Title: Dynamics and ergodic theory in negative curvature.
Speaker: Daniel Thompson
Abstract: The aim of this lecture is to give an accessible introduction to this active area of research, with a view to describing the interesting dynamical phenomena which are typical in spaces with negative curvature. We’ll start the lecture with a discussion of “What is Dynamics?” to give the bigger picture of where this area of research sits within mathematics. Geometry studies the properties of space, particularly shape and size. Dynamical systems and Ergodic Theory study the expected long-term behavior of systems which are changing with time. A natural dynamical system that arises in geometry is the geodesic flow: given an initial position and direction, we flow at unit speed along the path that locally minimize distance. This flow has special importance because of its relationship with the geometry and topology of the underlying space. If the space has non-trivial geometry the flow will often exhibit ergodic behavior. Consider a torus with more than one hole, which can be visualized as a pretzel. Paths that start close together will usually separate and take different and unpredictable journeys around these holes, and typically these journeys will eventually look independent of where they started. This is a model situation where the dynamics and geometry can be understood using the tools of ergodic theory. The theory is classical for compact manifolds of negative curvature. Recent work by myself, Vaughn Climenhaga, and former OSU graduate student Tianyu Wang has developed this theory for non-compact manifolds with negative curvature allowed to approach 0 and -∞.
