Mathematical Sciences: Hyperbolic 3-Manifold Theory

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9626780 Adams Hyperbolic 3-manifold theory has had a tremendous impact on the field of low dimensional topology. Outstanding conjectures have fallen, one after the other, with the application of hyperbolic techniques. There has been a shift in the entire approach to low dimensional topology, as researchers now use rigid geometric arguments to prove topological theorems. The investigators intend to continue their research, both individually and jointly, on the theory and computation of hyperbolic 3-manifolds. Through a set of projects, many relying heavily on their ability to compute examples on the computer, the researchers plan to further the understanding of the connections between hyperbolic 3-manifolds, the invariants that come out of hyperbolic 3-manifold theory, and the traditional topological invariants in 3-manifold theory. A sphere, a torus and a 2-holed torus are examples of topologically distinct 2-dimensional surfaces. For about 100 years mathematicians have known that the only possible (finite) surfaces are the n-holed tori and a corresponding family of nonorientable surfaces (e.g., the nonorientable surface corresponding to a 1-holed torus is a Klein bottle). In contrast, the topological possibilities for a 3-dimensional space are vastly richer. The beautiful relationships among the possible 3-dimensional spaces have been greatly clarified since 1978, when William Thurston discovered that the best way to understand the topology of the spaces was to give them shapes of constant curvature. The present project continues to explore the relationships between the curvature and the topology. Because the shapes can be computed by hand only in a very few cases, the project also continues to develop and support computer software for creating the spaces and studying their properties. This NSF funded software is freely available, and it is used not only by mathematicians, but also by cosmologists seeking to model the shape of the real universe, and by physicists modelling the behavior of various dynamical systems. ***