Séance du 1er avril 2016

Lieu: IHP, salle 201


11:00  Valentine Roos (ENS)
Viscosity and variational solutions of the evolutive
Hamilton-Jacobi equation.

Abstract: Two different notions of weak solutions were introduced for the
evolutive Hamilton-Jacobi equation, they coincide when the Hamiltonian is
convex in the fiber. The talk will point out examples where the two notions
differ even for small time and quadratic integrable Hamiltonians.
Then we will present the construction of an operator giving the variational
solutions for small time, by applying a minmax selector
to the explicit Chaperon's generating family of the geometric solution.

14:15  Manuel Rivera (Jussieu) 
String topology coproduct: chain level transversality and algebraic models.
Abstract: I will describe a chain level formulation for a “secondary" coproduct on a suitable chain model for the free loop space of a manifold. This operation- which combines a 1-parameter family of self intersections on a family of loops- was originally described by Goresky and Hingston at the level of the (relative) homology by using a finite dimensional approximation of Morse for the free loop space. The operation is also analogue to a coproduct described by Abbondandolo and Schwarz on (a version of) the symplectic Floer homology of the cotangent bundle.
To have a better grasp of this coproduct operation and its properties one would like to describe explicitly the transversality perturbations made at the chain level to obtain an operation parametrized by a nice geometric object. I will explain why this process is more subtle for this operation than for other string topology operations (such as the Chas-Sullivan loop product) and will outline how Dingyu Yang and I have achieved this in work in progress using the formalism of De Rham chains. There is also a rich algebraic theory behind this secondary coproduct. I also plan to describe some of the algebraic theory in the context of Hochschild complexes of dg Frobenius algebras and explain how this provides algebraic models that may be of great help for calculating the coproduct operation explicitly in certain cases.

This talk will concentrate on the algebraic topology side of the string topology story and hopes to open up the discussion of the relation between this secondary operation on chains on the free loop space and the analogue operation on symplectic-Floer homology of the cotangent bundle with those in the audience who are experts in the latter story

16:00  Margherita Sandon (Strasbourg)
Floer homology for translated points.
Abstract: A point p of a contact manifold (M,\xi) is said to be a translated point of a contactomorphism \phi, with respect to a contact form \alpha for \xi, if p and \phi(p) are in the same Reeb orbit and \phi preserves \alpha at p. In 2011 I conjectured that if M is compact then any contactomorphism which is contact isotopic to the identity has at least as many translated points as the minimal number of critical points of a function on M. In this talk I will describe a Floer homology theory for translated points that allows to prove the non-degenerate version of this conjecture in the case where there are no closed contractible Reeb orbits (recovering results of Albers-Fuchs-Merry and Meiwes-Naef) and on any contact manifold if the contactomorphism is generated by a contact Hamiltonian of small oscillation. I will also discuss a couple of immediate applications to orderability.

Prochaines séances: 13/05 (G. Liu, S. Guillermou, ?), 3/06 (F. Balacheff, S. Suhr, ?)
Autre activité symplectique à Paris: groupe de travail faisceaux.