Conference in Mathematical General Relativity 2016

Synopsis: 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. In recent years, a monumental breakthrough in mathematical GR and geometric analysis occurred with the study of black hole formation, showing that through the focusing of gravitational waves a closed trapped surface and subsequently a black hole will form. This research has launched new activities in areas of hyperbolic PDE. Progress has been made in investigating marginally outer trapped surfaces. They were proven to be useful for studying spacetime incompleteness, black hole uniqueness, and topological restrictions on asymptotically Euclidean initial data sets. At the same time, initial data engineering has made huge progress through gluing techniques. In general, the study of the constraint equations and the conformal method have produced interesting results. New insights were gained on quasilocal definitions of energy. The longstanding conjecture of strong cosmic censorship has been investigated very intensely. An area of very active research in the past years has been the stability of black hole spacetimes. The stability of Minkowski spacetime, the study of the evolution equations or the Penrose inequality have pushed further mathematical and physical research, thereby having created more challenging questions for the future. On the cosmological side, investigations have concerned small scale inhomogeneities, the accelerated expansion of the Universe and primordial gravitational waves. This conference discusses recent developments in these areas.

Organizers

NameUniversity
Lydia BieriUniversity of Michigan
Shiu-Yuen ChengTsinghua University
David GarfinkleOakland University
James IsenbergUniversity of Oregon
Shing-Tung YauHarvard University