Résumé : In this talk, we consider the following problem : Given the source-to-solution map for a relativistic Boltzmann equation on a neighbourhood V of an observer in a Lorentzian spacetime (M,g) and knowledge of the metric g restricted to V, can we determine (up to diffeomorphism) the spacetime metric g on the domain of causal influence for the set V ?
We will show that the answer is yes. The problem we consider is a so-called inverse problem. We will briefly motivate the mathematical area of inverse problems. We will also introduce the relativistic Boltzmann equation and comment on the existence of solutions to this nonlinear PDE given some initial data. Then, we present a broad sketch of the key ideas in the proof of our result. One such key point is that the nonlinear term in the relativistic Boltzmann equation which describes the behaviour of particle collisions captures information about a source-to-solution map for a related linear problem. We use this relationship together with an analysis of the behaviour of particle collisions by classical microlocal techniques to determine the set of locations in V where we first receive light particle signals from collisions in the unknown domain. From this data we are able to parametrize the unknown region and determine the metric.
The new results presented in this talk are joint work with Antti Kujanpää, Matti Lassas, and Tony Liimatainen, (University of Helsinki).
A preprint can be found here : https://arxiv.org/abs/2011.09312

Lieu : visioconférence

Résumé : Nous considérerons l’opérateur H_0 de Schrödinger 1d agissant sur l’espace des suites l^2(Z), et des perturbations
compactes V (non) autoadjointes. Nous relierons ensuite les propriétés de régularité de V à différentes propriétés spectrales
de l’opérateur perturbé H_0 + V. En particulier, la structure du spectre discret et des valeurs propres plongées sera étudiée.
Nos résultats seront basés sur une combinaison adéquate des méthodes de dilatations ou distorsions analytiques et de la
théorie des résonances. Si le temps le permet, nous aborderons les possibles extensions de nos résultats au cas des graphes.
Travail en collaboration avec Olivier Bourget et Amal Taarabt de l’Université Catholique de Santiago du Chili.

Lieu : visioconférence

Résumé : Ce travail avec M. Briane reprend un des problèmes les plus élémentaires de la théorie de l’homogénéisation. Nous montrons que l’abandon
d’une seule des conditions nécessaires à l’approche standard, la très forte ellipticité, introduit de nombreuses pathologies.
En particulier, nous engendrons ainsi par homogénéisation un éther anisotrope dans la tradition des physiciens du XIXème siècle.

Notes de dernières minutes : Attention horaire modifié 14h30-15h30

Résumé : 95 years after the publication of the Schrödinger equation, the quantum many-body problem still remains a mathematical challenge. The derivation of effective evolution equations provides a tool for an approximate solution, deriving physical predictions in certain scaling limits. I will discuss the coupled mean-field and semiclassical scaling limit for high-density fermionic systems, and sketch the derivation of the time-dependent Hartree-Fock equation in this limit. I will also present some recent results that can be seen as a next-order correction to Hartree-Fock theory, based on a new bosonization method.

Lieu : visioconférence

Résumé : Minimizing harmonic maps (i.e., minimizers of the Dirichlet integral) with prescribed boundary conditions are known to be smooth outside a singular set of codimension 3. I will consider mappings from an n-dimensional domain with values in the two dimensional sphere. I will present an extension of Almgren and Lieb’s linear law on the bound of the singular set. Next, I will investigate how the singular set is affected by small perturbations of the prescribed boundary map and present a stability theorem, which is an extension of Hart and Lin’s result. I will also discuss possible extensions to different target manifolds and the optimality of our assumptions. This is joint work with Michał Miśkiewicz and Armin Schikorra.

Lieu : visioconférence

Résumé : In this talk, I will present results regarding the observability properties of the Grushin and Kolmogorov equations. For these degenerate parabolic equations, in certain geometric configurations, a minimal time is required for an observability inequality to hold.
In a recent work with Karine Beauchard and Sylvain Ervedoza, we have obtained the minimal time for the 2d Grushin equation observed from one side of the domain. In a joint work with Julien Royer, we have obtain bounds on the minimal time for the 2d Kolmogorov equations. I’ll explain the strategy we use in both cases, which relies on Carleman estimates, localization of eigenvalues, and Agmon type estimates for the corresponding eigenfunctions.