Small variations of doubled real algebraic varieties is a classical construction technique which has led to many interesting results, for instance regarding the topology of real algebraic surfaces in the three-dimensional real projective space. We highlight a slight variation of this technique, which roughly consists in applying it two times in a row, and discuss some of its applications. In particular, we explain how this method can be used to obtain different topological types of maximal quartic hypersurfaces in the four-dimensional real projective space.

Quantum mechanics is described through solutions to the Schrödinger equation, which is the quantum analog of the Hamiltonian equations in classical dynamics. Stationary states are given by eigenfunctions of an elliptic operator, which in this talk will be the Laplace-Beltrami operator in a compact Riemannian manifold. Its eigenfunctions that correspond to very large eigenvalues are somewhat described by the Hamiltonian classical dynamics, which in our case is the geodesic flow. The nature of this correspondence is elusive, and depends on global geometric properties of the manifold. We will try to introduce the audience to this area of research through several simple, yet important examples, focusing mainly on manifolds with completely integrable geometry and their perturbations.

Le sujet de mon exposé sera les valeurs propres du laplacien magnétique dans un domaine borné du plan, pour un champ magnétique uniforme et des conditions au bord de Neumann. Nous nous intéresserons en particulier à l'inégalité de Faber-Krahn inverse conjecturée par Søren Fournais et Bernard Helffer , selon laquelle parmi tous les domaines simplement connexes d'aire donnée, le disque maximise la première valeur propre. Nous verrons que l'inégalité est vérifiée pour un champ magnétique suffisamment faible (tel que la fonction propre correspondante sur le disque soit radiale). Ces résultats ont été obtenus en collaboration avec Bruno Colbois, Luigi Provenzano et Alessandro Savo.

Do there exist 13 lines in the complex projective plane that only meet at triple points? I will discuss some topological ideas around this question and higher-degree variants.

1) Introduction

2) Exposé de André-Jean Glière :
"Les systèmes de nombres (Zahlensysteme) de Hermann Hankel (1867)"

3) Exposé de François Plantade :
"L'influence de Hankel sur l'édition de 1874 de la théorie élémentaire des quantités complexes de Houël"

4) Discussion générale

Fin à 17h.