Mathematical Sciences: Motion of Phase Interfaces by Mean Curvature and Related Problems

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The principal investigator will study several weak formulations of the motion of a singular surface by its mean curvature. Such motion arises in the theory of phase transitions. The work is part of an ongoing project to reconcile the various mathematical models for the motion of phase boundaries. The principal investigator will study the existence of unit density flows, the convergence of systems of phase-field equations, and the existence and properties of self-similar solutions. This award will support research in the general area of differential geometry and global analysis. Differential geometry is the study of the relationship between the geometry of a space and analytic concepts defined on the space. Global analysis is the study of the overall geometric and topological properties of a space by piecing together local information. Applications of these areas of mathematics in other sciences include the structure of complicated molecules, liquid-gas boundaries, and the large scale structure of the universe.