# Symplectix

Groupe de travail sur la topologie symplectique.

### Séance du 13 mai 2016

Lieu: IHP, salle

Autre activité symplectique à Paris: Séminaire Nantes-Orsay.

**01****11:00**Stéphane Guillermou (Grenoble)**The three cusps conjecture.***Abstract:*Arnol'd's three cusps conjecture is about the fronts of Legendrian curves in the projectivized cotangent bundle of the 2-sphere. It says that the front of a generic Hamiltonian deformation of the fiber over a point has at least three cusps. We will recall some results of the microlocal theory of sheaves of Kashiwara and Schapira and see how we can use them to prove the conjecture.**14:15**Guogang Liu (Nantes)**On positive loops of loose Legendrian embeddings****.***Abstract:*Positive Legendrian/contact isotopies relate to the orderability of the universal cover of contactomorphism groups. In fact, there does not exist positive loops based in any Legendrian.I will explain that for any loose Legendrian there always exists a positive loop based in it. For the case of a loose sphere, I will construct a positive loop by hand. While for the other cases, I will do it by h-principle.**16:00**Peter Uebele (Augsburg)**Periodic Reeb flows and products in symplectic homology.***Abstract:**For contact manifolds with periodic Reeb flow, symplectic homology can be computed with Morse-Bott methods. The computation indicates that its chain groups are periodic in the degree. We will show that, under a certain index assumption, this periodicity also holds on homology and has a natural explanation in terms of the pair-of-pants product. This also reveals a good part of the ring structure of symplectic homology, in particular that it is finitely generated as a algebra. The proof uses the V-shaped symplectic homology introduced by Cieliebak-Frauenfelder-Oancea and the action of a loop of Hamiltonian diffeomorphisms on Floer homology introduced by Paul Seidel.*Autre activité symplectique à Paris: Séminaire Nantes-Orsay.

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