CAREER: Geometric and Topological Approaches to Group Actions in Low Dimensions

Brown University, Providence RI

Investigators

Abstract

This project concerns actions of infinite groups on manifolds, an area bridging and uniting many areas of mathematics (topology, geometric group theory, foliation theory, and topological dynamics). The Principal Investigator (PI) will study basic mathematical objects by studying their symmetries, and by studying how the objects change under transformations: this is the study of group actions. The project will describe and distinguish rigid behavior, where small perturbations to a system of transformations do not qualitatively change the long-term outcome, versus unstable or chaotic behavior, which is highly sensitive to perturbation. Variations on this problem arise all around us, as we seek to understand the long-term behavior of mathematical models of real world objects ranging from weather patterns, to ocean currents, to the configuration space of a mechanical system. When the objects of interest are highly complex or not easily parametrized, these problems are difficult to approach, and the PI's program centers on several new techniques to render rigidity problems tractable. The project also involves the introduction of active-learning courses for undergraduate mathematics students, including a program to train graduate students in teaching methods, and a major workshop on effectively communicating mathematics across research areas with the aim of increasing communication between mathematicians in disparate fields. A major guiding principle in the PI's work is that rigidity of group actions, in the sense of topological dynamics, is often the result of an underlying geometric structure. One example of this phenomenon is the PI's recent results on foliated circle bundles over surfaces, where it is shown that every rigid such bundle is geometric. This project builds on this success of this program, extending this theme to new contexts and applications. The PI will continue work on flat bundles and rigid monodromy group actions, using techniques from foliation theory and coarse geometry in negative curvature to study boundary actions. Another strand of this project involves an investigation of rigidity of infinite discrete groups acting on the circle through the topology of spaces of circular orders, following work of A. Navas. Finally, the PI will adapt techniques from geometric group theory to the new context of non-locally compact groups, applying this this to study the dynamics of group actions on surfaces, and refining the notion of distortion and growth in transformation groups introduced by Gromov. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.