Mathematical Sciences: Minimal Surfaces, Complex and Symplectic Geometry

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The principal investigator will develop a theory of minimal surfaces for Kahler manifolds. Special emphasis will be paid to the relationship between the complex tangent points and the Riemannian geometry and stability properties of these surfaces. He will also investigate the relationship between the characteristic forms of vector bundles and the singularities of bundle maps. And he will extend his past work on harmonic maps to infinite Grassmannians. A new study of symplectic topology will begin. Minimal surfaces are the mathematicians' soap films. Exotic minimal surfaces can be formed by dipping ordinary curved wires into a soap and water solution. Great progress toward a complete understanding of these surfaces has been made in recent years. Now, the principal investigator and other geometers are using these new theories to build connections with the emerging geometrical physics of Kahler manifolds.