Institut de recherche mathématique avancée

Résumé : Inspired by Gromov's pioneering idea in the proof of his non-squeezing theorem, one can seek sharp numerical obstructions to symplectic embeddings by constructing pseudo-holomorphic curves with carefully controlled symplectic area. In the case of symplectic embeddings of ellipsoids, this approach leads to the problem of producing pseudo-holomorphic curves of arbitrarily large degree, subject to specific geometric and asymptotic constraints in the complex projective spaces. In higher dimensions, constructing pseudo-holomorphic curves with the correct symplectic area is often highly challenging due to the area constraint. To circumvent this difficulty, we propose a different perspective: rather than seeking curves of a prescribed symplectic area, one observes that the existence of a symplectic embedding typically implies the existence of some rigid pseudo-holomorphic curves. The count of such curves can be viewed as an obstruction to the existence of certain symplectic embeddings. Using this idea, I will explain a lower bound for the embedding capacity of higher-dimensional symplectic ellipsoids. This is based on work in progress.

Résumé : Abstract : The lifted TASEP is a variant of the totally asymmetric exclusion process where at each time-step, instead of trying to move forward a uniformly chosen particle, we try to move forward a marked particle which then may pass the marker to another particle. It was introduced by physicists as a toy model for non-reversible event-chain Monte-Carlo algorithms, which are expected to reach equilibrium faster than reversible dynamics. We will study the behaviour of this system on the integer line by evidencing a connexion with true self-avoiding walks, yielding timescales of the dynamics. This is based on joint work with Clément Erignoux, Werner Krauth, François Simenhaus and Cristina Toninelli.

Résumé : Dans cet exposé, je présenterai tout d'abord les résultats de prolongements uniques connus pour les équations de type ondes. J'expliquerai les difficultés pour obtenir des résultats globaux sous des hypothèses géométriques naturelles. Par la suite, je présenterai un résultat, en collaboration avec Cristobal Loyola, où nous prouvons le prolongement unique pour des équations d'ondes semilinéaires sous l'hypothèse de contrôle géométrique. Une étape cruciale est la propagation globale de l'analyticité en temps à partir d'ouverts vérifiant la condition de contrôle géométrique. La preuve utilise des méthodes de contrôle associées à des idées de Hale-Raugel concernant la régularité de l'attracteur.