Is there an asymptotic Chern-Simons series for ergodic vector fields in the 3-sphere?
- Date: Tuesday 6 June 2017, 11:00 – 12:00
- Location: Mathematics Level 8, MALL 1, School of Mathematics
- Type: Applied Mathematics seminars, Leeds applied nonlinear dynamics seminars, Seminar series
- Cost: Free
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