Tag - Symplectic geometry

In this talk I will discuss a joint project with Yuanpu Liang in which we establish several properties of the sequence of symplectic capacities defined by Gutt and Hutchings for star-shaped domains using S¹-equivariant symplectic homology. Among the results discussed will be the fact that, unlike the first of these capacities, the others all fail to satisfy the symplectic version of the Brunn Minkowski established by Artstein-Avidan and Ostrover. We also show that the Gutt-Hutchings capacities, together with the volume, do not constitute a complete set of symplectic invariants even for convex bodies with smooth boundary. The examples constructed to prove these results are not exotic. They are convex and concave toric domains. The main new tool used is a significant simplification of the formulae of Gutt and Hutchings for the capacities of such domains, that holds under an additional symmetry assumption. This allows us to compute the capacities in new examples and to identify and exploit blind spots that they sometimes share.

In this talk, I will present two results relating the qualitative dynamics of non-degenerate Hamiltonian isotopies on surfaces to the structure of their Floer complexes.

The first will be a topological characterization of those Floer chains which represent the fundamental class in CF_∗(H,J) and which moreover lie in the image of some chain-level PSS map. This leads to a novel symplectically bi-invariant norm on the group of Hamiltonian diffeomorphisms, which is both C⁰-continuous and computable in terms of the underlying dynamics.

The second result explains how certain portions of the Hamiltonian Floer chain complex may be interpreted geometrically in terms of positively transverse singular foliations of the mapping torus, with singular leaves given by certain maximal collections of unlinked orbits of the suspended flow. This construction may be seen to provide a Floer-theoretic construction of the 'torsion-low' foliations which appear in Le Calvez's theory of transverse foliations for surface homeomorphisms, thereby establishing a bridge between the two theories.

A generalization of the cartesian product and the free sum of two convex domains is the p-product operation. We investigate the behavior of symplectic capacities with respect to symplectic p-products, and we give applications related to Viterbo's volume-capacity conjecture and to p-decompositions of the symplectic ball.

The notion of positive (non-negative) contact isotopy, defined by Eliashberg and Polterovich, leads to two relations on the group of contactomorphisms. These relations resemble the causal relations of a Lorentzian manifold. In this talk we will introduce a class of Lorentzian distance functions on the group of contactomorphisms, that are compatible with these relations. The Lorentzian distance functions turn out to be continuous with respect to the Hofer-norm of a contactomorphism defined by Shelukhin.

We show that, for closed Legendrians in 1-jet bundles, when there is a sheaf with singular support on the Legendrian, then (1) its self Reeb chords are bounded from below by half the sum of Betti numbers, and (2) the Reeb chords between itself and its Hamiltonian push off is bounded from below by Betti numbers when the C⁰-norm of the Hamiltonian is small. I will show how to visualize Reeb chords/Lagrangian intersections in sheaf theory, and then explain the duality exact triangle and the persistence structure used in the proof. If time permits, I will state a conjecture on the relative Calabi-Yau structure that arises from the duality exact triangle.