Mathematical Sciences: Topics in Low-Dimensional Topology

对3流形的研究通常通过在流形上施加额外的结构来丰富：几何结构，如负曲率的黎曼度量，或拓扑动力学结构，如流或层压。两位研究者将继续研究3流形支撑层和伪阿诺索夫流。他们还将对3流形的结构和更一般的非负弯曲空间进行新的研究。在一个联合项目中，Mosher和Oertel将探索一种分析此类空间的新技术。任何这样的空间都可以用来支持某种二维测量层压。这种层压的性质将被研究和使用，试图揭示这些空间的一般结构。在一个单独的项目中，Mosher将继续研究3流形的Thurston同调范数，以及使用伪anosov流计算该范数的技术。他还将进行双曲型3-流形末端的计算机研究。在另一个独立的项目中，Oertel将继续研究一类称为层压流形的3-流形。第一个目标是将哈肯流形的方法和定理扩展到更大的层压流形。第二个目标是表明，在某种意义上，“大多数”3-流形是层压的。令人惊讶的是，尽管我们生活在一个三维空间，一个所谓的三维流形中，因此我们对这些几何物体有一种天生的直觉，但最终这并没有把我们带得像我们所期望的那样远，因为在三维的情况下，那些已经通过高维流形的代数计算解决的问题仍然令人困惑。其中最著名的是庞加莱在世纪之交提出的关于三维球体的著名猜想，正是在这个猜想中，最初的三维情况是唯一仍然开放的。研究人员正在研究关于三维流形的各种问题，其中一些带有一些奇怪的距离概念，即所谓的双曲度量，但这些问题一次又一次地被证明与具有更熟悉的距离概念的流形的情况有明确的关联。

The study of 3-manifolds is often enriched by imposing extra structure on the manifold: geometric structure such as a Riemannian metric of negative curvature, or topological-dynamical structure such as a flow or lamination. The two investigators will continue research into 3-manifold supporting laminations and pseudo-Anosov flows. They will also undertake new research into the structure of 3-manifolds and more general spaces which are not negatively curved. In a joint project, Mosher and Oertel will explore a new technique for analyzing such spaces. Any such space can be made to support a certain kind of 2-dimensional measured lamination. The properties of this lamination will be studied and used in an attempt to shed light on the general structure of these spaces. In a separate project, Mosher will continue a study of Thurston's homology norm for a 3-manifold, and techniques for computing this norm using pseudo-Anosov flows. He will also pursue a computer study of ends of hyperbolic 3- manifolds. In still another separate project, Oertel will continue research into a class of 3-manifolds called laminated manifolds. The first goal is to extend methods and theorems from the well-known class of Haken manifolds to the larger class of laminated manifolds. The second goal is to show that, in some sense, "most" 3-manifolds are laminated. It is a surprising fact that, although we live in a three dimensional space, a so-called 3-manifold, and so are blessed with a natural intuition about such geometric objects, in the end this does not carry us as far as we might have expected, for questions which have been settled by algebraic calculations for higher dimensional manifolds still remain baffling in the 3- dimensional case. The most famous of these is the celebrated conjecture of Poincare from around the turn of the century concerning 3-dimensional spheres, where precisely the original 3- dimensional case is the only one still open. The investigators are pursuing a variety of questions about 3-dimensional manifolds, some with slightly strange notions of distance on them, so-called hyperbolic metrics, but time and time again these questions have been shown to have clear relevance to the case of manifolds with a more familiar notion of distance.