Séance du Vendredi 5 octobre 2012.
Lieu: IHP, Salle 201.
- 11h: Rémi Leclercq (Université Paris Sud),
Uniqueness of continuous Hamiltonians
- 14h: Oliver Fabert (Université de Freiburg),
New algebraic structures in Floer theory
Connecting orbits for family of Tonelli Hamiltonians
11h: Rémi Leclercq: Uniqueness of continuous Hamiltonians
I will consider limits of Hamiltonian flows for the spectral and Hofer's distances. Namely, I will show that a continuous function obtained as a uniform limit of smooth Hamiltonians whose flows converge to the identity must vanish. As a key-step of the proof, I will first establish a new energy-capacity-type inequality relating Oh-Schwarz spectral invariants and the Hofer-Zehnder capacity. Finally, I will explain some consequences of the main result in C0-symplectic topology. This is joint work with Vincent Humilière and Sobhan Seyfaddini.
14h: Oliver Fabert: New algebraic structures in Floer theory
Floer cohomology for symplectomorphisms with its pair-of-pants product can be viewed as a generalization of the small quantum cohomology ring. In my talk I will show how the rich algebraic structures of rational Gromov-Witten theory beyond the small quantum product translate to the Floer theory of a symplectomorphism. In order to define a big version of the pair-of-pants product, the corresponding generalization of Frobenius manifolds and the resulting integrable systems, I make use of the rich algebraic structures from rational symplectic field theory. As an application of these new algebraic structures in Floer theory, I prove a new big product axiom for the Oh-Schwarz spectral invariants, as well as a new axiom involving the Gromov-Witten integrable hierarchy. Finally I sketch how transversality is established without polyfolds.
16h: Vito Mandorino: Connecting orbits for family of Tonelli Hamiltonians
Let M be a compact manifold without boundary. We consider the dynamical system generated by the time-one maps of a family of one-periodic Tonelli (or optical) Hamiltonians on T*M. Such a system is also called à polysystem. By using the variational framework of weak KAM theory, we try to construct orbits exhibiting some prescribed qualitative behaviors, such as connecting orbits between two different, far apart regions of T*M. This is related to the phenomenon of Arnold diffusion in near-integrable Hamiltonian systems, and generalize some results of Moeckel and Le Calvez for a family of exact-symplectic twist maps on the cylinder.