Mathematical Sciences: Geometric Variational Problems and Related PDE Questions

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Two investigators will analyze problems arising from the study of geometric partial differential equations. One will study questions related to asymptotics for geometric extrema, including questions about growth properties of entire solutions of the minimal surface equation and more general quasilinear elliptic equations. He will also work toward a solution of the long-standing Willmore conjecture. The second investigator will study the structure of singular sets of minimal surfaces, asymptotic behavior of harmonic maps near singularities, and topological types of minimal submanifolds. Both investigators will analyze problems which involve understanding how partial differential equations act on minimal surfaces. These equations include derivatives of functions of more than one variable and frequently have their origins in applied mathematics. Minimal surfaces are the mathematician's model of common soap films. In recent years the use of partial differential equations in connection with minimal surfaces has seen an explosion of research activity.