The focus of this project is the study of the local and global behavior of solutions of geometric hyperbolic PDE arising in general relativity and models used in gauge theories. In general relativity the full nonlinear system of the Einstein equations will be considered in both weak and strong gravitational field regimes. The principal investigator will continue his work on the bounded curvature conjecture, asserting that a metric solution of the Einstein vacuum equations can be locally extended as long as its curvature tensor is bounded in the square integrable norm. A strong gravitational field regime will be studied in the context of extensions and developments of the short pulse method recently introduced by Christodoulou in the problem of formation of trapped surfaces. The project will continue a rigorous mathematical study of linear and nonlinear waves on black hole backgrounds, where the stability of the Kerr solution remains an outstanding open problem. It will also pursue further study of the stable regime of singularity formation for the Wave Map and Yang-Mills equations.
The Einstein equations of general relativity provide the main classical description of evolution of the physical space-time continuum. The study of mathematical and physical phenomena arising in general relativity is of the fundamental importance. While the physical understanding of the subject has made rapid advancement and generated a number of outstanding conjectures, the rigorous mathematical picture has yet to emerge. The latter fact is, to a large extent, due to the highly nonlinear nature of the Einstein equations and a lack of the mathematical tools to deal with it. The development of a satisfactory mathematical approach lies on the interface between analysis, geometry and partial differential equations.
