Geometric Analysis and General Relativity

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General Relativity is one of the fundamental theories of modern physics. Since its formulation by Einstein in 1915, it has been a cornerstone of our understanding of the universe. Mathematical research on the problems of general relativity brings together many important areas of research in partial differential equations, differential geometry, dynamical systems and analysis. The fundamental mathematical questions which arise in general relativity have stimulated the development of substantial new tools in analysis and geometry. Conversely, the introduction of modern tools from these fields have recently led to significant developments in general relativity, shedding new light on a number of deep and far reaching conjectures. The area of interaction between the analysis of the Lorentzian Einstein equation, the field equation of general relativity, and other geometric partial differential equations is large. Important classes of hyperbolic equations such as the wave maps equation and the Yang-Mills equation are connected with the Einstein equation at a fundamental level, and harmonic analysis methods used in the study of those equations are being applied to the Einstein equation. Fluids as well as kinetic models such as the Vlasov equation appear as matter models. The Einstein equation shares issues of convergence, collapse and stability with important geometric evolution equations such as the Ricci flow and the mean curvature flow. Asymptotic behavior at singularities as well as questions related to asymptotic geometrization are being intensely studied in the case of the Einstein equation as well as in the cases of the Ricci and the mean curvature flows. There is the potential for continued cross fertilization between these fields. Elliptic problems and techniques arise in studying the initial data sets in the context of the Cauchy problem. The analysis of the Riemannian Einstein equation plays a fundamental role in modern Riemannian geometry and geometric analysis. Recent developments in General Relativity have brought many ideas and techniques which have been developed in the Riemannian context to bear on the Lorentzian Einstein equations. An example of this type of development is the realization that apparent horizons, which are relevant for our understanding of quasilocal aspects of black holds, have many properties in common with minimal surfaces. Many recent results in this area have been inspired in part by this connection. Other areas of geometry which have played a significant role in general relativity in recent years include the geometrization of three-manifolds; convergence and collapsing phenomena in Riemannian manifolds with bounds on their curvatures, the study of manifolds admitting metrics of positive scalar curvature, and curvature flows for hypersurfaces.