Description du projet
Faire progresser le domaine de la topologie symplectique continue
La topologie symplectique continue étudie les analogues continus d’objets symplectiques lisses, tels que les homéomorphismes symplectiques et hamiltoniens, et étudie la persistance de divers phénomènes symplectiques sous des limites et des perturbations uniformes. Le projet HSD, financé par l’UE, explorera la topologie symplectique continue sous deux angles différents. La première est une perspective topologique symplectique qui s’inspire de la géométrie symplectique douce et dure de Gromov. La seconde est motivée par les interactions récentes entre la topologie symplectique continue et les systèmes dynamiques, qui relèvent du nouveau domaine de la dynamique symplectique.
Objectif
The subject of this proposal is the field of continuous symplectic topology. This is an area of symplectic topology which defines and studies continuous analogues of smooth symplectic objects such as symplectic and Hamiltonian homeomorphisms and asks questions about persistence of various symplectic phenomena under uniform limits and perturbations.
Our aim is to explore, and further develop, continuous symplectic topology from two different perspectives: The first is a symplectic topological perspective which is informed by Gromov’s soft and hard view of symplectic topology. The second is motivated by the recent interactions of continuous symplectic topology and dynamical systems and it falls under the new field of symplectic dynamics.
We outline an extensive research program in line with the above two viewpoints. On the one hand, we propose to develop new tools for the advancement of the field via the medium of barcodes which will serve as a replacement of Floer homology for homeomorphisms. On the other hand, we propose new approaches towards several important questions in the field including the symplectic four-sphere problem which asks if non-symplectic manifolds, such as the four-sphere, could admit the structure of a topological symplectic manifold, and the simplicity conjecture which asks if the group of compactly supported area-preserving homeomorphisms of the disc is a simple group.
Champ scientifique
Programme(s)
Thème(s)
Régime de financement
ERC-STG - Starting GrantInstitution d’accueil
75794 Paris
France