Hyperbolic Geometry, Heegaard Surfaces, Foliation/Lamination Theory, and Smooth Four-Dimensional Topology

低维拓扑学研究四维或更少维的流形，是数学的一个中心、活跃领域。许多成功用于研究高维流形的数学工具并不适用于低维流形，尽管在过去四十年中取得了进展，但基本的、重要的问题仍未解决。低维拓扑学是许多数学分支的联系。来自几何、最小曲面理论和分析、群论、数论、动力系统和理论计算机科学的方法为其发展做出了贡献，相反，低维拓扑的研究也促进了这些领域的进步。本研究课题涉及双曲几何、层积与叶理理论、Heegaard理论和光滑四维流形理论。研究中的许多问题对研究生和新研究人员都是开放的，其中一些人将参与该项目。研究者将继续研究末端层合空间的拓扑结构，小尖头和小体积双曲3-流形的分类，以及Heegaard分裂的分类。通过一种新颖的方法，该项目将研究光滑的四维舍恩菲斯猜想：在标准四球边界内的任何光滑嵌入的三球都是光滑的标准四球。

Low-dimensional topology, which studies manifolds of four or fewer dimensions, is a central, active area of mathematics. Many of the mathematical tools successfully used to study high-dimensional manifolds do not apply in low dimensions, and despite advances in the last forty years, fundamental, important problems remain unresolved. Low-dimensional topology is at the nexus of many branches of mathematics. Methods from geometry, minimal surface theory and analysis, group theory, number theory, dynamical systems, and theoretical computer science have contributed to its development, and conversely, research in low-dimensional topology stimulates advances in these areas. This research project addresses topics in hyperbolic geometry, lamination and foliation theory, Heegaard theory, and smooth 4-dimensional manifold theory. Many of the questions under study are accessible to graduate students and new researchers, several of whom will be involved in the project.The investigator will continue research on the topology of ending lamination space, the classification of both small cusped and low volume hyperbolic 3-manifolds, and the classification of Heegaard splittings. Through a novel approach, the project will investigate the smooth 4-dimensional Schoenflies conjecture: any smoothly embedded 3-sphere in the standard 4-sphere bounds a smoothly standard 4-ball.