New Analytic Techniques in Group Theory

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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.