Mathematical Sciences: Singularities of Geometric Heat Flows

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Essential computer studies will be used by the principal investigator to formulate theories concerning formation of singularities of maps between smooth Riemannian manifolds. These maps evolve both by harmonic map heat flow and mean curvature heat flow. Specifically, he will investigate time durations for singularity formation, the dependence of blow-up times on the singularity's codimension, and whether such heat flows may be extended in unique fashion past the singularity. The work extends known theories of nonsingular heat flows and the special case of non-positively curved target manifolds. Computers will be used extensively in this project. Equations modeling physical systems such as phase boundaries in chemical reactions and liquid crystals will be the subject of a systematic study. The principal investigator will use numerical studies to suggest quantities which in turn will be used as barrier functions for theoretical estimates.