Low-dimensional Geometry and Topology

提案：DMS-0072540 PI：William Thurston摘要：三维流形上的各种结构对我们的理解有很大的贡献：值得注意的是，几何结构，不可压缩曲面，叶理，叠层，接触结构，三维流形基本群上的自动结构，以及三维流形的有限片覆盖格。 这些结构之间有许多连接，但已知的连接是零星的，往往只是松散的，虽然暗示更深的连接仍有待发现。 PI将研究这些不同的结构及其相互关系，重点是分析可计算性，并开发实际构建和计算示例的技术。 例如，有没有一种结构可以从一个绷紧的叶理或一个基本的层压结构转变为一个几何分解结构？反过来说，是否每一个双曲三维流形都有叶理或至少有真正的层理？三维流形是三维空间拓扑互连的可能方式的数学描述。它们之所以重要，是因为它们是通过数学的每一个角落中的几何模型产生的。 现代三流形拓扑学的一个中心主题是几何化猜想，它是所有三流形都由局部齐次部分组成的猜想，也就是说，三流形具有一个几何，其中任何一个点的邻域与任何其他点的邻域完全相同，直到某个固定半径。 这个猜想，大约20年前由PI提出，现在得到了大量理论和经验证据的支持。 尽管如此，许多基本问题仍然未知，包括著名的庞加莱猜想，这是几何化猜想的一个特例，它断言对于一个三流形只有一种可能的拓扑结构，其中每个回路都可以收缩到一个小球内。 除了三维流形的几何结构之外，还有许多其他有趣的结构对拓扑学有重要的影响，但它们之间的联系只是松散的。 这些结构包括不可压缩曲面、叶理（一种层状结构）、叠层（只存在于流形的一部分上的层状结构）、接触结构（与哈密尔顿力学有关）以及来自群论的各种组合结构。 PI将研究这些不同结构之间的联系，重点是可计算性和实例实际计算的技术。该项目得到数学科学部拓扑计划和计算机与信息科学与工程局数值、符号和几何计算计划的支持

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