CAREER: Universal Circles Between Dynamics and Geometry

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The mathematical notion of a dynamical system allows for every time-varying physical system to be considered in a uniform framework, as a "state space" that organizes the sea of possible instantaneous configurations, together with a "flow" that describes the evolution of states with time. From this viewpoint a predator-prey system bears a striking resemblance to a swinging pendulum: both have 2-dimensional state spaces whose axes correspond to either the total predator and prey populations or to the position and velocity of the pendulum point. This illustrates the unifying power of mathematical dynamics, which is the primary focus of this project. The PI's research program will connect the geometry of low-dimensional phase spaces with the dynamics of the systems that they support, with an eye towards applications in both geometry and dynamics. At the same time, the PI's educational program will bring the tools and viewpoint of mathematical dynamics to a wide audience across the STEM disciplines. The research component of this project is concerned with quasigeodesic and pseudo-Anosov flows on 3-manifolds. These flows give rise to lower-dimensional structures — flowspaces, leaf spaces, and universal circles — that entwine the large-scale dynamics of the flow with an action of the ambient manifold’s fundamental group, and act as a conduit between the underlying dynamics and geometry. The PI aims to show that one can use the universal circle action of a quasigeodesic flow to reconstruct the entire manifold, together with a pseudo-Anosov representative of the original flow, and characterize the group actions on circles that appear as universal circles. In addition the PI aims to use quasigeodesic flows as tools to probe the structure of 3-manifolds with potential applications towards the Cannon Conjecture and understanding the faces of the Thurston norm ball. For the educational component the PI proposes what he calls the State Space Project, an interdisciplinary working group (which includes faculty from engineering, chemistry, biology, physics and statistics) which includes graduate courses and a seminar. The PI also plans to continue broader impact activities related to student mentoring, curriculum development, involvement in Math Circles and editorial service.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.