GIG: Problems in Low-Dimensional Topology

GIG：低维拓扑问题

• 批准号: 9510505
• 负责人: Darren Long
• 金额: $ 10万
• 依托单位: University of California-Santa Barbara
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-08-01 至 1999-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9510505&HistoricalAwards=false
• 关键词: GIG; Problems; Low; Dimensional; Topology

9510505 Long This award is directed towards the infrastructure needs of the geometric topology group in the areas of graduate student support and in the hiring of postdoctoral students in the field. The group has a wide range of interests within and contiguous to the area of geometric topology. The following indicates the current directions of the researchers: 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 which 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 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 objects with precisely this property: anyone living in one would see his or her world as 3- dimensional. Such objects are called "3-manifolds", and the project aims to increase our understanding of them. 3- manifolds support interesting phenomena. One of these phenomena is "knotting", in which a simple object like a garden-hose (or a string of DNA) can be maneuvered so that its positioning in space is quite complex. More generally , objects like chemical molecules, usually though of abstractly as "graphs" (much like tinker toy models), can be put in 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 understand knotting and, conversely, understanding knotting (a second principal aim of the project) helps us 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 intereqt in braid theory and its connections to dynamical systems.

小行星9510505 该奖项是针对几何拓扑组在研究生支持领域的基础设施需求，并在该领域雇用博士后。该小组在几何拓扑学领域内和邻近领域有广泛的兴趣。以下是研究者们目前的研究方向：库珀，朗和沙勒曼专注于三维流形。库珀的项目之一是利用有限叶理和几何有限曲面之间的关系在双曲3流形。另一个是继续发展的性质的A-多项式的纽结所产生的表示理论。第三个涉及建筑物的理论应用于表示的辫子群。 长期的具体项目关注的研究有限foliations和由此产生的动力系统以及应用这些想法双曲3流形。他还致力于问题的代数几何和使用的程度猜想，以证明财产P，和有限维线性表示的辫子群。 Scharlemann的主要兴趣是Heegaard分裂的稳定化问题。 成功将有重要的影响一般分类问题的3流形。 通过纽结的“隧道数”的概念，它也与纽结理论有联系。关于我们周围世界的一个最基本的观察，几乎从我们出生时就很明显，那就是它是三维的。 因此，理解具有这种特性的物体是很有趣的：任何生活在其中的人都会把他或她的世界看作是三维的。 这样的物体被称为“三维流形”，该项目旨在增加我们对它们的理解。 3-流形支持有趣的现象。 这些现象之一是“打结”，其中一个简单的物体，如花园软管（或一串DNA）可以被缠绕，使其在空间中的定位非常复杂。 更一般地说，像化学分子这样的物体，虽然通常被抽象为“图形”（很像修补匠玩具模型），但如果人们把“棍子”想象成可以打结和交织的橡胶制成的话，它们可以以非常复杂的方式放在三维流形中。 正在开发的工具来理解三维流形帮助我们理解打结，反过来，理解打结（该项目的第二个主要目标）帮助我们理解三维流形。 例如，上面提到的“Heegaard分裂”指的是一种技术，其中一般3-流形的所有复杂性都被吸收到厚图中。 然后关于图的信息给出关于三维流形的信息。 一种更有规律的打结类型，称为编织，发生在例如托科马克型环面中的流体或等离子体流的轨迹中。 由于这种打结更有规律，因此有更多的工具可用于理解和分类这种打结。 因此，人们对辫子理论及其与动力系统的联系很感兴趣。