Joint IAS/Princeton/Montreal/Paris/Tel-Aviv Symplectic Geometry Zoominar

Alexia Corradini (University of Cambridge) : The Lagrangian Ceresa Cycle

In algebraic geometry, the Ceresa cycle provided one of the first examples of a nullhomologous cycle which is not algebraically trivial. I will explain how one can obtain a mirror statement about the Lagrangian Ceresa cycle, a nullhomologous Lagrangian living in a symplectic six-torus. This requires introducing a new equivalence relation on Lagrangians in a symplectic manifold, algebraic Lagrangian cobordism, inspired by algebraic equivalence.

Ibrahim Trifa (ETH Zurich) : A Local Quasimorphism Property for Link Spectral Invariants

Given a finite collection of disjoint Lagrangian circles on a symplectic surface satisfying some area constraints, Cristofaro-Gardiner, Humilière, Mak, Seyfaddini and Smith define a link spectral invariant, by computing the Lagrangian Floer homology of the product of the circles inside the symmetric product of the surface. When the surface is the sphere, this spectral invariant is a quasimorphism, however this is not the case for higher genus surfaces. In this talk, I will show that the link spectral invariants on higher genus surfaces are local quasimorphisms, i.e. that their restriction to Hamiltonian diffeomorphisms supported in any given topological disc inside the surface is a quasimorphism. This is a joint work with Cheuk Yu Mak.

Stefan Matijević (Ruhr-Universität Bochum) : Systolic $S^1$-index and characterization of non-smooth Zoll convex bodies

We define the systolic $S^1$-index of a convex body as the Fadell–Rabinowitz index of the space of centralized generalized systoles associated with its boundary. We show that this index is a symplectic invariant. Using the systolic $S^1$-index, we propose a definition of generalized Zoll convex bodies and prove that this definition is equivalent to the usual one in the smooth setting. Moreover, we show how generalized Zoll convex bodies can be characterized in terms of their Gutt–Hutchings capacities and we prove that the space of generalized Zoll convex bodies is closed in the space of all convex bodies. As a corollary, we establish that if the interior of a convex body is symplectomorphic to the interior of a ball, then such a convex body must be generalized Zoll, and in particular Zoll if its boundary is smooth.