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

这个项目是在结理论和三维流形的拓扑领域。研究人员将研究3流形，使用两种观点来分析3流形中的表面。第一种方法使用组合技术结合结的“薄位置”的概念。这将用于考虑结理论中的一些老问题，如布线猜想，以及Seifert纤维空间的横向分裂的分类问题。薄位置曲面是Pitts-Rubinstein在研究Heegaard分裂时使用的极大极小曲面的组合版本，它们将用于解决类似于Rubinstein最近解决的3球识别问题的问题。第二种方法利用光滑最小曲面理论和双曲几何技术来研究3流形的拓扑结构。一个目标是证明任何3-流形只允许每个属的有限数量的Heegaard分裂，这是johnson最近为Haken流形建立的一个结果。另一个目标是证明非haken 3-流形是由它们的基本群决定的。最后，研究人员将探讨在流形及其边界上施加曲率条件所带来的拓扑结果。三维几何是我们生活的世界的几何，所以有人可能会认为这个领域的数学研究将有特别广泛的应用。事实确实如此，这个领域导致了物理学的应用，在研究控制物理现象的微分方程、群论和许多其他数学分支中。数学家在所有维度上研究被称为“流形”的几何物体，它们具有与我们生活的空间相似的性质。三维流形以其独特的特性成为当前研究的热点。这个项目的目标是研究这些3-流形的某些类别，以追求理解这些几何物体的总体目标。利用从研究打结曲线及其移动的复杂性中发展起来的技术，以及从肥皂膜和最小表面理论中发展起来的技术，研究人员的目标是为3-流形的分类问题做出贡献，并详细了解3-流形的特定类别。

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.