Low-dimensional Geometry and Topology

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Proposal: DMS-0072540PI: William ThurstonAbstract: A variety of structures on three-manifolds have contributed significantly to our understanding: notably, geometric structures, incompressible surfaces, foliations, laminations, contact structures, automatic structures on fundamental groups of three-manifolds, and the lattice of finite-sheeted coverings of three-manifolds. There are many connections among these structures, but nevertheless the known connections are sporadic and often only loose, although suggestive of deeper connections remaining to be discovered. The PI will investigate these various structures and their interrelationships, with an emphasis on analyzing computability, and developing techniques for actually constructing and computing examples. For example, is there a construction to go from a taut foliation or an essential lamination to a geometric decomposition? And conversely, does every hyperbolic 3-manifold admit a foliation or at least a genuine lamination?Three-manifolds are the mathematical descriptions of the possible ways for 3-dimensional space to be topologically interconnected. They are important because they arise through geometric models in every corner of mathematics. A central theme in modern three-manifold topology is the Geometrization conjecture, which is the conjecture that all three-manifolds are made up of locally homogeneous pieces, that is, three-manifolds that have a geometry in which a neighborhood of any one point is completely identical to a neighborhood of any other point, up to some fixed radius. This conjecture, proposed by the PI about 20 years ago is now supported by a great deal of theoretical and empirical evidence. Nonetheless, many basic questions remain unknown, including the famous Poincare conjecture a special case of the Geometrization conjecture which asserts that there is only one possible topology for a three-manifold in which every loop can be contracted to fit inside a small ball. Besides geometric structures for 3-manifolds, there are a number of other interesting structures that have important implications for topology, but only loose connections among them are understood. Among these structures are incompressible surfaces, foliations (a kind of layered structure), laminations (layered structures that only exist on part of the manifold), contact structures (related to Hamiltonian mechanics), and various combinatorial structures from group theory. The PI will investigate connections among these various structures, with an emphasis on computability and techniques of making actual computations of examplesThe project is supported by both the Topology Program in the Division of Mathematical Sciences and the Numeric, Symbolic, and Geometric Computation Program in the Computer and Information Science and Engineering Directorate