**01**

**11:00**Alvaro Del Pino Gomez (Madrid)

**An introduction to Engel structures.**

*Abstract:*A maximally non-integrable 2-plane field in a 4-manifold is called an Engel structure. Like contact structures, they possess a Darboux model, and hence they lack local invariants. At the same time, the lack of a Gray stability type theorem makes the construction of global invariants (up to deformation through Engel structures) a complicated matter.

The aim of the talk will be to review the recent advances in flexibility in Engel geometry. Namely, I will outline the proof of the existence h-principle for Engel structures proven in [CPPP] and the h-principle for horizontal and transverse immersions proven in [CP]. If time allows, I will discuss some work in progress regarding horizontal knots in standard Engel R

^{^4}.

[CP] A. del Pino, F. Presas. Flexibility for tangent and transverse immersions in Engel manifolds. In preparation.

[CPPP] R. Casals, J.L. Pérez, A. del Pino, F. Presas. Existence h-Principle for Engel structures . arXiv:1507.05342.

**14:15**Milena Pabiniak (Cologne)

**The contact Arnold Conjecture for lens spaces via a non-linear Maslov index.**

*Abstract:*Diffeomorphisms in symplectic category posses certain rigidity properties. An important manifestation of rigidity is given by the conjectures posed by V. Arnold describing a lower bound for the number of fixed points of a Hamiltonian diffeomorphism of a compact symplectic manifold, greater than what topological arguments could predict.

Arnold
Conjectures present a difficult problem and motivated a lot of
important research in symplectic geometry. It has been translated to the
contact geometry setting where one looks for a lower bound for the
number of translated points.

Givental's
construction of a quasimorphism, called the non-linear Maslov index,
allows one to prove the Arnold Conjecture for complex and real
projective spaces. Moreover, the properties of this quasimorphism imply
that the real projective space is orderable, has a non-displaceable
pre-Lagrangian and that its discriminant and oscillation norms are
unbounded.

In this talk I will describe my work joint with G. Granja, Y. Karshon and S. Sandon, aimed at constructing a quasimorphism for lens spaces (building on the ideas of Givental) and proving the corresponding statements for these spaces (Contact Arnold Conjecture, orderability, ... ). I will discuss the difficulties of
constructing such a generalization to lens spaces and the possibility of
generalizing these ideas even further: to prequantizations of symplectic toric
manifolds. **16:00**Yuichi Ike (Tokyo)

**Categorical localization for the coherent-constructible correspondence**

*Abstract:*The coherent-constructible correspondence is a version of homological mirror symmetry for toric varieties. It says that the derived category of coherent sheaves on a toric variety is equivalent to the derived category of constructible sheaves on the real torus whose microsupports are contained in some Lagrangian. We prove categorical localization for the constructible-side categories, which can be regarded as a microlocal counterpart of categorical localization for Fukaya categories. This is a joint work with Tatsuki Kuwagaki.

Prochaines séances: 4 novembre (F. Balacheff, U. Hryniewicz, ?), 2 décembre (S. Suhr, ?, ?).

Autre activité symplectique à Paris:

- Séminaire Nantes-Orsay,

- Doctorat Honoris Causa de Dusa McDuff : les 10 et 11 octobre à Paris 6.