Mathematical Sciences: Evolution Equations in Geometry

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Richard Hamilton will continue his exciting work on the heat equation in differential geometry. In recent work he has shown that on a compact four manifold with positive curvature operator the metric evolves to one of constant positive sectional curvature. In collaboration with Michael Gage, he showed that a convex curve in the plane shrinks under curve shortening to a point, remaining convex all the time and becoming asymptotically circular near the end. In continuing this work he will consider convergence of curvature operators in higher dimensions. Similarly he will investigate convergence of metrics on the two sphere in the case where the initial metric has some negative curvature. There are clearly many avenues to be explored and the probability of significant achievements seems very high.