Mathematical Sciences: Structure of Codimension one Foliations

该项目继续与劳伦斯W。康伦 （圣路易斯华盛顿大学）关于 余维一叶理，工作开始于1975年。 他们已经表明 每一个开放的表面都可以作为某种叶状的叶子出现， 问什么样的准等距类型的表面可以作为叶子出现。 Markov-exceptional局部极小集是行为良好的， 一个问题是，是否每个例外的最小集都在某个 感觉本质上是马尔可夫的，或者，如果不是，至少有许多 一个马尔可夫的性质。 在一系列引人注目的论文中， Gabai已经研究了结子补缀的叶理，但仍有许多 在这个领域要做的事情。 例如，首席研究员 意图构造更容易的结补语的叶理 想象然后用瑟斯顿球的某一秒 同调群用来描述某些纽结所有绷紧叶理- 互补。 流形的叶理是填充流形的一种方法 低维度的碎片。 在余维1的情况下， 叶理，这些碎片的尺寸小于 给定流形。 想想洋葱或洋蓟。 拓扑 一个流形的形状与它所处的叶理类型密切相关， 将支持，并在熟练的手，这种关系已经锻造 成为研究流形拓扑的有力工具。 这是一个不直观的事实，主要的代数工具， 研究流形的拓扑结构在以下情况下效果最好： 高维流形 提供的几何工具， 因此，在低海拔地区， 维流形 这方面的一个主要例子是 三维空间中纽结的补集研究 球体，这被证明是获取信息的重要方式 关于绳结本身

This project continues joint research with Lawrence W. Conlon (Washington University in St. Louis) on the structure of codimension-one foliations, work begun in 1975. They have shown that every open surface can occur as a leaf of some foliation and ask what quasi-isometry types of surfaces can occur as leaves. Markov-exceptional local minimal sets are well-behaved, and the question arises whether every exceptional minimal set is in some sense essentially a Markov one or, if not, at least has many of the properties of a Markov one. In a remarkable series of papers, Gabai has studied foliations of knot-complements, but much remains to be done in this area. For example, the principal investigator intends to construct foliations of knot-complements that are easier to visualize and then to use the Thurston ball of a certain second homology group to describe all taut foliations of certain knot- complements. A foliation of a manifold is a way of filling the manifold with lower dimensional pieces. In the case of a codimension-one foliation, these pieces are of dimension one less than that of the given manifold. Think of an onion or an artichoke. The topology of a manifold is strongly related to the kind of foliation which it will support, and in skillful hands this relation has been forged into a powerful tool for investigating the topology of manifolds. It is an unintuitive fact that the major algebraic tools for investigating the topology of manifolds work best in the case of high dimensional manifolds. The geometric tool afforded by foliations is thus particularly welcome in the case of low dimensional manifolds. A major instance of this is the investigation of the complement of a knot in the three-dimensional sphere, which turns out to be an important way to gain information about the knot itself.