Descrizione del progetto
Far progredire il campo della topologia simplettica continua
La topologia simplettica continua studia gli analoghi continui di oggetti simplettici lisci, come gli omeomorfismi simplettici e hamiltoniani, e approfondisce la persistenza di vari fenomeni simplettici in condizioni di perturbazione e limitazione uniforme. Il progetto HSD, finanziato dall’UE, esplorerà la topologia simplettica continua da due diversi punti di vista. Mentre il primo riguarda una topologia simplettica informata dalla geometria simplettica morbida e dura di Gromov, il secondo è motivato dalle recenti interazioni della topologia simplettica continua con i sistemi dinamici, che rientrano nel nuovo campo della dinamica simplettica.
Obiettivo
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.
Campo scientifico
Programma(i)
Argomento(i)
Meccanismo di finanziamento
ERC-STG - Starting GrantIstituzione ospitante
75794 Paris
Francia