La technique d’inflation de Lalonde-McDuff permet d’exploiter des calculs d’invariants de Seiberg-Witten, pour résoudre le problème des empilements de boules dans certaines 4-variétés (par exemple l’espace projectif, un produit de sphères, ou leurs éclatés). L’inflation singulière permet d’utiliser les mêmes quantités pour établir des résultats de plongements de nombreux domaines sources (par exemple des ellipsoides ou des domaines toriques concaves). J’expliquerai cette technique, et ses applications aux problèmes de plongements
In this talk we discuss a criterion that characterizes overtwisted contact structures. The result establishes seven equivalent properties that relate overtwistedness to loose Legendrians, adapted open books decompositions and thick neighborhoods of overtwisted submanifolds. We first provide the necessary definitions and properties involved in the criterion, and then present the proof of (part of) these equivalences. Finally, we explore possible consequences and applications. This is joint work with E. Murphy and F. Presas.
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.
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.
If we think of CP^3 as the space of triples of points on the sphere then the Chiang Lagrangian is the subspace of triples with centre of mass at the origin. We will see that it has non-vanishing Floer cohomology if and only if the coefficient ring has characteristic 5. This calculation is related to Hitchin's work on Poncelet polygons.
I will discuss a symplectic embedding problem for polydisks in dimension 4. Symplectic embeddings are a key phenomenon of symplectic rigidity. In the case I consider, the obstruction for embedding the polydisk comes from a certain Lagrangian torus and uses the theory of J-holomorphic foliations in dimension 4. This is joint work with Richard Hind.
I will talk about embedded contact homology capacities and some recent applications of it to obstructing symplectic embeddings. I will show how to compute ECH capacities of so-called concave toric domains and I will give some examples of when they give sharp obstructions to symplectic ball packings. In particular, I will discuss the case of the union of a ball with an infinite cylinder. This is joint work with K. Choi, D. Cristofaro-Gardiner, D. Frenkel and M. Hutchings.
