Descripción del proyecto
Impulso del ámbito de la topología simpléctica continua
La topología simpléctica continua estudia los análogos continuos de objetos simplécticos blandos, como homeomorfismos hamiltonianos y simplécticos, e investiga la persistencia de diversos fenómenos simplécticos en situaciones de alteraciones y límites uniformes. En el proyecto HSD, financiado con fondos europeos, se explorará la topología simpléctica continua desde dos perspectivas diferentes. La primera es una perspectiva topológica simpléctica, basada en la geometría simpléctica dura y blanda de Gromov. La segunda está motivada por las recientes interacciones de la topología simpléctica continua y los sistemas dinámicos, que pertenecen al nuevo campo de la dinámica simpléctica.
Objetivo
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.
Ámbito científico
Programa(s)
Régimen de financiación
ERC-STG - Starting GrantInstitución de acogida
75794 Paris
Francia