Institut de recherche mathématique avancée

Résumé : When an object is photographed, the resulting image is pixelated. The position of a point in such an image is described by integer coordinates, unlike that of a point on the original object, which is described by real coordinates. This transition from the usual Euclidean geometry describing the original object to the discrete geometry describing the obtained image, called digitization, causes significant information loss. If the resolution of the discrete image is too low compared to the level of detail of the original object, topological information and geometric quantities can be lost. It then becomes necessary to impose certain assumptions on this real object to allow the reconstruction of this information.

By modeling the digitization process, it is possible to ensure the reconstruction of the topology and geometric quantities of objects that meet certain assumptions. However, currently in digital geometry, the assumptions on real objects that guarantee the reconstruction of all this information are quite restrictive and do not allow the simultaneous inclusion of shapes whose boundary is a smooth curve and those whose boundary is a polygon.

To simultaneously address these two families of shapes, we propose a new assumption based on the concept of total curvature introduced by Milnor in 1950. This consists of locally limiting this total curvature on the boundary of the real object. This assumption, which includes shapes with smooth or polygonal boundaries, guarantees the reconstruction of topology and allows for bounding the errors of discrete estimators of geometric quantities.

About the speaker: Étienne Le Quentrec is assistant professor (maître de conférences) at ICube, member of IMAGeS team since September 2022. He defended his PhD thesis in 2021 on digital Geometry at ICube. He was also student at UFR in mathematics where he obtained his "agrégation" in 2016. His main research interests are digital topology and discrete estimation.

Résumé : Résumé : Dans cet exposé, on s'intéresse à des estimations de décroissance locale pour l’équation des ondes dans un cadre asymptotiquement Euclidien. En dimensions paires, on va au-delà de la décroissance optimale en fournissant le profil asymptotique à long terme, donné par une solution de l’équation des ondes libres. En dimensions impaires, on améliore les meilleures estimations connues. En particulier, on obtient un taux de décroissance qui dépasse la décroissance optimale en dimensions paires. L’analyse repose principalement sur une comparaison de la résolvante correspondante avec la résolvante du problème libre pour les basses fréquences. De plus, tous les résultats s’appliquent à l’équation des ondes amorties avec un indice d’absorption à courte portée. Il s'agit d'un travail en collaboration avec J. Royer.

Résumé : Many important moduli spaces in modern p-adic Hodge theory mix the theory of rigid analytic geometry, which is a p-adic analog of the theory of complex analytic spaces, with the topological theory of profinite sets. In the usual approaches based on perfectoid rings, the profinite directions often eliminate the possibility of obtaining an interesting differential theory for these objects. In this talk, we discuss some progress towards constructing a better differential theory, and how it connects to other recent advances in p-adic geometry and p-adic Hodge theory.

Résumé : Morse homology is a homology theory that allows to say something about the critical points of a smooth function on a manifold. It can be used for example to prove lower bounds on the number of critical points, called the Morse inequalites. Inspired by these, Arnold conjectured a generalization concerning the number of periodic orbits of certain vector fields. I will explain how one gets the Morse inequalites using Morse homology, and how one can prove Arnold conjecture thanks to an infinite dimensional version of Morse theory, called Floer theory.

Résumé : It is a guiding paradigm is higher category theory (known as Grothendieck’s homotopy hypothesis) that the homotopy theory of spaces should be the same as the homotopy theory of infinity-groupoids, i.e. such infinity-categories, in which every 1-morphism has an inverse. This conceptual equivalence can be realized in terms of the fundamental infinity groupoid of paths, a higher categorical analogue of the fundamental group. Here, I want to talk about the stratified analogue of this correspondence. First, I will be talking about the homotopy theory of stratified spaces, explain why it is an excellent homotopy theoretic setting to work in and why it is well connected with geometric examples of stratified spaces. Then, I will discuss a presentation of the stratified analogue of the homotopy hypothesis in terms of a (Quillen) equivalence using exit-path categories.