Séance du 8 janvier 2016

Lieu: IHP, salle 201

14:15  Mélanie Bertelson (Bruxelles)
A correspondance à la Sullivan between contact structures and cone structures.
Abstract: In his paper Cycles for the Dynamical Study of Foliated Manifolds and Complex Manifolds, Sullivan shows that existence of a symplectic structure on a closed manifold is equivalent to that of an ample cone structure of bivectors that satisfies a certain homological condition. As will be explained in the talk, this equivalence can be adapted to the contact case. .

16:00  Renato Vianna (Cambridge)
Infinitely many monotone Lagrangian tori in Del Pezzo surfaces.
Abstract: In 2014, we showed how the Chekanov torus arises as a fiber of an al-
most toric fibration and how this perspective enable us to describe an in-
finite range of monotone Lagrangian tori. More precisely, for any Markov
triple of integers (a, b, c) - satisfying a² + b² + c² = 3abc - we get a mono-
tone Lagrangian torus T (a² , b² , c² ) in CP² . Using neck-stretching tech-
niques we are able to get enough information on the count of Maslov in-
dex 2 pseudo-holomorphic disks that allow us to show that for (d, e, f ) a
Markov triple distinct from (a, b, c), T (d² , e² , f² ) is not Hamiltonian iso-
topic to T (a² , b², c²).
In this talk we will describe how to get almost toric fibrations for all del
Pezzo surfaces, in particular for CP² #kCP² for 4 ≤ k ≤ 8, where there
is no toric fibrations (with monotone symplectic form). From there, we
will be able to construct infinitely many monotone Lagrangian tori. Some
Markov like equations appear. They are the same as the ones appearing
in the work of Haking-Porokhorov regarding degeneration of surfaces to
weighted projective spaces..


Prochaines séances: 5/02 (M. Khanevsky, J. Latschev, G. Spano).
Autre activité symplectique à Paris: groupe de travail faisceaux.