Workshop "C0-symplectic geometry and Hamiltonian dynamics",
IHP, Paris, January 2016, 18th-22nd

General presentation.

C0-rigidity is a well-established feature of modern symplectic geometry, with classical theorems like Eliashberg-Gromov C0-rigidity, rigidity of the Poisson bracket (Entov-Polterovich-Zapolsky, Buhovsky...), or dynamics of area-preserving surface homeomorphisms (Le Calvez, Franks-Handel). Further results have appeared in the recent years, that have extended, refined, and generalized to other areas these by-now classical theorems, starting to draw the border between the C0-flexible and C0-rigid properties of symplectic world. The aim of this workshop is to gather researchers that have taken a part in these results, and uncover new horizons in the field.

Organization of the meeting.

We will encourage discussion sessions as much as we can. Talks will mainly take place in the morning, while the afternoon will be dedicated to discussions. The workshop will take place in Paris, at the "Institut Henri Poincaré (IHP)", room 421.

Financial support and registration.

Unfortunately, we cannot provide any financial support, except to the speakers, and there is no registration. Those who want to participate are welcome.

Speakers.

(Preliminary) Program.

Monday Tuesday Wednesday Thursday Friday
9h-10h30 M. Khanevsky
(abstract) 		R. Leclercq
(abstract) 		E. Opshtein
(abstract) 		F. Le Roux
(abstract) 		C. Viterbo
(abstract)
10h30-11h BREAK
11h-12h30E. Shelukhin
(abstract) 		F. Zapolsky
(abstract) 		V. Humilière
(abstract) 		B. Bramham
(abstract) 		N. Vichery
(abstract)
12h30-14h30 LUNCH
14h30-15h30 Discussion session Discussions session Discussion session
15h30-16h30 A.F.A
(Almost Free Afternoon)

Titles and abstracts

Barney Bramham : An unlinked Floer Homology and Poincaré surfaces
I will describe how the concept of maximally unlinked collections of periodic orbits, as introduced by LeCalvez, is a good condition to define a Floer-like homology theory on surfaces. In particular this allows to construct foliations by pseudoholomorphic curves that project to foliations transverse to the dynamics.

Vincent Humilière : A C0 counter example to the Arnold conjecture.
I will explain the construction of a Hamiltonian homeomorphism with only one fixed point on any closed symplectic manifold of dimension at least 4. This uses the Buhovsky-Opshtein "quantitative h-principle" technics. This joint work with Lev Buhovsky and Sobhan Seyfaddini.

Misha Khanevsky : Non-continuity of surface quasimorphisms in the Hofer metric
There are several constructions of quasimorphisms on the Hamiltonian groups of surfaces that were proposed by Gambaudo-Ghys, Polterovich and Py. These constructions are based on topological invariants either of individual orbits or of orbits of finite configurations of points and the quasimorphisms compute the average value of these invariants in the surface. We show that many quasimorphisms that arise this way are not Hofer continuous.

Rémi Leclercq : Reduction of symplectic homeomorphisms.
I will briefly explain why symplectic homeomorphisms, in the sense of Gromov-Eliashberg, preserve the coisotropic nature of submanifolds as well as their characteristic foliation. Thus, they descend to the reduction and I will show that, in particular cases, the reduced homeomorphisms preserve a symplectic capacity. This is joint work with Vincent Humilière and Sobhan Seyfaddini.

Frédéric Le Roux : Arnold's conjecture on surfaces (and more?) via transverse foliations
The topological version of Arnold's conjecture on surfaces says that a hamiltonian homeomorphism on a compact surface has at least as many fixed points as a gradient-like foliation. I will explain Le Calvez's proof using transverse foliations.

Emmanuel Opshtein: An h-principle that is suited to C0-symplectic geometry
The basic question of this talk is the following. Given two subsets of a symplectic manifold which are C0-close and hamiltonian isotopic, under which conditions can we find a C0-small Hamiltonian diffeomorphism that brings one to the other ? I will also show an example of application, the C0-rigidity of the reduction of a coisotropic submanifold. This is a joint work with Lev Buhovski.

Egor Shelukhin : Lagrangian cobordisms and metrics
We describe a natural way of measuring the distance between two (possibly non-isotopic!) Lagrangian submanifolds based on the notion of a Lagrangian cobordism, and study the non-degeneracy or degeneracy properties of the resulting pseudo-metrics, reflecting the rigidity or flexibility of the class of cobordisms considered. This is a joint work with Octav Cornea.

Nicolas Vichery :Non convex Aubry Mather theory, invariant measures and rotation vectors.
After a few reminders about symplectic homogenization, we will present an extension of the fundational result of Mather about the existence of invariant mesures with rotation vector prescribed by the subdifferential of the effective Hamiltonian. This can be done by replacing the effective hamiltonian by the symplectic homegenization. We will finish by some applications in the context of classical KAM theory.

Claude Viterbo
Title: tba

Frol Zapolsky : "Spectral invariants for contactomorphisms of prequantization bundles and applications"
I'll sketch the construction of spectral invariants for contact isotopies of prequantization bundles over monotone symplectic manifolds of minimal Chern number at least 2. The construction uses Lagrangian spectral invariants in non-convex symplectic manifolds. Applications include a Floer-theoretic construction of a quasi-morphism on Cont_0 of the real projective space, and other things. Joint work in progress with Peter Albers and Egor Shelukhin.

Organizers :

Vincent Humilière, Frédéric Le Roux, Emmanuel Opshtein.

Sponsors :

ANR hameo, Labex IRMIA, ERC-Starting Grant STEIN-259118.