Geometry, foliations and flows

The field of topology dates back at least to Poincare in the early 1900's, who was well aware of the interaction between geometric shapes ("manifolds") and dynamical systems, which evolve with time according to set rules. These insights, initially motivated by the dynamics of the solar system, became a far-reaching theory in areas of math both near and far to physics. Today mathematicians and scientists often study flows in manifolds, in which points travel in intricate orbits constrained by the overall shape of the manifold, as well as foliations, which are decompositions of a manifold into lower-dimensional slices like leaves of a layered pastry. This project aims to study such phenomena and build the understanding of the intricate connections between geometry and dynamics which occur especially in "low dimensional" settings such as surfaces and three-dimensional manifolds. Experience has shown that these low-dimensional settings, amenable to visual and geometric intuition, are also a valuable testing ground and provide inspiration for many more general phenomena within and outside of mathematics. Graduate students funded by the grant will, in addition to their research work, be trained in mathematics education and outreach, preparing them for contributions to society through higher education and other leadership roles. Additional activities of the PI and his students and collaborators will include public educational events for local school children and their families, bringing cutting-edge mathematical knowledge to the greater New Haven community. In this project, the principal investigator (PI) will explore a number of connections among foliations, flows, dynamics and hyperbolic geometry in low dimensions. One topic will be the topological dynamics of horospherical flows in infinite-volume manifolds. Building on recent successes of the PI and coauthors in the case of Z-covers of compact surfaces, the project will seek to extend results to higher rank abelian covers, higher dimensions and other generalizations. The PI also plans to revisit connections between two mature subjects in the setting of 3-manifolds: hyperbolic geometry on one hand and the topology and dynamics of taut foliations and pseudo-Anosov flows on the other. The goal here is to obtain a more robust and uniform theory relating topological features of foliations and flows to geometric features of the hyperbolic metric. In a third direction, the PI will study problems associated to Thurston's skinning map and the construction of uniform models for hyperbolic 3-manifolds. A motivating conjecture here is a uniform bound on Thurston's skinning map under certain topological hypotheses, for which the PI and coauthors have a promising line of attack. 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.