Research Interests

My mathematical research concerns disparate aspects of geometry. I began by studying minimal surfaces -- surfaces of general dimensions which, like soap films, minimize area. These surfaces arise as solutions of a variational problem and satisfy a certain nonlinear system of differential equations. I found closed solutions of these having arbitrary topological type in the three-dimensional sphere. I then used higher dimensional solutions to probe the architecture of more complicated geometric objects. Complex analytic spaces are sets defined by functions satisfying the Cauchy-Riemann equations in several variables. Reese Harvey and I completely characterized the boundaries of these spaces, thereby extending old analytic results of Hartogs and Bochner. We also introduced a theory of calibrations which extended complex analytic geometry to constellations of other absolute volume-minimizers. This theory is useful in various areas of mathematics and is currently of interest in physics. In the field of topology I constructed the first codimension-1 foliations of spheres in dimensions $> 3$. I also worked in group actions with S. T. Yau. We used the index theorem to establish the complete asymmetry of certain exotic spheres. I have made some contribution to the study of non-abelian gauge field theory, in particular by showing that stable solutions of the Yang-Mills equations were necessarily self-dual and therefore energy minimizing. Misha Gromov and I used the Dirac equations to study Riemannian manifolds of positive scalar curvature (spaces which are very, very weakly "spherical''). We succeeded in classifying these manifolds in terms of modern topological invariants. Recently I have been studying families of solutions of algebraic equations inside of algebraically defined spaces. I determined the global topological structure of these families in fundamental cases. The methods and results have developed into a richly structured theory for algebraic varieties with applications to the field of topology.

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Section 11: Mathematics