Low-dimensional Geometry and Topology

摘要：三维流形上的各种结构对我们的理解有很大贡献：值得注意的是几何结构、不可压缩曲面、叶层、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联系，但已知的联系是零星的，通常只是松散的，尽管暗示着还有更深层次的联系有待发现。PI将调查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紧凑的叶状结构或基本的层状结构过渡到几何分解？反之，是否每个双曲三维流形都有一个层状结构，或者至少有一个真正的层状结构？三维流形是对三维空间拓扑互连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各个角落的几何模型产生的。现代三维流形拓扑学的一个中心主题是几何猜想，它是一个猜想，即所有的三维流形都是由局部齐次的片断组成的，也就是说，三维流形具有这样的几何：其中任何一点的邻域与任何其他点的邻域完全相同，直到某一固定半径。这一猜想是由PI在大约20年前提出的，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Poincare猜想，这是几何化猜想的一个特例，该猜想断言，对于一个三维流形，只有一个可能的拓扑，其中每个圈都可以收缩到一个小球中。除了三维流形的几何结构外，还有许多其他有趣的结构对拓扑学有重要的影响，但我们只了解它们之间的松散联系。在这些结构中有不可压缩的表面、叶层(一种层状结构)、层状结构(只存在于流形的一部分上的层状结构)、接触结构(与哈密顿力学有关)，以及来自群论的各种组合结构。PI将调查这些不同结构之间的联系，重点是可计算性和进行实例实际计算的技术。该项目得到了数学科学部的拓扑学计划和计算机和信息科学与工程局的数值、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