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## Seminar on** **

**Mathematical General Relativity**

* * *Organizers: *Sergiu Klainerman *(Princeton)*

#### Philippe G. LeFloch *(Univ. Pierre et Marie Curie) * Gabriele Veneziano *(Collège de France)*

#### With the financial support of the **Fondation des Sciences Mathématiques de Paris**

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**Monday March 28, 2011**

### Laboratoire Jacques-Louis Lions

#### Université Pierre et Marie Curie

** Lecture rooms: 15/25 101 (morning) and 15-/25 102 (afternoon)**

**Lecture rooms: 15/25 101 (morning) and 15-/25 102 (afternoon)**

**11h Rafe MAZZEO (Stanford) The constraint equation and cylindrical ends**

Abstract. This talk will give a closer look at some of the results discussed in last month’s seminar by Piotr Chrusciel concerning our joint work on the existence and classification of solutions of the constraint equations on manifolds with cylindrical ends. I will also describe some new work, with Akutagawa and Carron, concerning the Yamabe equation on stratified spaces, and some regularity theorems for all of these problems.

**14h Jared SPECK (Princeton and MIT) The global stability of the Minkowski spacetime solutions to the Einstein-nonlinear electromagnetic system in wave coordinates**

Abstract. The Einstein-nonlinear electromagnetic system is a coupling of the Einstein field equations of general relativity to nonlinear electromagnetic field equations. In this talk, I will discuss the family of covariant electromagnetic models that satisfy the following criteria: i) they are derivable from a sufficiently regular Lagrangian, ii) they reduce to the familiar Maxwell model in the weak-field limit, and iii) their corresponding energy-momentum tensors satisfy the dominant energy condition. I will mention several specific electromagnetic models that are of interest to researchers working in the foundations of physics and in string theory. I will then discuss my main result, which is a proof of the global nonlinear stability of the 1 + 3 dimensional Minkowski spacetime solution to the coupled system. This stability result is a consequence of a small-data global existence result for a reduced system of equations that is equivalent to the original system in a wave coordinate gauge. The analysis of the spacetime metric components is based on a framework recently developed by Lindblad and Rodnianski, which allows one to derive suitable estimates for tensorial systems of quasilinear wave equations with nonlinearities that satisfy the weak null condition. The analysis of the electromagnetic fields, which satisfy quasilinear first-order equations, is based on an extension of a geometric energy-method framework developed by Christodoulou, together with a collection of pointwise decay estimates for the Faraday tensor that I develop. Throughout the analysis, I work directly with the electromagnetic fields, thus avoiding the introduction of electromagnetic potentials.