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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 November 3, 2011**

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

** Lecture room 15-25 326 (third floor)**

**Lecture room 15-25 326 (third floor)**

**14h Charles Boubel (Strasbourg) Germs of Lorentzian metrics and holonomy**

**Abstract. **The holonomy group of a pseudo-Riemannian metric g -so e.g. a Riemannian or a Lorentzian metric- is a subgroup of O(g) which indicates, in a certain sense, how much its Levi-Civita connection fails to be flat. A central task related to those groupes is to determine the list of the subgroups of O(g) arising as holonomy, and for each item, to parametrize the set of corresponding metrics and build global examples (i.e. complete or compact). In the Riemannian case, this work is now done. We will see that, regarding holonomy matters, Lorentzian metrics behave totally differenly from Riemannian ones. I will review works of L. Bérard-Bergery, A. Ikemakhen, T. Leistner, A. Galaev, and myself, that together deal with the local aspect of the question.

**15h30 David Parlongue (Nice) Breakdown criteria and extendibility in general relativity**

**Abstract. **We will begin this talk by reviewing a geometric breakdown criterion for Einstein’s vacuum equations introduced by S. Klainerman and I. Rodnianski and various improvements (non-vacuum case, integral conditions, various gauge choices). We will then examine a spacetime localization of these criteria. We will focus on consequences in terms of formation of singularities, extendibility of spacetimes, and local regularity of foliations.