Group Actions, rigidity and geometry

Indiana University, Bloomington IN

Investigators

Abstract

Abstract Award: DMS-0541917 Principal Investigator: David Fisher The proposed research lies between dynamical systems and geometry. Work of Furstenberg, Mostow, Margulis and others shows that certain dynamical systems are an important tool for studying geometric properties of important geometric spaces, particularly symmetric and locally symmetric spaces. . One important property of symmetric spaces is that they have non-positive curvature. The investigator plans to extend this work in several inter-related directions. The study of symmetric spaces will be generalized to include more general spaces of non-positive curvature as well as more general maps between spaces with an emphasis on infinite dimensional spaces. The investigator will attempt to exploit these relationships to prove long-standing conjectures of Zimmer. The group of diffeomorphisms of a compact manifold acts on the space of square-integrable Riemannian metrics, which is naturally an infinite-dimensional space of non-positive curvature. In previous work the PI has extensively studied dynamics of certain groups acting on ``flat" infinite dimensional spaces, and some present work can be viewed as generalizing those result to spaces that have curvature. Typically if one is a perturbing a dynamical system one already understands, the study of nearby dynamical systems can be reduced to properties of a flat infinite dimensional space. However, if one wants to classify dynamical systems which one does not assume are perturbations of ones that are well-understood, one is forced in a space that has curvature. Dynamical systems is a young and important field of mathematics that investigates the evolution of physical or mathematical systems over time (e.g. fluid flow) while differential geometry is a more classical field that tends to study static configurations of curves and shapes in space. New ideas from dynamical systems theory such as chaos and fractals have had a profound impact on our perception of the world. One of the deepest and most influential mathematical applications of dynamical systems has been to the study of geometric properties of spaces with "many symmetries". The PI's research can be viewed as part of a general development in modern mathematics in which ideas from dynamics and differential geometry interact to lead to both proofs of old conjectures and exciting new discoveries. A key idea that appears repeatedly is that spaces that possess many symmetries must actually be homogeneous, i.e. any space with enough symmetry actually looks the same at every point. The PI's work on these topics has relationships with diverse areas of research including computer science (expander graphs and property (T) of Kazhdan) and celestial mechanics (KAM theory and stability of the solar system).