Mathematical Sciences: Studies in Symplectic & Lorentzian Geometries

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Paul Ehrlich, Gerard Emch and Paul Robinson will work on problems in differential geometry and quantum theory. The questions to be addressed include geodesic incompleteness of Lorentzian manifolds, symplectic geometry of geodesics and geometric quantization. These problems are of interest in both mathematics and physics. They are fundamental to the mathematical aspects of relativity and the relationships between classical mechanics and quantum theory. Ehrlich's work focusses primarily on Lorentzian geometry and will build on recent advances made towards establishing the splitting theorem. This is related to rigidity phenomena in the singularity theory of general relativity. Emch and Robinson will investigate spinors in both the symplectic and the orthogonal contexts. These arise naturally in the attempt to geometrize the passage from classical mechanics to quantum theory where the behavior of symmetries plays an important role.