Abstract

Einstein's general relativistic field equations govern the Universe, in particular phenomena in cosmology, astrophysics and notably gravitational waves. The study of these equations has led to thriving new mathematical research in the areas of geometric analysis, nonlinear partial differential equations (PDE) of hyperbolic and elliptic character, differential geometry as well as in scattering theory and the analysis of asymptotic behavior of solutions. Purely analytic and numerical methods complement each other on this road. Modern mathematical breakthroughs allow to attack and solve physical problems that have been a challenge for the last century. Among these are the study and detection of gravitational waves. These are produced in mergers of black holes or neutron stars or in core-collapse supernovae. Our era faces the verge of detection of these waves, which we can think of as fluctuations in the curvature of the spacetime. Mathematically gravitational waves are investigated by means of geometric analysis as well as numerics. Through geometric analysis Christodoulou's findings of a nonlinear memory effect of gravitational waves, displacing test masses permanently has sparked new research leading to insights into this very effect for other fields coupled to Einstein equations. Moreover, Christodoulou's results on black hole formation have likewise launched abundant activities in hyperbolic PDE. Further the stability of Minkowski spacetime, the stability of black hole spacetimes, the study of the constraint equations, the evolution equations or the Penrose inequality have pushed further mathematical and physical research, thereby having created more challenging questions for the future. This conference discusses recent developments in these areas.