The Einstein Equations have seen extraordinary developments in the last few years, such as results on the formation of black holes, investigations into the uniqueness and stability of the Kerr solution, and advances in our understanding of cosmic censorship, quasi-local mass, deformations of initial data sets and properties of marginally trapped surfaces. Highly fruitful connections with other geometric PDE's, such as Yang Mills, wave maps, the Yamabe equation, various parabolic flows, as well as connections to other nearby areas, such as harmonic analysis, have led to an array of exciting new discoveries. This activity has attracted many geometric analysts to the wealth of problems and potential applications of the geometrical techniques developed. The special session on Mathematical Relativity organized by the PI's at the fifth conference on Complex Analysis and Dynamical Systems in Akko, Israel, May 22 to 26, aims to gather leading specialists and younger researchers in mathematical relativity and geometric PDE's, to discuss these problems and recent results in order to promote collaborations and further progress.
With the advance of several experimental projects on relativistic gravitational physics, it is highly important todevelop a wealth of theoretical models. In order to assure that mathematical relativity continues to have a strong impact on, and interaction with, the experimental efforts, these models should mature and be able to make quantitative predictions on a number of subjects such as gravitational collapse, gravitational waves and other phenomena associated with strong gravitational fields. This special session will encourage further collaborative research into these topics and also draw young talent to this field. The participation of graduate students and young Ph.D.'s will ensure the continued vitality of the discipline.
General Relativity is Einstein's geometric theory of gravity, which is now almost 100 years old. The Einstein field equations describe how our universe curves due to the presence of matter. This curvature is reponsible for the effects of gravity. There have been, in the past few years, very exciting breakthroughs in the study of the Einstein equations, such as work on the developement of trapped surfaces and the formation of black holes, and progress on the question of the stability of astrophysical black holes. Important advances have been made in our understanding of the cosmic censorship conjecture (which asserts that naked singularties cannot not form in the cosmos), the issue of quasi-local mass, the deformations of initial data sets and properties of marginally trapped surfaces. Much of this activity is relevant to the analysis and interpretation of data coming in from LIGO and other gravitational wave experiments, and to the study of modern cosmological models, which incorporate dark energy and dark matter. All this has created a very fertile area of scientifc research which has attracted scores of researchers from various areas such as geometric analysis and partial differential equations. This grant provided travel support for leading U.S. researchers in the area to participate in a Special Session on Mathematical Relativity at the conference on Complex Analysis and Dynamical Systems V held in May 2011, in Akko, Israel. This session provided participants with the opportunity to present their latest research and to discuss the newest developments in this fundamental area of gravitational physics. The session included presentations of new results on the postivity of mass (which includes gravitational field energy), Penrose-type inequalities, quasi-local mass, initial data manifolds and solution of the constraints, and the stability of black holes.