Einstein’s field equation of general relativity is one of the most important geometric partial differential equations. Over the past decade, the mathematical research on Einstein equation has made spectacular progress on many fronts (Cauchy problem, cosmic censorship, asymptotic behavior). These developments have brought into focus the deep connections between the Einstein equation and other important geometric PDE’s, including the wave map equation, Yang-Mills equation, Yamabe problem, as well as Hamilton’s Ricci flow. The field is of growing interest for mathematicians and of intense current activity, as is illustrated by major recent breakthrough, concerning the uniqueness and stability of the Kerr black hole model, the formation of trapped surfaces, and the bounded L2 curvature problem. Specifically, the themes of mathematical interest that will be developed in the present Program and are currently most active include:

• The initial value problem for Einstein equation and the causal geometry of spacetimes with low regularity, formation of trapped surfaces

• Techniques of Lorentzian geometry: injectivity radius estimates, geometry of null cones; construction of parametrix

• Geometry of black hole spacetimes: uniqueness theorems, censorship principles

• Coupling of Einstein equation for self-gravitating matter models, weakly regular spacetimes, nonlinear stability of Minkowski space with matter