In this talk, I will present several recent results, in collaboration with Nicolas Burq, on the control theory of Schrödinger propagators on tori. Our goal is to address the following conjecture: on a torus of arbitrary dimension, Schrödinger propagators with bounded potentials are observable, and therefore controllable, from arbitrary space-time domains of positive Lebesgue measure.
Using a scheme that combines (1) approximation of rough functions by continuous functions, (2) the cluster structure of lattice points near paraboloids, and (3) mathematical induction on the dimension, we reduce the conjecture to certain integrability bounds for linear Schrödinger waves. These bounds are weaker than Bourgain’s conjectured periodic Strichartz estimates but remain nontrivial. In particular, our criteria imply the observability conjecture for the one-dimensional torus.
Applications of our results include Cantor--Lebesgue type theorems and uniform nonvanishing estimates for quantum limits.
Que devient un système dynamique avec le temps ? Où se dirigent ses points, où retournent-ils, et quelle est la complexité de leurs trajectoires ? À travers des exemples, nous explorerons des notions fondamentales comme la périodicité, l’intégrabilité et l’entropie. Nous verrons comment la géométrie du système peut contraindre le mouvement. Si le temps le permet, nous ferons un petit tour dans le monde de la dynamique symplectique.
Aucun prérequis n’est nécessaire, si ce n’est une licence en math. Des notions en géométrie différentielle peuvent être utiles, mais pas indispensables.
Dans cet exposé, je proposerai une introduction à l’ergodicité quantique. On considère une variété riemannienne compacte lisse, ainsi que l’opérateur de Laplace–Beltrami associé, qui possède un spectre discret et une base hilbertienne de fonctions propres. Du point de vue de la mécanique quantique, les densités de probabilité associées à ces fonctions propres décrivent la probabilité de présence d’une particule en un point de la variété. Une question centrale est de comprendre comment ces mesures se répartissent lorsque l’énergie tend vers l’infini, et en quoi ce comportement reflète la dynamique du flot géodésique.
Afin d’illustrer ces notions, je présenterai un exemple dans un cadre euclidien muni d’un champ magnétique, où apparaissent concrètement les différentes notions évoquées.
In this talk I will introduce the quenched Edwards--Wilkinson equation, which models the growth of an interface among an obstacle
field. Due to the elasticity effect of the laplacian, obstacle may slow down or stop the growth of the interface. When the driving force
is low and there is enough disorder of the obstacle field, the interface may get pinned. But for a large enough driving force, there
is a positive speed of propagation of the interface. I will give the intuition for this phenomenon, mention what is done in the literature
and then will turn to this equation with a Gaussian disorder, which is white in the spatial component. Due to the irregularity we need tools from Rough Paths, like the (stochastic) sewing lemma and regularisation by noise in order to show well-posedness. I will
explain the idea behind these tools and how we apply them. This is joint work with Toyota Matsuda and Jaeyun Ji.
Consider a particle randomly moving in a bounded (planar) domain starting at any given point within. Assume it bounces against the boundary and consider
$\Sigma$, a small part of that boundary. What is the expected time we need to wait before the particle hits $\Sigma$ ? This question is known as the narrow escape problem. We can also consider the related question : what is the probability that the particle hits $\Sigma$ before another given subset of the boundary $\Gamma$ ? In this talk, I will address these questions and give quantitative answers in the asymptotic regime where the lengths of the windows tend to 0. To tackle the problem, I will prove a Feynman-Kac formula, linking the stochastic process studied to a deterministic PDE which has the form of a Poisson equation with mixed boundary conditions. Then, constructing appropriate quasimodes to this PDE, we are able to derive sharp asymptotics for the expected time and probabilities.
