Mathematical Sciences: Problems in Low-Dimensional Topology

小行星9504438 库珀，长期和Scharlemann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。 另一个是继续发展从表示论中产生的节点的A-多项式的性质。 第三个涉及建筑物的理论应用于表示的辫子群。 龙的具体项目涉及研究有限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，并对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”概念，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，理解具有这种性质的空间是很有趣的： 任何生活在太空中的人都将看到他们的世界是三维的。 这样的空间被称为“三维流形”，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，一个本质上简单的物体，如花园软管（或DNA串），可以被打结，使其在空间中的位置非常复杂。 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”（很像修补模型），如果人们把“棒”看作是由橡胶制成的，可以打结和交织，就可以以非常复杂的方式放入三维流形中。 正在开发的工具来理解3-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3-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。 ***

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. ***