Topological approaches to Anosov flows in dimension 3

Anosov flows are an important and fundamental class of dynamical systems. They give mathematical models for chaotic motion (sensitivity of outcomes or trajectories to initial conditions), but at the same time exhibit robust global behavior. This confluence of chaos and stability makes them particularly interesting for study and an important mathematical entry-point to understanding the theory behind such physical phenomena. This project will describe and classify all the Anosov flows on three-dimensional spaces, a problem which has been open and of interest since the 1960s. There are already many known examples in the three dimensional space, and the project will show how the dynamics of these various examples of flows are related to geometrical features of the space. Broader impacts include mentorship and training of early career researchers, building new bridges between different areas of mathematics, such as geometry, topology and dynamics, and a workshop to improve communication by mathematicians. The technical goals of this project focus on constructing a dictionary relating the topology and geometry of the foliations associated to an Anosov (or pseudo-Anosov) flow on a 3-manifold with the structure of the 3-manifold and the dynamics of the flow. This builds on previous work of the PI, which used a dimension-reduction technique to translate the classification problem for flows into one for "Anosov-like group actions" on the plane, and then translate results back from the plane to the 3-manifold. The project will carry out a detailed study of such actions to produce new invariants for classification and solve open conjectures such as the finiteness problem. In doing so, the project will also deepen our understanding of the relationship between actions of discrete groups on the plane and the circle, and continuous flows on 3-manifolds, providing new links between dynamics and low-dimensional topology. 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.