Low-dimensional Geometry and Topology

摘要：三流形上的各种结构对我们的理解有重要的贡献：特别是三流形基本群上的几何结构、不可压缩表面、叶理、层合、接触结构、自动结构以及三流形有限片覆盖的晶格。这些结构之间有许多联系，但已知的联系是零星的，往往只是松散的，尽管暗示着更深层次的联系仍有待发现。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，是否存在从紧绷的叶状结构或基本层压到几何分解的结构？反过来说，是否每个双曲3流形都有叶理，或者至少是真正的层压？三流形是三维空间在拓扑上相互连接的可能方式的数学描述。它们很重要，因为它们出现在数学各个角落的几何模型中。现代三流形拓扑的一个中心主题是几何化猜想，即所有三流形都是由局部齐次块组成的猜想，也就是说，三流形的几何结构中，任何一点的邻域都与任何其他点的邻域完全相同，直至某个固定半径。这个猜想是由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，许多基本问题仍然是未知的，包括著名的庞加莱猜想——几何化猜想的一个特例，它断言三流形只有一种可能的拓扑结构，其中每个环都可以收缩到一个小球里。除了3-流形的几何结构外，还有许多其他有趣的结构对拓扑学具有重要意义，但它们之间的松散连接被理解。这些结构包括不可压缩表面、叶状（一种层状结构）、层状（只存在于流形部分的层状结构）、接触结构（与哈密顿力学有关）以及群论中的各种组合结构。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。该项目由数学科学部的拓扑计划和计算机与信息科学与工程理事会的数字、符号和几何计算计划共同支持

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