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## Seminar on** **

**Mathematical General Relativity**

*Organizers:*

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

*Ghani Zeghib (Ecole Normale Supérieure, Lyon)*

**ANR Project**

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

**Thursday April 5, 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)**

**10:00 am Lars Andersson (Potsdam) **Hidden symmetries and conserved charges

**Abstract. **A proof of decay estimates for test fields with non-zero spin, e.g. Maxwell and linearized gravity, on the Kerr background is an important step towards understanding the black hole stability problem. Fields with non-zero spin on Kerr admit non-radiating modes which must be eliminated in order to prove decay. In this talk I will discuss the relation between conserved charges and hidden symmetries for linearized gravity on Minkowski space and vacuum spaces of Petrov type D, and outline the application of these ideas in proving estimates for the higher spin fields on the Kerr background.

**11:30 am François Béguin (Orsay) **On the BKL conjecture for vacuum spatially homogeneous models

**Abstract. **About forty years ago, Belinski, Kalatnikov, and Lifschitz proposed a rather complete description (based on heuristic arguments) of the asymptotic behavior of spatially homogeneous spacetimes close to their initial singularity. In particular, a “generic” vacuum spatially homogeneous spacetime is expected to exhibit a “chaotic” oscillatory behavior closed to its initial singularity. It was only in 2010 that the first rigorous mathematical results going in this direction were established. I will review here what is known (from a rigorous mathematical viewpoint) on the behavior of vacuum spatially homogeneous spacetimes close to their initial singularity. I will also try to explain what one can (or cannot) hope to prove about this asymptotic behavior and will sketch the proofs.

**2:00 pm Simone Calogero (Granada) **Dynamics of spatially homogeneous cosmological models

**Abstract. **Understanding the asymptotic dynamics of the Universe is one of the main goals of theoretical cosmology. In the context of spatially homogeneous (SH) cosmological models, where the Einstein equations of general relativity reduce to a system of ordinary differential equations, our knowledge on this problem has increased substantially over the years, thanks to the application of methods from the theory of finite dimensional dynamical systems. After reviewing the results known for the dynamics of perfect fluid cosmological models, this talk will focus on the more complicated case of a space-time filled with anisotropic matter. In the latter case the qualitative global behavior of SH solutions depends strongly on the asymptotic values of the principal pressures in the limit toward the singularity. Such behavior has been completely characterized for locally rotationally symmetric (LRS) solutions in the class A of Bianchi models. It has been found that there exist matter models, compatible with the standard energy conditions, for which Bianchi type IX LRS solutions are singularity-free for an initial data set with positive measure, and others for which the approach toward the singularity of generic solutions is oscillatory. This talk is based on a series of works in collaboration with Mark Heinzle.

**3:30 pm**** Ghani Zeghib (Lyon) **Projective transformation groups

**Abstract. **To pseudo-Riemannian (in particular Riemannian or Lorentzian) metric one naturally associates its Levi-Civita connection. It is also natural to ask whether, conversely, the connection determines the metric, that is, (essentially) whether two metrics having the same geodesics coincide? More strongly, two metrics are said to be projectively equivalent if their geodesics coincides as geometric non-parametrized curves. One then asks when projectively equivalent metrics coincide? The answer to this classical problem is `no’, but only for very special metrics. This confirms a real interest for this problem since it allows the emergence of special structures. We will deal here with a parallel (i.e. weaker) rigidity question: classify metrics admitting an essential projective transformation, that is, diffeomorphism preserving, non-parametrized geodesics without being an isometry?