We define fractional twists, a generalization of Dehn twists, for higher dimensional symplectic manifolds and look at some of their properties using contact topological methods. In particular, we will describe their role in invariant contact structures and discuss the difference between right-handed and left-handed twists.

We will survey some of the history of the three-body problem, and then discuss how contact geometry can be used to get some new insights. In particular, we will describe how finite energy foliations can be used to find global surfaces of section. We use these global surfaces of section to display some interesting dynamics.

Quand deux algèbres de dimension finie ont-elles la même catégorie dérivée? Le cas où les deux algèbres ont dimension globale $\leq 1$ est une question classique traitée par Dieter Happel dans les années 80. Dans cet exposé, j'expliquerai comment ce résultat peut se généraliser dans le cas où la dimension globale est $\leq 2$. Puis je donnerai une grande famille d'exemples: les algèbres de surface, dans lesquels le résultat devient beaucoup plus précis.

Je vais présenter des exemples de sous-groupes discrets de PU(2,1), le groupe des isométries holomorphes du plan hyperbolique complexe. Ce dernier peut-être vu comme la boule unité de C^2, et apparaît donc comme une généralisation naturelle du disque de Poincaré, ou de l'espace hyperbolique réel de dimension 3. Je m'intéresserai principalement aux groupes de surfaces. Si le temps le permet j'évoquerai certains exemples de structures CR sphériques sur les variétés de dimension 3 associés.

Embedded contact homology is an invariant of a three-manifold isomorphic to Heegaard Floer homology and Seiberg-Witten Floer homology. However, ECH chain complex depends on the choice of a contact form on the manifold and is of interest for studying symplectic geometric properties (e.g. ECH capacities). Extending the work of Hutchings-Sullivan, we combinatorially describe  the ECH chain complexes of toric contact manifolds such as T^3 with a T^2-invariant contact form.

Rationally and polynomially convex domains in $\C^n$ are fundamental objects of study in the theory of functions of several complex variables. After defining and illustrating these notions, I will explain joint work with Y.Eliashberg giving a complete characterization of the possible topologies of such domains in complex dimension at least three. The proofs are based on recent progress in symplectic topology, most notably the h-principles for loose Legendrian knots and Lagrangian caps.

Lagrangian Floer cohomology is the best tool to study the intersection theory of Lagrangian submanifolds of a given symplectic manifold. Unfortunately it cannot be defined in general, instead there is a more complicated algebraic invariant, a filtered A-infinity algebra, known as the Fukaya algebra, introduced by Fukaya-Oh-Ohta-Ono . In this talk we will review its constructions and describe the Fukaya algebra of the product of two Lagrangians. For this we will first have to define the tensor product of A-infinity algebras.