Mathematical Sciences: Geometric Measure Theory and the Calculus of Variations

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The principal investigator will continue an ongoing project in the geometric calculus of variations and its applications. This will include an investigation into the geometric structure of surfaces minimizing integrals with geometric integrands such as area. Additionally, he will study the evolution of surfaces towards minimization and analyze the algorithms for their computation. And he will begin a theoretical study of general evolution processes such as would model the growth of a crystal as it freezes from its melt. Geometric measure theory provides the mathematical setting for these analyses and their applications in continuum mechanics. This project involves a variety of applications of geometric measure theory as well as extensions of the theory itself. Crystal growth and soap films are two physical phenomena to which this theory has already been applied. The principal investigator will study both the theoretical aspects of geometric measure theory and ways of developing software for its use in computer graphics.