By a theorem of Kontsevich and Lambrechts-Volić, the little disks operad is formal in every dimension, meaning that its rational homotopy type is faithfully represented by its cohomology. This theorem has had significant consequences in both high and low-dimensional topology (e.g., Vassiliev invariants, diffeomorphisms of the disk) as well as in deformation theory (e.g., quantization deformation of Poisson manifolds). In my talk, I will review these results and explain a recent result on the equivariant formality of the little disks operad.

This is joint work with Pedro Boavida de Brito and Joana Cirici.

We say that a link L in S^3 is negative amphichiral if there exists an orientation-reversing diffeomorphism of S3 that sends every component of L to itself with the opposite orientation. If such a map can be chosen to be an involution, then the link is said to be strongly negative amphichiral. Kawauchi proved that every strongly negative amphichiral link is rationally slice, i.e. it bounds a disjoint collection of disks in a rational homology 4-ball. In this talk, we prove that every negative amphichiral link is rationally slice, extending the aforementioned work of Kawauchi. Our proof relies on a careful analysis of the JSJ decomposition of the link complement of negative amphichiral links. This is joint work in progress with Jaewon Lee (KAIST, Daejeon) and Oğuz Şavk (CNRS, Nantes).

La simulation numérique peut être vue comme une retranscription informatique de modèles mathématiques utilisant un schéma numérique particulier. À mesure que l'on peaufine les modèles ou qu'on utilise des méthodes numériques spécifiques, on engendre une complexification du code demandant un véritable savoir-faire que le chercheur en mathématiques ou physique n'a pas nécessairement. Cette expertise est celle de l'ingénieur de recherche en calcul scientifique. Ainsi cet exposé a pour but de présenter ce métier.
On s'attardera sur la présentation de variations des méthodes Runge-Kutta, comment compléter leur étude (stabilité, précision, ordre, etc.), et les implémenter ainsi que les partager au près de la communauté.

A famous theorem of Laudenbach and Poénaru says that any diffeomorphism of the boundary of a 4-dimensional 1-handlebody extends to a diffeomorphism of the whole handlebody. I will present a new proof of this result using Heegaard splittings and a generalization to 4-dimensional compression bodies. I will also explain why this is essential in the theory of trisections.

Joint work with Trenton Schirmer.

Van de Ven a démontré en 1959 que les sous-variétés de l'espace projectif dont la suite du fibré normal est scindée sont des sous-espaces linéaires. Dans cet exposé j'expliquerai une généralisation de ce résultat aux variétés homogènes rationnelles : une sous-variété d'une variété homogène rationnelle dont la suite du fibré normal est scindée est une variété homogène rationnelle. Il s'agit d'un travail en collaboration avec Andreas Höring.

The presentation will be divided in two parts: (1) We will briefly introduce the linear inverse problem setting in which we seek to approximate an unknown function from measurements by an element of a linear space. However, linear spaces become ineffective for approximating simple and relevant families of functions, such as piecewise smooth functions, that typically occur in hyperbolic PDEs (shocks) or images (edges). We will then analyze which conditions can give us certified recovery bounds for inversion procedures based on nonlinear approximation spaces. (2) We will apply this framework to the recovery of general bidimensional shapes from cell-average data (high-resolution images from coarser cell averages). Then we will show how to build fast higher order methods to reconstruct interfaces as well as two strategies to deal with non-smooth interfaces presenting corners.