"Manifolds, knots and groups"

三维流形描述了我们宇宙的可能形状，也出现在许多应用中，如动力系统。我们的建议是开发代数和几何工具，以应用于我们对三维流形及其结构的更深层次的理解。这些工具在节点理论和其他拓扑现象中也很有用。 具体地说，一个流形或一个纽结有一个与之相关的代数系统，称为“群”。最近由佩雷尔曼解决的庞加莱的一个著名猜想是，如果群归结为单一元素，那么流形就被确定为简单的三维球体。更一般地，该组提供有关流形或结的整体形状的可计算信息。通过考虑组中对象的适当排序，我们可以锐化可用的信息。近年来，这种有序群揭示了纽结和三维流形的新性质，这是以前没有的。这是代数在几何拓扑学中应用的一个正在进行的、令人兴奋的新发展，还有许多有待发现的地方。

Three-dimensional manifolds describe the possible shapes of our universe, and also arise in many applications such as dynamical systems. The proposal is to develop algebraic and geometric tools to apply toward our deeper understanding of 3-manifolds and their structure. These tools are also useful in the theory of knots and other topological phenomena. Specifically, a manifold or a knot has an algebraic system, called a "group", associated with it. A celebrated conjecture of Poincare, which was solved recently by Perelman, is that if the group reduces to a single element, then the manifold is determined to be simply a 3-dimensional sphere. More generally this group gives computable information about the overall shape of the manifold or knot. By considering appropriate orderings of the objects in the group, we can sharpen the information available. Such ordered groups, in recent years, have uncovered new properties of knots and 3-dimensional manifolds not available before. This is an ongoing, exciting new development in the application of algebra to geometric topology, with much yet to be discovered.