Symplectic and Hamiltonian homeomorphisms – Hameo

Our tools are those of symplectic topology (e.g. Floer homology) as well as those specific to dimension 2 (e.g. Le Calvez theory).

Significant progress have been done in C° symplectic topology. In a collaboration with R. Leclercq (Paris 11) and S. Seyffadini (ENS), V. Humilière
shows that the property of being coisotropic is C°-rigid. E. Opshtein shows that symplectic homeomorphisms that preserve an hypersurface induce on the reduction homeomorphisms that preserve capacities.

The above mentionned results raise many question that we want to study.
By the way, the team members all work of the topological invariance in higher dimension of some symplectic invariants.

This project proposes to bring together several mathematicians from different mathematical fields: symplectic geometry for V. Humilière and E. Opshtein, topological dynamical systems for F. Le Roux. The purpose is to study some problems where these fields interact and which we now describe.

The first problem concerns the study of what is called “C^0 symplectic geometry”, in other words to investigate the rigidity properties of the C^0 topology which appear in a surprising way in the purely differentiable context of symplectic geometry. In particular, there exist natural notions of symplectic homeomorphisms (Gromov-Eliashberg) or of Hamiltonian homeomorphisms (Oh-Muller), but these objects are very poorly understood. Our second problem deals with the groups of area preserving transformations on surfaces. The structure of these groups remains still largely unexplored, but there has been some progress due to various recent works (in particular by Polterovich, Entov-Polterovich, Franks-Handel, Oh), giving place to new directions of research. These two problems are actually intertwined. In particular, the above mentioned concepts of symplectic and hamiltonians homeomorphisms plays a central part in both cases. The Hofer energy of isotopies, which is still quite mysterious, also turns up at several places in these problems.

We hope to exploit the complementarity of our mathematical cultures to carry out substantial progress on these problems. We also propose to organize an international conference on these themes, gathering specialists from both fields.