Mathematical Sciences: Problems in Low-Dimensional Topology

数学科学：低维拓扑问题

• 批准号: 9504438
• 负责人: Martin Scharlemann
• 金额: --
• 依托单位: University of California-Santa Barbara
• 依托单位国家: 美国
• 项目类别: Continuing grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-07-15 至 1999-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9504438&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Problems; Low; Dimensional

9504438 Scharlemann Cooper, Long and Scharlemann focus on 3-manifolds. One of Cooper's projects is to utilize the relationship between finite foliations and geometrically finite surfaces in hyperbolic 3-manifolds. Another is to continue developing the properties of the A-polynomial for knots that arise from representation theory. A third involves the theory of buildings applied to representations of the braid group. Long's specific projects concern the study of finite foliations and the resulting dynamical systems, as well as the application of these ideas to hyperbolic 3-manifolds. He also is working on problems in algebraic geometry and the use of the degree conjecture to prove Property P, and on the finite dimensional linear representations of the braid groups. Scharlemann's main interest is the stabilization problem for Heegaard splittings. Success would have important implications for the general classification problem for 3-manifolds. There are connections to knot theory as well, via the notion of "tunnel number" for a knot. One of the most basic observations about the world around us, apparent almost from our birth, is that it is 3-dimensional. So it is of interest to understand spaces with precisely this property: anyone living in the space would see their world as 3-dimensional. Such spaces are called "3-manifolds," and this project aims to increase our understanding of them. 3-manifolds support interesting phenomena. One of these phenomena is "knotting," in which an intrinsically simple object like a garden-hose (or a DNA string) can be maneuvered so that its positioning in space is quite complex. More generally, objects like chemical molecules, usually thought of abstractly as "graphs" (much like tinkertoy models), can be put into a 3-manifold in extraordinarily complex ways if one thinks of the "sticks" as made of rubber which can be knotted and interweaved. Tools which are being developed to understand 3-manifolds help us und erstand knotting and, conversely, understanding knotting (a second principal aim of the project) helps us to understand 3-manifolds. For example, the "Heegaard splittings" mentioned above refer to a technique in which all the complexity of a general 3-manifold is absorbed into a thick graph. Then information about the graph gives information about the 3-manifolds. A more disciplined type of knotting, called braiding, occurs, for example, in the trajectories of fluid or plasma flow in a Tokomak-type torus. Since this knotting is more disciplined, more tools are available for understanding and classifying such knotting. Hence the interest in braid theory and its connections to dynamical systems. ***

9504438 Scharlemann Cooper、Long 和 Scharlemann 专注于 3 流形。 Cooper 的项目之一是利用双曲 3 流形中的有限叶状结构和几何有限表面之间的关系。 另一个是继续开发由表示论产生的结的 A 多项式的属性。 第三个涉及应用于辫子组表示的建筑物理论。 Long 的具体项目涉及有限叶状结构和由此产生的动力系统的研究，以及这些思想在双曲 3 流形中的应用。 他还致力于研究代数几何问题和使用度猜想来证明性质 P，以及辫子群的有限维线性表示。 沙勒曼的主要兴趣是赫加德分裂的稳定问题。 成功将对 3 流形的一般分类问题产生重要影响。 通过结的“隧道数”概念，也与结理论有联系。 对我们周围世界最基本的观察之一，几乎从我们出生起就显而易见，它是三维的。 因此，理解具有这种属性的空间是很有趣的：生活在该空间中的任何人都会将他们的世界视为 3 维的。 这样的空间被称为“三流形”，这个项目旨在增加我们对它们的理解。 3-流形支持有趣的现象。 其中一种现象是“打结”，其中本质上简单的物体如花园软管（或 DNA 绳）可以被操纵，从而使其在空间中的定位相当复杂。 更一般地说，像化学分子这样的物体，通常被抽象地认为是“图形”（很像修补玩具模型），如果人们认为“棍子”是由橡胶制成的，可以打结和交织，则可以以极其复杂的方式放入三流形中。 正在开发的用于理解 3 流形的工具可以帮助我们理解打结，相反，理解打结（该项目的第二个主要目标）可以帮助我们理解 3 流形。 例如，上面提到的“Heegaard splittings”是指将一般 3 流形的所有复杂性都吸收到厚图中的技术。 然后有关图的信息给出有关 3 流形的信息。 例如，托科马克型环面中的流体或等离子体流的轨迹中会发生一种更严格的打结类型，称为编织。 由于这种打结更加规范，因此有更多工具可用于理解和分类此类打结。 因此人们对辫子理论及其与动力系统的联系产生了兴趣。 ***