Projektbeschreibung
Fortschritte auf dem Gebiet der kontinuierlichen symplektischen Topologie
Die kontinuierliche symplektische Topologie untersucht kontinuierliche Analoga von glatten symplektischen Objekten, wie symplektische und Hamiltonsche Homöomorphismen, und prüft die Persistenz verschiedener symplektischer Phänomene unter einheitlichen Grenzen und Störungen. Das EU-finanzierte Projekt HSD wird die kontinuierliche symplektische Topologie aus zwei verschiedenen Perspektiven erforschen. Die erste ist eine symplektische topologische Perspektive, die von Gromovs weicher und harter symplektischer Geometrie geprägt ist. Die zweite ist motiviert durch die jüngsten Wechselwirkungen zwischen kontinuierlicher symplektischer Topologie und dynamischen Systemen, die unter das neue Gebiet der symplektischen Dynamik fallen.
Ziel
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.
Wissenschaftliches Gebiet
Programm/Programme
Thema/Themen
Finanzierungsplan
ERC-STG - Starting GrantGastgebende Einrichtung
75794 Paris
Frankreich