Is there an asymptotic Chern-Simons series for ergodic vector fields in the 3-sphere?

João Faria Martins, University of Leeds. Part of the Leeds applied nonlinear dynamics seminars series.

A fruitful way to address the geometry of vector fields in the 3-sphere is to look at the way integral curves knot and entangle. For instance, the helicity of a vector field, also known as the asymptotic Hopf number, measures the average asymptotic linking number of a pair of integral curves. The linking number (also called Gauss number) is just the order one case of a much more general link invariant: the Chern-Simons perturbation series. The latter can be constructed in a similar way to the Gauss number as a configuration space integral.

In this talk, I will review the concept of asymptotic knot invariant of an ergodic vector field in the 3-sphere, explain how the helicity arises as the asymptotic Gauss number and address how this could in principle be extended to handle some instances of the more general Chern-Simons perturbation series.

Topological methods in hydrodynamics. By VI Arnold and BA Khesin, Springer.
Knots and Dynamics. By É Ghys. Institute Fourier.

João Faria Martins, University of Leeds