Abstract of Proposed Research Igor Rodnianski
The focus of this research project is the study of the local and global behavior of solutions of geometric hyperbolic PDEs arising in General Relativity and other equations arising in Quantum Field Theory and gauge theories. One project is to study the solutions of the full nonlinear system of Einstein equations in a weak gravitational wave regime. We will continue our work on the L2 curvature conjecture, asserting that a solution, metric g, of the Einstein vacuum equations Ric(g) = 0 can be locally extended whenever its curvature tensor is bounded in L2 . We will also study problem of stability of Minkowski space for models where gravity is coupled to matter fields. Another direction of investigation is to continue rigorous mathematical study of linear and nonlinear waves in the regime of strong gravitational fields. The project will also investigate problems of stability of topological solitons in classical field theories connected with models of Quantum Field Theory and Gauge Theory. We will concentrate on development of robust analysis methods for these models without requiring any complete integrability assumptions. We will investigate connections between infinite dimensional dynamics, generated by the above models, and geodesic motion induced on finite dimensional moduli spaces of static solutions associated with these 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 fundamental importance for cosmology. While the physical understanding of the subject has made rapid advancement and generated a number of outstanding conjectures; so far there have been relatively few rigorous mathematical results. To a large extent, this is due to the highly nonlinear nature of the Einstein equations and a lack of mathematical tools for analyzing them. This project will pursue the development of such mathematical tools and results and involves many mathematical issues at the interface of Analysis, Geometry and Partial Differential Equations.
