La topologie des courbes complexes compactes (dimension réelle 2) est bien comprise : depuis Riemann, on sait que la topologie d'une telle courbe est décrite par un entier naturel : son genre. Mais les choses se compliquent lorsqu'on cherche à comprendre la topologie des surfaces complexes (dimension réelle 4). Dans cet exposé je présenterai un outil important utilisé par Poincaré pour étudier la topologie des surfaces algébriques complexes : les pinceaux (et les fibrations) de Lefschetz.

At the starting point of this talk are two integrable Hamiltonian systems in infinite space dimension. The first one is the Benjamin--Ono equation and was introduced about forty years ago in Fluid Mechanics. The second one is the Szegö equation and was introduced about ten years ago as a model of a non dispersive Hamiltonian evolution. Both systems admit a Lax pair structure, involving operators on the Hardy space of the disk enjoying special commuting properties with the shift operator : Hankel operators and Toeplitz operators. I will focus on the inverse spectral problems for these Lax operators, on the similarities in the strategy for solving them and on the dramatically different outputs.

We study two different invariants of framed oriented links. Augmentations are rank one representations of a non-commutative algebra, whose definition is motivated by Floer homology. Sheaves in microlocal theory can be thought of as generalizations of link group representations. We will demonstrate two constructions going back and forth between these invariants. We will also tell a motivating story behind the scene, using SFT and microlocalization correspondence in symplectic topology.

The numerical simulation of elastic thin-walled bodies immersed in an incompressible viscous fluid is an essential ingredient in the mathematical modeling of many living systems: From the opening and closing dynamics of heart valves to the wings of a bird interacting with the air or the fins of a fish moving in water.
The numerical methods for the simulation of these systems generally fall into one of the following two categories: fitted and unfitted mesh methods. Fitted mesh methods are known to deliver optimal accuracy for moderate interface displacements, but they become cumbersome or lose efficiency in presence of topological changes (e.g., due to contacting solids). Unfitted mesh methods, such as the Immersed Boundary/Fictitious Domain methods or the recently developed Nitsche-XFEM method, allow for arbitrary interface displacements but this flexibility comes at a price: the mismatch between the fluid and solid meshes complicates the interface coupling. In this talk, we will review some of these approaches by comparing them on some known FSI benchmarks involving moving interfaces and topology changes. We will also introduce a new time splitting scheme for a particular class of fictitious domain approximations, which invokes the fluid and solid solvers only once per time-step without compromising stability and accuracy.

En 1993, K. Kuperberg construit des exemples lisses et même analytique réels de flots sans points fixes et sans orbites périodiques sur toute variété fermée de dimension 3. Ces exemples sont à ce jour les uniques exemples de flots ayant ces propriétés. Il sont construits à l’aide de pièges. Un piège est une variété à bord et à coins, nous pouvons penser au produit d'un disque de dimension 2 par un intervalle, qui est munie d’un flot dont les orbites peuvent sortir. Il a la propriété de piéger des orbites : il y a des orbites qui rentrent dans le piège et ne ressortent jamais.

Une orbite piégée s’accumule sur un ensemble fermé invariant à l’intérieur du piège, celui-ci doit contenir un ensemble minimal du flot. Je vais présenter certains aspects de l’étude de l’ensemble minimal des exemples de K. Kuperberg. A ma connaissance, celui-ci est le premier ensemble minimal exceptionnel de dimension topologique deux. Les résultats présentés ont été obtenus en collaboration avec Steve Hurder.

In this talk, I give a formula for counting complex rational curves passing through certain configuration of points on the 3- dimensional projective space $\mathbb{C}P^1\times\mathbb{C}P^1\times\mathbb{C}P^1$ by using the counting on the 2-dimensional projective space $\mathbb{C}P^1\times\mathbb{C}P^1$ blowing-up at 2 points. That gives rise to the fancy formula for Gromov-Witten (GW) invariants in the Fano threefolds of index 2 with related to the GW invariants in one specific type of surfaces which realise one half of their first Chern class.

Classical and flag paraproducts arise naturally in the study of nonlinear PDEs. While multi-parameter paraproduct has been studied thoroughly, known estimates for flag paraproducts only involve single parameter. We will state $L^p$ estimates for a particular case of the bi-parameter flag paraproduct on some restricted function spaces. We will discuss the key ingredient of the proof - a stopping-time argument which combines information from subspaces to obtain estimates on the entire space.