Mathematical Sciences: The Topology and Geometry of 3- Dimensional Manifolds

这个项目是在结理论和拓扑领域 三维流形 调查人员将研究 三维流形，使用两个观点来分析曲面， 三歧管 第一种方法使用组合技术 结合结的“薄位置”的概念。 这 将被用来考虑纽结理论中的一些老问题，如 就像布线猜想和分类问题一样， Seifert纤维空间的Heegaard分裂 薄位 曲面作为极大极小曲面的组合形式 在他们对海加德的调查中， 分裂，它们将被用来解决类似于 3-球体的识别问题，最近由 鲁宾斯坦 第二种方法使用光滑极小 曲面理论和双曲几何来研究 3-流形的拓扑 一个目标是证明任何三维流形 每个亏格只允许有限个Heegaard分裂， Johannson最近为哈肯流形建立的结果。 另一个目标是证明非Haken三维流形是确定的 他们的基本群体。 最后，调查人员将 探索由强加于人而产生的拓扑后果 流形及其边界上的曲率条件。 三维的几何学是 我们生活的世界，所以人们可能会认为数学研究 这一领域将有特别广泛的应用。 这是 事实上，这一领域在物理学中的应用， 在研究控制物理的微分方程时， 现象，群论，和许多其他分支的 数学 数学家们在各个维度上研究 称为“流形”的几何物体， 类似于我们生活的空间。 维数流形 三个表现出独特的特点，使他们的研究， 目前研究的热点领域。 的目的 项目是研究这些三维流形的某些类别，在 追求理解这些几何的总体目标， 对象 利用从研究打结曲线发展起来的技术 以及它们的复杂性，从肥皂的理论来看， 电影和最小的表面，调查人员的目标是贡献 三维流形的分类问题， 具体类别的三维流形的详细信息。

This project is in the area of knot theory and the topology of3-dimensional manifolds. The investigators will study 3-manifolds, using two points of view to analyze surfaces in a 3-manifold. The first method uses combinatorial techniques combined with the concept of 'thin position' of a knot. This will be used to consider some old problems in knot theory, such as the cabling conjecture, as well as the classification problem for Heegaard splittings of Seifert fibered spaces. Thin position surfaces serve as a combinatorial version of the minimax surfaces used by Pitts-Rubinstein in their investigations of Heegaard splittings, and they will be used to attack problems similar to the recognition problem for the 3-sphere, recently solved by Rubinstein. The second method uses techniques of smooth minimal surface theory and hyperbolic geometry to investigate the topology of 3-manifolds. One goal is to show that any 3-manifold admits only a finite number of Heegaard splittings of each genus, a result recently established for Haken manifolds by Johannson. Another goal is to show thatnon-Haken 3-manifolds are determined by their fundamental groups. Finally, the investigators will explore the topological consequences that follow from imposing curvature conditions on a manifold and its boundary. The geometry of three dimensions is the geometry of the world we live in, so one might expect that the mathematical study of this area would have particularly broad applications. This is indeed the case, and the area leads to applications in physics, in the study of the differential equations governing physical phenomena, to group theory, and to many other branches of mathematics. Mathematicians study, in all dimensions, the geometric objects called "manifolds," which have properties similar to those of the space we live in. Manifolds in dimension three exhibit unique features which make their study a flourishing area of current research. The objective of this project is to study certain categories of these 3-manifolds, in pursuit of the overall goal of understanding these geometric objects. Using techniques developed from studying knotted curves and the complexity of their raveling, and from the theory of soap films and minimal surfaces, the investigators aim to contribute to the classification problem for3-manifolds and to understand specific classes of 3-manifolds in great detail.