Mathematical Sciences: Topics in Low-Dimensional Topology

三维流形的研究常常通过引入额外的 流形上的结构：几何结构，例如 负曲率黎曼度量或拓扑动力学度量 结构，如流动或层压。 两个调查者 将继续研究三歧管支撑叠片， 伪Anosov流 他们还将进行新的研究， 三维流形和更一般的空间的结构， 而不是负向弯曲。 在一个联合项目中， 将探索一种新的技术来分析这样的空间。 任何此类 可以使空间支持某种二维 测量层压。 这种层压的性质将是 研究和使用，试图阐明一般 这些空间的结构。 在另一个项目中，莫舍将 继续研究三维流形的Thurston同调范数， 以及使用伪Anosov流计算该范数的技术。 他还将从事双曲3- 流形 在另一个单独的项目中，Oertel将 继续研究一类称为层合的三维流形 流形 第一个目标是将方法和定理从 著名的哈肯流形类到更大的类， 叠层流形 第二个目标是表明，在某些情况下， 从某种意义上说，“大多数”三维流形是层压的。 这是一个令人惊讶的事实，虽然我们生活在一个三 三维空间，一个所谓的三维流形，等等， 对这些几何物体有一种自然的直觉， 这并不像我们所期望的那样，因为 这些问题已经通过代数计算解决， 高维流形仍然在3- 维度案例 其中最著名的是著名的 世纪之交的庞加莱猜想 关于三维领域，正是原来的3- 维度的案子是唯一一个还没结的 调查人员 正在研究各种各样的三维问题 流形，有些距离的概念有点奇怪， 他们，所谓的双曲线度量，但一次又一次，这些 问题已被证明与案件有明确的相关性， 流形与更熟悉的距离概念。

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.