Hyperbolic 3-Manifold Theory

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9803362 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 two principal 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 explicit examples on a 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. They will also consider applications of 3-manifold theory to cosmological models of the spatial universe, working in conjunction with cosmologists. Undergraduates will be involved in original research projects related to this research. A 2-dimensional surface may have positive curvature like a sphere, zero curvature like a flat Euclidean plane, or negative curvature like a saddle. Almost all surfaces can be given negative curvature, meaning that locally, they behave like the hyperbolic plane discovered by Lobachevsky, Gauss, and Bolyai in the last century. An analogous understanding of finite 3-dimensional spaces eluded researchers for decades, but in the mid-1970's W. P. Thurston conjectured that 3-manifolds, the analog of 2-dimensional surfaces one dimension up, can be cut into pieces, each of which has one of eight geometries. He proved the conjecture for vast quantities of 3-manifolds. By far the most commonly occurring geometry for 3-manifolds is hyperbolic geometry. The resulting "hyperbolic structures" are rich and have been used to solve a great many topological and geometrical questions. T he present project continues this work both theoretically and computationally. The theoretical advances make the computations possible, and the explicitly computed examples guide the further development of the theory. In the past year the project has found a new and very exciting application. The spatial universe within which we live is an example of a 3-manifold. Data from NASA's Microwave Anisotropy Probe (MAP) may, in the year 2001, reveal the spatial universe to be finite in extent. If so, algorithms from the present project will be used to determine the exact shape of the universe. The investigators are currently extending their software to tolerate the high levels of noise that will be present in the MAP data, and to handle the spherical and Euclidean cases as well as the hyperbolic one. Preliminary data suggests the real universe is hyperbolic, but the other possibilities cannot yet be ruled out. ***