Surfaces in low-dimensional topology

研究者建议使用二维叠层和叶理作为基本工具来研究三维流形的拓扑。主要目的是展示如何使用这些对象来攻击低维拓扑中的几个长期存在的拓扑结构。特别是，他建议使用的基本分层结构，由于研究者和其他人，分类的特殊德恩手术交替结。他计划使用叠层来证明一些大类的3流形满足虚哈肯猜想。他还将继续致力于更好地理解缝合流形分解的结构，这是建立和使用拉紧叶理的基本技术，通过开发具有大深度的双曲结的例子。我们生活在一个三维宇宙中，并在其中移动，我们认为它在各个方向上无限延伸。但最近的观测暗示，空间可能会自我缠绕，这样我们就生活在一个有限的宇宙中。几何拓扑学的核心正是这种空间自我缠绕的思想，其目标是理解局部看起来像普通欧几里得空间的物体的全局结构。研究者计划通过使用二维表面来研究可以发现它们的空间，研究旨在揭示三维空间的整体结构模式的几个问题。主要思想是使用表面作为将空间切割成碎片的方法，通过研究所产生的碎片，我们可以洞察三维空间的大尺度结构。

The investigator proposes to study the topology of 3-dimensional manifolds,using 2-dimensional laminations and foliations as the basic tools. The maingoal is to show how these objects can be used to attack several long-standingconjectures in low-dimensional topology. In particular, he proposes to useconstructions of essential laminations, due to the investigator and others,to classify the exceptional Dehn surgeries on alternating knots. Heplans to use laminations to show that some large classes of 3-manifoldssatisfy the Virtual Haken Conjecture. He will also continue work aimed at providing a better understanding of the structure of sutured manifolddecompositions, a fundamental technique in building and using taut foliations,by developing examples of hyperbolic knots with large depth.We live in and move through a 3-dimensional universe, which we perceiveas extending infinitely in all directions. But recent observations hint at thepossibility that space may instead wrap around itself, so that we live in a finite universe. It is this very idea of a space wrapping around itself that lies at theheart of geometric topology, whose goal is to understand the global structure of objects which locally look like ordinary Euclidean space. The investigator plans to study several questions aimed at uncovering patterns in the global structure of 3-dimensional spaces, by using 2-dimensional surfaces to study the spaces in which they can be found. The main idea is to use surfaces as away of cutting the space into pieces; by studying the resulting pieces, we can gain insight into the large-scale structure of the 3-dimensional space.