Packing stability, ECH capacity asymptotics, and simplicity of the Hamiltonian diffoemorphism group
Summary:
Recently, Oliver Edtmair posted to arxiv a pair of papers proving packing stability for all symplectic four-manifolds with smooth boundary. The proof relies on a refinement of an old theorem by Banyaga, stating that the group of hamiltonian diffeomorphisms is simple. I start by describing the ethos of ball packing problems, and discuss how symplectic ball packing controls the asymptotics of ECH capacities. Then, I introduce the the toy example from the introduction of Oliver’s paper, a class of manifolds where the simplicity of the Hamiltonian diffeomorphism group provides an explicit decomposition of the manifold into balls and polydiscs.
Presented at:
- UC Berkeley Symplectic Geometry Seminar, Fall 2025
🔗 Link to file
Sources
The two papers by Oliver Edtmair:
- Packing stability and the subleading asymptotics of symplectic Weyl laws
- Using a refinement of simplicity of the Hamiltonian diffeomorphism group, explicitly decomposes any symplectic four manifold with smooth boundary into balls and polydiscs. Then, uses this decomposition to construct a full filling of the four manifold by a single ellipsoid. Since ellipsoids have packing stability, the full manifold has packing stability. From this, oliver extracts an $O(1)$ bound of the error term for ECH asymptotics.
- Smooth perfectness of Hamiltonian diffeomorphism groups
- Proves a quantitative refinement of the simplicity of the Hamiltonian diffeomorphism group, needed for the other paper.