CAREER: New Analytic Techniques in Group Theory

点击翻译按钮获取中文摘要

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.