CAREER: New Analytic Techniques in Group Theory

Indiana University, Bloomington IN

Investigators

Abstract

Abstract DMS 0643546 PI: David M. Fisher 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) (b) developing harmonic map techniques in rigidity theory, particularly in the context of infinite dimensional target spaces and (c) developing a normal form theory for group actions based on new generalizations and variants of hard implicit function theorems. All research lies in the broad interdisciplinary area of rigidity in dynamics, geometry and topology. The grant will also support summer schools devoted to new developments in rigidity theory. The field is broad and not entirely well defined, new developments frequently involve ideas from other areas of mathematics. These summer schools will provide students and young scientists the opportunity to learn about new developments quickly and to build the professional networks required to remain abreast of future new developments. In 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 PI investigates ways of characterizing, describing and studying spaces with many symmetries in various dynamical, geometric and topological settings. These questions are interdisciplinary in nature and often require learning, adapting and applying ideas and techniques from many areas of mathematics. The PI's work has connections with diverse areas of mathematics: from celestial mechanics (KAM theory and stability of the solar system) to theoretical computer science (expander graphs, Kazhdan's property (T) and coarse embedding problems). Many proposed projects involve applying analytic ideas and techniques to problems traditionally studied by dynamical, geometric or topological methods.