Rigid Behavior of Group Actions and Geometric Structure
Kathryn P. Mann, Mathematics, studies basic mathematical objects. She investigates their symmetries and looks at how the objects change under transformations. This is the study of infinite groups on manifolds, which bridges many areas of mathematics (topology, geometric group theory, foliation theory, and topological dynamics). Mann is working to 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, ranging from weather patterns, to ocean currents, to the configuration space of a mechanical system.
A major guiding principle in Mann’s work is that rigidity of group actions, such as with topological dynamics, is often the result of an underlying geometric structure. One example of this phenomenon is Mann’s recent results on foliated circle bundles over surfaces, where she shows that every rigid bundle is geometric. Mann is building on the success of this project, extending the theme to new contexts and applications. She is continuing to work on flat bundles and rigid monodromy group actions, using techniques from foliation theory and coarse geometry in negative curvature to study boundary actions.
She is also investigating rigidity of infinite discrete groups acting on the circle through the topology of spaces of circular orders. Mann is adapting techniques from geometric group theory to the new context of non-locally compact groups, applying this to study the dynamics of group actions on surfaces, and refining the notion of distortion and growth in transformation groups introduced by Mikhail Leonidovich Gromov.