Mathematics

Helicity is an invariant of divergence free vector fields on a three-manifold. One of its fundamental properties is invariance under volume preserving diffeomorphisms. Arnold, having derived an ergodic interpretation of helicity as an asymptotic Hopf invariant, asked whether helicity remains invariant under volume preserving homeomorphisms, and more generally, whether it admits an extension to topological volume preserving flows. In this talk, I will present an affirmative answer to both questions for non-singular flows. The proof draws on recent advances in C^0 symplectic geometry, in particular regarding the algebraic structure of the group of area preserving homeomorphisms, which I will also survey. This is based on joint work with Sobhan Seyfaddini