Laminations, Foliations and Flows in 3-Manifolds

摘要：“三维流形中的层理、层理和流动”本质层理和无叶层理是研究三维流形的一个基本和基本的工具。它们的应用得到了关于三维流形拓扑的深刻结果，也与三维流形的几何猜想有关。一个目标是分析三维流形，特别是双曲三维流形中层合的存在性问题。PI最近证明了存在无穷多个不允许本质分层的双曲3-流形。一个自然的问题是，基本的薄片有多常见。该项目将研究Dehn手术对圆环束和一般表面束以及特殊类型的火焰/叶状结构的影响。该项目还将考虑更一般的结构，如松散的层合板和它们具有的特性。该项目的另一部分是了解双曲三维流形中的叶状结构和横向伪Anosov流的几何行为--这是一类非常大的流形。重点将放在通用盖中的大尺度几何行为上，这是此类流形的基础。一个重要的问题是，这种流动是否是准奇异性的--在某些情况下，PI和Lee Mosher之前的工作证明了这一点。一个目标是使用流的准偶极线性质来获得关于横切于流的叶理的渐近行为的信息。这在一般有限深度叶面的情况下尤其有希望。另一个目标是将这些性质与叶面的普适圆联系起来--在叶面的整体结构和普适覆盖的全球几何之间建立联系。另一个项目是研究双曲流形中有限叶集的性质。该项目旨在更好地理解三维流形。三维流形在科学中被广泛应用：例如，结及其性质与DNA研究非常相关。3-流形也被用来获得大脑和其他可视化技术的映射。了解3-流形可以在这些其他领域取得进展。该项目专注于层片和叶状结构，这是低维拓扑中一个基本的和当前非常动态的领域。拟议的项目将对该领域的一些主要研究方向产生影响。一个重要的目标是在这个令人兴奋的领域吸引学生和博士后研究人员。

Abstract."Laminations, foliations and flows in 3-manifolds" Essential laminations and Reebless foliations are a basic and fundamental tool in the study of 3-manifolds. Their use yields deep results concerning 3-manifold topologyand they are also related to the geometrization conjecturefor 3-manifolds. One goal is to analyse the existencequestion for laminations in 3-manifolds, particulary forhyperbolic 3-manifolds. The PI has recently shown thereare infinitely many hyperbolic 3-manifolds which do not admit essential laminations. A natural question is how common are essential laminations. The project will investigate Dehn surgery on torus bundles over the circleand general surface bundles and also special types oflaminations/foliations. The project will also considermore general structures, such as loosesse laminations andwhich properties they have. Another part of the project isto understand the geometric behavior of foliations andtransverse pseudo-Anosov flows in hyperbolic 3-manifolds - which is a very large class of manifolds. The focus will be on the large scale geometric behavior in the universal cover, which is fundamental for such manifolds. An important question is whether such flows are quasigeodesic - this is true in certain cases by previous work of the PI and Lee Mosher. One goal is to use the quasigeodesic property for the flows to derive information about the asymptotic behavior of the foliation transverse to the flow. This is speciallypromising in the case of general finite depth foliations.Another goal is to connect these properties with the universalcircle of the foliation - establishing a link between the global structure of the foliation and the global geometry of theuniversal cover. Another project is to study properties oflimit sets of leaves of foliations in hyperbolic manifolds. The project aims to better understand 3-manifolds.3-manifolds are widely used in the sciences: for exampleknots and their properties are very relevant to DNAresearch. 3-manifolds are also used to get mappings of thebrain and in other visualization techniques. Understanding3-manifolds can lead to progres in these other areas.The project focuses on laminations and foliations, whichare an essential and currently extremely dynamic area inlow dimensional topology. The proposed projects will have an effect on some main research directions in the area.One important goal is to attract students and postdoctoral researchers in this exciting area.