve ideas from other areas of mathematics. Thesesummer schools will provide students and youngscientists the opportunity to learn about newdevelopments quickly and to build the professionalnetworks required to remain abreast of future newdevelopments.In the study of mathematical objects, a key role isoften played by the symmetries of the object --particularly when the object has many symmetries. The PIinvestigates ways of characterizing, describing andstudying spaces with many symmetries in variousdynamical, geometric and topological settings. Thesequestions are interdisciplinary in nature and oftenrequire learning, adapting and applying ideas andtechniques from many areas of mathematics. The PI's workhas connections with diverse areas of mathematics: fromcelestial mechanics (KAM theory and stability of thesolar system) to theoretical computer science (expandergraphs, Kazhdan's property (T) and coarse embeddingproblems). Many proposed projects involve applyinganalytic ideas and techniques to problems traditionallystudied by dynamical, geometric or topological methods.