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

点击翻译按钮获取中文摘要

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.