Mathematical Sciences: Mathematical Studies of Gravity and Other Fields in Spacetime

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James Isenberg will apply the methods of modern geometrical analysis to examine certain questions arising in the study of gravitational and other fields in spacetime. The techniques to be applied will arise from Lie group theory, symplectic geometry, algebraic geometry, and the theory of elliptic and hyperbolic partial differential equations. Amongst the questions to be studied are 1) Is it true that all spacetimes which satisfy Einstein's equations and are extendible across a compact Cauchy horizon must contain Killing vector fields? 2) Can the class of spatially homogeneous solutions of the Einstein Yang-Mills equations be treated as a finite dimensional dynamical system? 3) Can the behavior of constant mean curvature surfaces or hypersurfaces be systematically analyzed near singular regions in the ambient manifold?