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## Seminar on

**Mathematical General Relativity**

*Organizers:*

* *Philippe G. LeFloch *(Paris)*

*Jérémie Szeftel (Paris) *

*Ghani Zeghib (Lyon)*

**ANR Project**

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

**May 28, 2014**

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

**Lecture room ** 1525-321

**Lecture room**

**14h Erwann Delay (Avignon)** A study of some curvature operators near the Euclidian metric

Abstract. We will show that some curvature operators of Ricci (or Einstein) type are locally invertible, in some weighted Sobolev spaces on Rn, near the euclidian metric. In the smooth case, we then deduce that the image of some Riemann-Christoffel type operators are smooth submanifolds in the neighborhood of the Euclidian metric.

**15h30 Mahir Hadzic (London)** Stability problem in the dust-Einstein system with a positive cosmological constant

Abstract. The dust-Einstein system models the evolution of a spacetime containing a pressureless fluid, i.e. dust. We will show nonlinear stability of the well-known Friedman-Lemaitre-Robertson-Walker (FLRW) family of solutions to the dust-Einstein system with positive cosmological constant. FLRW solutions represent initially a quiet fluid evolving in a spacetime undergoing accelerated expansion. We work in a harmonic-type coordinate system, inspired by prior works of Rodnianski and Speck on Euler-Einstein system, and Ringstrom’s work on the Einstein-scalar-field system. The main new mathematical difficulty is the additional loss of one degree of differentiability of the dust matter. To deal with this degeneracy, we commute the equations with a well-chosen differential operator and derive a family of elliptic estimates to complement the high-order energy estimates. This is joint work with Jared Speck.