Evolution Equations in Geometry

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Abstract Proposal: DMS-9803550 Principal Investigator: Richard Hamilton Richard Hamilton will work on non-linear parabolic partial differential equations and systems of equations in science and geometry, including the evolution of a Riemannian metric by its Ricci curvature: the motion of a surface by its mean curvature or its Gauss curvature, the flow of gas in a porous mechanism, the motion of a liquid crystal, and others. Special emphasis will be on the formation of singularities, using precise models to show their explicit structure. These equations model many interesting phenomena in science where diffusion occurs. As examples: applications occur in material science to the motion of the boundary between a solid and a liquid, the wear of rocks under impact at a random angle, and the diffusion of oil in shale; in biology to the reproduction of sparse species and the migration of solitary animals; in photography to image sharpening; and in general relativity to the shape of the universe.