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## Seminar on** **

**Mathematical General Relativity**

*Organizers:*

* *Philippe G. LeFloch *(Univ. Pierre et Marie Curie) *

*Ghani Zeghib (Ecole Normale Supérieure, Lyon)*

#### With the financial support of the** ****ANR Project**

#### “Mathematical General Relativity. Analysis and geometry of spacetimes with low regularity”

**Thursday February 9, 2012**

### Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris

**Lecture room 15-25 101 (first level)**

**Lecture room 15-25 101 (first level)**

**14h Alan Rendall (AEI, Potsdam) ***Singularity formation in solutions of the Einstein-Vlasov system*

**Abstract. **Important questions in mathematical relativity are when singularities form in solutions of the Einstein equations coupled to matter and, in cases where they do form, what their qualitative nature is. A type of matter model which apparently rarely loses smoothness in the absence of black hole formation is collisionless matter modelled by the Vlasov equation. This contrasts with dust, a type of matter popular among relativists. In this talk I describe recent work with Juan Velazquez where we try to obtain new insights about the dynamics of the Einstein-Vlasov system by interpolating between smooth Vlasov and dust in a suitable way. We have shown that for certain mildly singular initial data a curvature singularity can form. It is constructed by means of a shooting argument for a system of ordinary differential equations. From the point of view of physics it would be desirable to improve this solution in various ways and I will report briefly on work in progress on doing this.

**15h30 François Filastre (Cergy-Pontoise) ***Brunn–Minkowski theory in Minkowski spacetime *

**Abstract. **The Brunn–Minkowski theory deals with the relations between the addition and the volume of convex bodies of the Euclidean space. Convex bodies are described by function on the sphere. The main result of the theory is that the volume is log-concave. We establish an analog result for a class of convex sets in the Minkowski spacetime. The compactness is replaced by a global invariance property under the action of particular groups of linear isometries. In particular, these convex sets can be described by functions on compact hyperbolic manifolds and, in this case, the volume is convex.