Mathematical Sciences: Differential Geometry

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The principal investigator will study several problems from the theories of partial differential equations and differential geometry. He will construct complete Riemannian manifolds with special holonomy and study their relationship with the physical theories of "supermembranes" and "supergravity." Degenerate partial differential equations which arise in the geometry of 3- manifolds and the theory of exterior differential systems will be the subjects of two further investigations. The interplay of ideas between the physics of gravitation and subatomic particles, and the mathematics of differential geometry and geometric partial differential equations, has become intense in recent years. Geometric structures which have been under development for many decades are now sufficiently powerful to handle many new problems of modern physics. The principal investigator will solve a problem in holonomy of Riemann manifolds which relates to recent theories of "supergravity."