Mathematical Sciences: Minimal Surfaces and the Topology of 3-dimensional Manifolds

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This project is concerned with applications of techniques of differential geometry to the study of 3-dimensional manifolds. In particular, it aims to investigate the use of minimal surfaces to study non-Haken 3-manifolds and their diffeomorphisms. It also aims to attack problems on the Heegaard splittings of 3-manifolds and on harmonic diffeomorphisms of surfaces. The point is to take natural but complicated three-dimensional objects and cut them apart into simpler objects by means of surfaces of least area (like soap films). This has proved a useful way to understand 3-manifolds in the past and will probably continue to be.