Mathematical Sciences: Seifert Fiber Spaces

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A Seifert manifold is a topological space with a bundle-like structure; that is, it has a parameter space and over every point in the parameter space, there sits another space called "a fiber." The fibers vary slightly from point to point in the parameter space. Spaces of this class occur naturally in many different branches of mathematics, for example, topology, differential geometry and algebraic geometry, as well as in physics (crystallographic groups). The investigator will study the geometry and topology of these spaces as well as their symmetries. The problems to be studied vary from general questions concerning the geometric structures of these spaces to specific problems connected with the existence of symmetric and asymmetric Seifert manifolds. This will greatly enhance our knowledge of this important class of spaces, their intrinsic and extrinsic geometry, structure and properties. The broad spectrum of techniques and methods which will be employed in this research include methods of algebraic and geometric topology, differential geometry, Lie group theory and group cohomology.