Mathematical Sciences: Research in Pseudo-Riemannian Geometry and Mathematical Relativity

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The principal investigator will study problems in Lorentzian, Riemannian, and pseudo-Riemannian geometry. He will build from his earlier work involving a Lorentzian analogue of the Cheeger-Gromoll splitting theorem of Riemannian geometry. This work will entail an analysis of the existence of timelike lines in space-times obeying suitable curvature, causality, and completeness conditions. In another vein, he will use minimal surface theory to demonstrate the simplicity of spacial topology near a black hole. Lastly, he will investigate as to whether compact pseudo-Riemannian manifolds have closed geodesics. The principal investigator will study several problems related to the geometry of space-time. Einstein was the first to suggest that the universe is a four-dimensional dynamic hypersurface which both restricts the movement of masses and is curved by their gravity. In this project, mathematics developed for the study of more general hypersurfaces will be applied to the specific hypersurfaces which are used to model our space- time.