CAREER: New Analytic Techniques in Group Theory

摘要DMS 0643546 PI：大卫M.费希尔提出的研究的主要目的是在群论和群体行动的研究中引入和发展新的分析技术。 主要项目包括：（a）研究准等距使用一个新的概念粗微分（介绍了在最近的联合工作与Eskin和怀特）（B）开发调和映射技术在刚性理论，特别是在无限维目标空间的背景下，（c）开发一个正常形式理论的基础上新的推广和变种的群体actionshardimplicit函数定理。格兰披治大学还支持致力于刚性理论新发展的暑期学校。这个领域很广泛，而且还没有完全明确的定义，新的发展经常涉及到数学其他领域的思想。暑期学校将为学生和计算机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--特别是当对象有许多对称性时。PI研究了在各种动力学、几何学和拓扑学环境中表征、描述和研究具有许多对称性的空间的方法。这些方程本质上是跨学科的，通常需要学习，适应和应用来自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）和粗嵌入问题）。许多建议的项目涉及applyinganalytic思想和技术的问题traditionallystudied动力学，几何或拓扑方法。

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.