Flows, circles, and dynamics at infinity

Washington University, Saint Louis MO

Investigators

Abstract

A pendulum can be described as a flow on a 2-dimensional "phase space," whose coordinates are given by position and velocity. The mathematical system that describes the motion of a pendulum is a dynamical system. Another example is fluid dynamics, which for instance, is used to study the flow of air over an airplane wing. Often, the structure of a phase space has implications for the dynamical systems that it supports. For example, any flow on the sphere has a stationary point, which means that the corresponding dynamical system has a state that never changes. This project is concerned with the obverse of this idea: What does a flow on a space say about the space itself? The spaces we consider are 3-dimensional, like the space surrounding us; the flows are used as a tool to "flatten" the space into simpler 1- and 2-dimensional spaces, where the geometric structure of the original space is reflected in the symmetries of these flattened spaces. In addition to its implications within mathematics, this project has applications to applied dynamics, where the flattened spaces can be used to understand the stability of fluid flows. Quasigeodesic and pseudo-Anosov flows are product covered, which means that the orbit space in the universal cover is a topological plane. This plane has a natural circle at infinity, with an action of the fundamental group, which reflects the large-scale dynamics of the flow and the geometry of the manifold. This project uses techniques from dynamics, 3-manifolds, geometric group theory, and classical analysis situs to understand the relationship between a flow, its underlying manifold, and the circle at infinity. In particular, it aims to prove Calegari's conjecture that every quasigeodesic flow may be deformed into a pseudo-Anosov flow, characterize the circle actions that come from such flows, and use flows to understand cubulations and essential surfaces in hyperbolic 3-manifolds.