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Explore the topological invariance of helicity in this advanced mathematics lecture from the IAS/Princeton/Montreal/Paris/Tel-Aviv Symplectic Geometry Zoominar. Delve into helicity as an invariant of divergence-free vector fields on three-manifolds and examine its fundamental property of invariance under volume-preserving diffeomorphisms. Learn about Arnold's ergodic interpretation of helicity as an asymptotic Hopf invariant and his question regarding whether helicity remains invariant under volume-preserving homeomorphisms and admits extension to topological volume-preserving flows. Discover an affirmative answer to both questions for non-singular flows through a proof that utilizes recent advances in C^0 symplectic geometry, particularly concerning the algebraic structure of the group of area-preserving homeomorphisms. Gain insights into cutting-edge research in symplectic geometry and topological dynamics through this collaborative work between Oliver Edtmair from ETH Zurich and Sobhan Seyfaddini.
