From Geometry to Numerics: interdisciplinary aspects in mathematical and numerical relativity

This article reviews some aspects in the current relationship between mathematical and numerical General Relativity. Focus is placed on the description of isolated systems, with a particular emphasis on recent developments in the study of black holes. Ideas concerning asymptotic flatness, the initial value problem, the constraint equations, evolution formalisms, geometric inequalities and quasi-local black hole horizons are discussed on the light of the interaction between numerical and mathematical relativists.Comment: Topical review commissioned by Classical and Quantum Gravity. Discussion inspired by the workshop "From Geometry to Numerics" (Paris, 20-24 November, 2006), part of the "General Relativity Trimester" at the Institut Henri Poincare (Fall 2006). Comments and references added. Typos corrected. Submitted to Classical and Quantum Gravit

Summary of sessions B1/B2 and B2: relativistic astrophysics and numerical relativity

The numerical relativity session at GR18 was dominated by physics results on binary black hole mergers. Several groups can now simulate these from a time when the post-Newtonian equations of motion are still applicable, through several orbits and the merger to the ringdown phase, obtaining plausible gravitational waves at infinity, and showing some evidence of convergence with resolution. The results of different groups roughly agree. This new-won confidence has been used by these groups to begin mapping out the (finite dimensional) initial data space of the problem, with a particular focus on the effect of black hole spins, and the acceleration by gravitational wave recoil to hundreds of km s?1 of the final merged black hole. Other work was presented on a variety of topics, such as evolutions with matter, extreme mass ratio inspirals and technical issues such as gauge choices