Institut de recherche mathématique avancée

Résumé : Ce mini-cours vise à introduire la méthode de concentration–compacité. Cette méthode repose sur des travaux pionniers de P.-L. Lions en calcul des variations et cherche dans un premier temps, à décrire les défauts de compacité des célèbres injections de Sobolev. L’exposé donne un aperçu de toute l’envergure de cette notion fine, notamment à travers sa version moderne appelée décomposition en profils et ses applications aux EDP non-linéaires, comme la percée de Kenig et Merle dans la description de la dynamique des solutions de l’équation de Schrödinger à énergie critique.

Résumé : The diversity among galaxies showcases a broad spectrum of mass and angular momentum distributions. During formation, gravitational forces drive these celestial bodies to collapse, acquiring angular momentum via torques. Thanks to dissipation, the result post virialization is typically a flattened rotating structure. Understanding how this geometry and kinematics affects the response of galaxies is important to explain their long-term evolution.
Such endeavor has been attempted through costly N-body simulations. However, a worthy alternative is to follow the path of Kalnajs and compute the linear response of such systems. Historically, the complexity of studying generic three-dimensional systems posed significant challenges (six-dimensional phase space fully coupled via self-gravity). However, modern computers can now model more complex shapes or kinematics, opening the prospects of also extending our understanding beyond the spherical or razor thin geometries.

About the speaker :
Kerwann Tep is a postdoctoral researcher at the Observatoire astronomique de Strasbourg working on the exascale project. Prior to joining the observatory, he did his PhD at the Institute d'astrophysique de Paris and a postdoctoral stay at the University of North Carolina at Chapell Hill working on Galactic dynamics and stability of gravitational systems

Résumé : I will first introduce the moduli spaces of Higgs bundles that appear in the Hausel-Thaddeus topological mirror symmetry conjecture, present its different proofs and generalisations. In particular I will explain the p-adic integration approach of Groechenig, Wyss and Ziegler. Finally, I will present my result, which is a generalisation beyond the original coprime case of the key intermediate step of this approach, which we call a non-archimedean topological mirror symmetry.

Résumé : Numerous PDEs can be written of the form y'=Ay+f(y) where A is a linear operator and f represents a nonlinearity. To deal with this kind of PDE, we use to treat first the linear case which comes in the form y'=Ay. The aim of the talk will be to study the issue of solving such PDEs using the notion of semigroups. I will introduce the notion of unbounded operators on Hilbert spaces and give some of their properties. This will lead us to the semigroups and to the Hille-Yosida's Theorem. If we have enough time, we will see several applications.