Lagrangian links on surfaces and the Calabi invariant

Abstract: The identity component in the group of area-preserving homeomorphisms of a compact surface admits a `mass-flow’ (or flux) homomorphism to the reals. We will prove that the kernel of this homomorphism is not simple (extending earlier results of Cristofaro-Gardiner, Humilière and Seyfaddini in the genus zero case), resolving a question of Fathi from the late 1970s. The proof appeals to a new family of Lagrangian spectral invariants associated to Lagrangian links on the surface, which are used to probe the small-scale geometry of the surface; their crucial feature is that they can be used to recover the classical Calabi invariant of a Hamiltonian. The Floer cohomology theory behind these spectral invariants is a close cousin of the knot Floer homology of Ozsváth-Szabó and Rasmussen. This talk reports on joint work with Dan Cristofaro-Gardiner, Vincent Humilière, Cheuk Yu Mak and Sobhan Seyfaddini.

algebraic geometrydifferential geometrygeometric topologysymplectic geometry

Audience: researchers in the topic

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