Fluid Dynamics and Contact Topology

Dr. Tanya Cofer

Department of Mathematics

Northeastern Illinois University

Fluid dynamics are complex and intriguingly intractable. This is evident even when studying the enormously restricted class of vector fields which "everywhere twist about themselves" (Beltrami fields). Such fields comprise an important and highly non-trivial class of fluid velocity fields. Unfortunately, "important" has not yet translated into "well-understood". Until recently, little was known (rigorously) about the dynamics of these beguiling fields beyond a few limited results and numerical experiments. New results have emerged from an unexpected connection between Beltrami fields and an ostensibly esoteric branch of topology known as contact topology. Contact topology studies the topology of 3-dimensional manifolds by looking at certain plane fields that "everywhere twist too much" to be tangent to any embedded surface. I will give a general overview of how Beltrami fields arise as certain steady solutions to the Euler equations, the connection between these fields and contact structures, and some interesting results obtained by exploiting this connection.