Mathematical Sciences: Evolution Equations in Geometry

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Hamilton 9504528 The proposed research is in the area of nonlinear parabolic partial differential equations - for example, the heat equation - arising in differential geometry and the physical sciences. The proposer has previously obtained a number of results on the quantitative and qualitative aspects of the solutions to these parabolic differential equations; these results, in turn, led to applications in smooth manifold theory. In particular, the proposer's work on the diffeomorphism classification of four-manifolds with positive isotropic curvature is considered an important result in the field. Applications of the proposed research in the physical sciences include movement of solid-liquid interfaces of materials, wear of material bodies under random impart, motion of lubricating fluids, and evolution of liquid crystals.