Global Existence of the Critical Einstein-Equivariant Wave Map System

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A fundamental and interesting aspect of mathematics is that it provides a consistent language for expressing our understanding of the phenomena occurring in nature. An example is the wave equation which models many natural phenomena like vibrating strings, acoustic waves, water waves, electromagnetic waves and linearized Einstein's equations of general relativity. A geometric and nonlinear generalization of the wave equation is the wave maps equation. The motivation to study wave maps is that they occur naturally in Einstein's equations of general relativity and are proven tools to understand the nonlinear effects of Einstein's equations.The aim of the proposed project is to prove global existence of equivariant wave maps coupled to Einsteins equations of general relativity. The proposed techniques for the proof are based on Friedlander's representation formula and its recent improvements by Vincent Moncrief. In addition to introducing new techniques in the study of nonlinear wave equations, the project has implications for one of the fundamental questions in general relativity: the global regularity of Einstein's equations and consequently the cosmic censorship conjectures of Roger Penrose.