Séminaires

La conjecture standard de type Hodge prédit la positivité de certaines formes d’intersections sur les cycles algébriques. Cette conjecture est connue en caractéristique 0 via les relations de Hodge-Riemann bilinéaires. Dans cet exposé, on s’intéressera à des nouveaux cas de la conjecture en caractéristique positive, en particulier au cas des puissances de variétés abéliennes de dimension 3.

It is known since Wierzba–Wisniewski’s work in 2003 that a birational map between holomorphic symplectic fourfolds is a composition of Mukai flops. Hu and Yau conjectured that in any dimension, a birational map is a composition of Mukai elementary transformations in codimension two, that is, maps which locally look like a product of a 4-dimensional Mukai flop and the identity on a polydisc, « up to codimension three or higher ». Call  a birational map f from X to X’ a « Hu-Yau transformation » , if there are proper closed subsets Z, Z’ of codimension three or higher such that f induces a Mukai transformation in codimension two between X\Z and X’\Z’. We show that any birational map between irreducible holomorphic symplectic manifolds is a product of Hu-Yau transformations, and give an example showing that a stronger version of the conjecture cannot be true: it is in general not possible to decompose a given map into a product of Mukai elementary transformation after discarding some codimension three closed subsets from the source and the target.

A couple of years ago, Ammann, Kröncke and Müller proposed a construction producing initial data sets for spacetimes with lightlike parallel spinor as studied by Baum, Leistner and Lischewski in the context of Lorentzian special holonomy. The input data for the AKM-construction essentially consists of a curve in the moduli space of Ricci-flat metrics on a closed manifold Q together with a parallel spinor for a metric representing its starting point. It remained unclear, however, to which extent all initial data for spacetimes with lightlike parallel spinor can be obtained by this construction for a fixed codimension 2 topology Q. In this talk, based on joint work with Bernd Ammann and Klaus Kröncke, we seek to improve the construction. It turns out that there is mainly only one additional freedom: to prescribe a single geometrically meaningful scalar function.

In this talk, I will present the meeting of different dynamical systems: tiling billiards, the wind-tree model and the Eaton lenses. The three of them are motivated by physics.
In the beginning of the 2000’s, physicists have conceived metamaterials with negative index of refraction. Tilling billiards’ trajectories consist of light rays moving in an arrangement of metamaterials with opposite index of refraction. The wind-tree model was introduced by Paul and Tatyana Ehrenfest to study a gaz: a particle is moving in a plane where obstacles are periodically placed, on which the particle bounces. The Eaton lenses are a periodic array of lenses in the plane, in which we consider a light ray that is reflected each time it crosses a lens.
After having introduced these dynamical systems, I will consider a mix of them: an arrangement of rectangles in the plane, like in the wind-tree model, but made of metamaterials, like for tiling billiards. I study the trajectories of light in this plane. They are refracted each time they cross a rectangle.
Using dynamics on half-translation surfaces and their moduli space, I show that these trajectories are trapped in a strip, for almost every parameter. This behavior is similar to the one of the Eaton lenses.

titres et résumés à venir

titres et résumés à venir

Geometric methods in computational complexity