Mathematical Sciences: Low-Dimensional Geometry and Topology

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9704135 Thurston This project is a multi-pronged investigation into low-dimensional geometry and topology with the central long-range goal of supporting and establishing the Geometrization Conjecture, namely, that every 3-dimensional manifold has a canonical decomposition into pieces with locally homogeneous Riemannian metrics. The first of these prongs is an investigation, partially aided by computer, of geometric Dehn filling spaces for 3-manifolds. Geometric Dehn fillings interpolate between manifolds with distinct topology. A suitable variation of this idea has the potential to connect all possible 3-manifolds and equip them with geometric decompositions. The second prong is an investigation of the geometry of taut foliations and essential laminations and their relationships to hyperbolic structures and other geometric structures. This approach has the promise to give a common generalization of the canonical decompositions of surface homeomorphisms developed by Thurston in 1976, and the geometrization of Haken manifolds developed by Thurston in the late 1970's and early 1980's. Additional prongs of this project involve contact structures, confoliations, and geometric group theory. In addition, the project will explore connections of low-dimensional geometry and topology to biology, particularly genetics. Low-dimensional geometry and topology is a beautiful area of mathematics that has undergone tremendous growth and transformation over the last two decades. Paralleling this internal development of the subject, lagging slightly behind, has been the external development of new connections and strengthened connections to other areas of mathematics and of science. Some of these connections are direct, like the cosmological issue of the shape of our universe or the more down-to-earth issue of the shapes of crystals. Other connections are indirect, to diverse topics such as databases, group theory, genetics, or computational chemistry. Any topic tha t can be formulated in a quantitative way can be studied with geometric methods (alongside the more prevalent symbolic, algebraic and analytic methods), so that in any particular case, the powerful discoveries and tools of low-dimensional geometry and topology have a reasonable chance to have a direct bearing. The aim of this project is further development of the tools of low-dimensional geometry and topology to make them more portable, more powerful and more universal. ***