Mathematical Sciences: Problems in Low-Dimensional Topology

Cooper， Long和Scharlemann专注于3流形。Cooper的一个项目是利用双曲3流形中有限叶形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结点的a多项式的性质。第三种是将建筑理论应用于辫子群的表现。Long的具体项目涉及有限叶化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。他也在研究代数几何中的问题，利用度猜想证明性质P，以及辫群的有限维线性表示。Scharlemann的主要兴趣是heegard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”的概念，结理论也有联系。关于我们周围世界的最基本的观察之一，几乎从我们出生开始就显而易见，那就是它是三维的。因此，理解空间的这种特性是很有趣的：任何生活在空间中的人都会把他们的世界看作是三维的。这样的空间被称为“3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”，在这种现象中，一个本质上很简单的物体，如花园软管（或DNA线），可以被操纵，从而使其在空间中的位置相当复杂。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”（很像工匠模型），如果把“棒”想象成橡胶制成的、可以打结和交织的“棒”，就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）有助于我们理解3-流形。例如，上面提到的“Heegaard分裂”指的是一种将一般3流形的所有复杂性都吸收到厚图中的技术。那么关于图的信息给出了关于3流形的信息。例如，在托科马克型环体中流体或等离子体流动的轨迹中，会发生一种更有规律的打结，称为编织。由于这种打结更有纪律，更多的工具可用于理解和分类这种打结。因此人们对编织理论及其与动力系统的联系产生了兴趣。＊＊＊

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. ***