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## Seminar on

**Mathematical General Relativity**

*Organizers:** *

* *S. Klainerman *(Princeton)*

#### P.G. LeFloch *(Paris)*

*A. Zeghib (Lyon)*

**Fondations des Sciences Mathématiques de Paris**

**ANR Project**

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

**Thursday June 14, 2012**

### Laboratoire J-L Lions

### Université Pierre et Marie Curie, Paris

**Lecture room 15-25 104**

**Lecture room 15-25 104**

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**Jean-Philippe Nicolas (Brest) Perspectives in conformal scattering**

**Abstract. **The origins of conformal scattering are to be found in a paper by Friedlander in 1980 “Radiation fields and hyperbolic scattering theory”, in which he realized that the Lax-Phillips theory was in fact providing an interpretation of scattering theory as the well-posedness of the characteristic Cauchy problem for the conformally rescaled wave-equation on null infinity. He clearly saw that the method provided an interesting geometrical short-cut to define a scattering operator with the advantage that all the analytical structure can be recovered a posteriori. The true power of the conformal approach to scattering theory lies in its complete indifference to time dependence. This talk will review the essential features of Lax-Phillips theory and its intimate link with conformal infinity via the Radon transform and the Whittaker formula, then describe the pinciples of conformal scattering with the crucial importance of the precise resolution of the Goursat problem; we will present some results (actual scattering constructions and studies of the Goursat problem) and explain the necessary steps of the extension of the method to black hole spacetimes, which is currently under development.

**Jérémie Szeftel (Paris) The bounded L2 curvature conjecture in general relativity**

**Abstract. ** In order to control locally a spacetime which satisfies the Einstein equations, what are the minimal assumptions one should make on its curvature tensor? The bounded L2 curvature conjecture roughly asserts that one should only need an L2 bound on the curvature tensor on a given space-like hypersurface. This conjecture has its roots in the remarkable developments of the last twenty years centered around the issue of optimal well-posedness for nonlinear wave equations. In this context, a corresponding conjecture for nonlinear wave equations cannot hold, unless the nonlinearity has a very special nonlinear structure. I will present the proof of this conjecture, which sheds light on the specific null structure of the Einstein equations. This is joint work with S. Klainerman and I. Rodnianski.