GIG: Problems in Low-Dimensional Topology

9510505 Long该奖项是针对几何拓扑组在研究生支持和该领域招聘博士后方面的基础设施需求而设立的。该小组在几何拓扑学领域内具有广泛的兴趣，并与几何拓扑学领域相邻。下面指出了研究人员目前的研究方向：Cooper、Long和Scharlemann专注于三维流形。库珀的项目之一是利用双曲3-流形中有限叶层和几何有限曲面之间的关系。另一个是继续发展由表示理论产生的纽结的A-多项式的性质。第三个涉及应用于辫子群表示的建筑物理论。龙的具体项目涉及有限叶系和由此产生的动力系统的研究，以及这些思想在双曲3-流形上的应用。他还致力于代数几何中的问题，并使用次数猜想来证明性质P，以及辫子群的有限维线性表示。Scharlemann的主要兴趣是Heegaard分裂的稳定化问题。这一研究的成功将对三维流形的一般分类问题产生重要的影响。与纽结理论也有联系，通过纽结的“隧道数”的概念。关于我们周围世界的最基本的观察之一，几乎从我们出生起就很明显，就是它是三维的。因此，准确地理解具有这种性质的物体是很有趣的：任何生活在其中的人都会认为他或她的世界是三维的。这类物体被称为“3-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。其中一种现象是“打结”，在这种情况下，一个简单的物体，如花园软管(或一串DNA)可以操纵，使其在空间中的位置相当复杂。更广泛地说，像化学分子这样的物体，通常虽然抽象地被认为是“图形”(很像修补玩具模型)，但如果人们认为“棍子”是由橡胶制成的，可以打结和交织，就可以以极其复杂的方式将其放入3-歧管中。正在开发的理解3-流形的工具帮助我们理解打结，反过来，理解打结(该项目的第二个主要目标)帮助我们理解3-流形。例如，上面提到的“Heegaard Splitings”指的是一种技术，在这种技术中，一般三维流形的所有复杂性都被吸收到一个粗图中。那么关于图的信息就给出了关于3-流形的信息。例如，一种更有规律的打结类型，称为编织，发生在托克马克型环体中的流体或等离子体流动的轨迹中。由于这种打结更有纪律性，所以有更多的工具可以用来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感兴趣。

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.