Lieu: IHP, salle 201.
11:00 Alexandru Oancea (Jussieu)
Dualité de Poincaré dans les espaces de lacets
Résumé: La dualité de Poincaré est une symétrie remarquable entre quantités homologiques et cohomologiques, spécifique aux variétés de dimension finie. Une telle symétrie a été observée depuis longtemps pour les principes variationnels qui détectent les géodésiques fermées, formulés dans des espaces de lacets. J’expliquerai comment formaliser la dualité de Poincaré dans ce cadre. Il s’agit d’un travail en cours avec Kai Cieliebak et Nancy Hingston. Alors même que le problème d’origine est riemannien, notre solution utilise des méthodes symplectiques.
14:15 Michael Hutchings (Berkeley)
Two or infinitely many Reeb orbits
Abstract: We show under two assumptions that a contact form on a closed connected three-manifold has either two or infinitely many Reeb orbits. The two assumptions are that the contact form is nondegenerate (which holds generically), and that the first Chern class of the contact structure is torsion (which holds for example when the three-manifold is a rational homology sphere). The idea of the proof is to assume that there are only finitely many Reeb orbits, and use embedded contact homology to show that most of the three-manifold has a foliation by genus zero holomorphic curves. One can then use a theorem of Franks to show that there are exactly two Reeb orbits. Joint work with Dan Cristofaro-Gardiner and Dan Pomerleano.
16:00 Mads Bisgaard (ETH Zurich)
Symplectic Mather theory
Abstract: I will discuss two different approaches to systematically studying invariant sets of Hamiltonian systems. The first approach builds heavily on results due to Viterbo and Vichery, except it uses pseudo holomorphic curve techniques. I will discuss how an analogue of Mather's alpha-function arises from homogenized Floer homological Lagrangian spectral invariants and how it gives rise to the existence of an analogue of Mather measures (from Aubry-Mather theory) to general symplectic manifolds. Unlike what happens in the Tonelli case, I will show that the support of these measures can be extremely "wild" in the non-convex case. I will explain how this phenomenon is closely related to diffusion phenomena such as Arnold' diffusion. The second approach builds on work due to Buhovsky-Entov-Polterovich and provides a C^0-analogue of Mather measures for Hamiltonians on symplectic manifolds where every compact subset is displaceable. I will discuss applications to Hamiltonian systems on twisted cotangent bundles and R^2n.
Autre activité symplectique à Paris:
Séminaire Nantes-Orsay
Conférence en l'honneur de Jean Cerf (inscription à l'avance requise pour faciliter le travail des organisateurs)
