Mathematical Sciences: Geometry and Topology of Foliations and Flows in 3-Manifolds

小行星9612317 在三维流形的拓扑学研究中，叶层是一个基本的工具。 闭3-流形M中的Reebless-dimensional foliation F通过泛覆盖中的拓扑平面提升到foliation（F1）。 本计画拟研究F1的拓扑结构、F1的几何性质，以及它们与M的拓扑与几何的关系。 特别感兴趣的是研究的大规模几何和渐近行为（当M是双曲）的叶F1。 要考虑的第二个主题包括流动3歧管和大规模的几何形状的流线时，解除了普遍的覆盖。 主要研究者已经构建了许多例子度量有效（称为quasigodesic）伪Anosov流在双曲3流形，现在提出研究的后果quasigodesic属性的几何形状的稳定和不稳定的叶理与这些流量和有限深度的叶理，这些quasigodesic流量几乎是横向的。 叶理是一个n维的对象（称为流形）分解成一个不相交的维数小于n的对象的并集（称为叶理的叶子），就像一本书（3维）是它的页面（2维）的并集一样。 在数学中，这些对象是很常见的;例如，没有固定轨道的流是一维的叶（叶子是流线，它们具有一维）。 流形理论的一个一般原则是：维数越小，越容易理解。 因此叶理是非常有用的;理解叶以及它们如何组合在一起形成流形，人们就部分地理解了流形。这个项目试图理解树叶的几何形状如何与流形的几何形状相互作用;在一本书的情况下，树叶平放在流形上，但一般来说，它们的几何行为要复杂得多。 首席研究员希望描述一种特殊的流形称为双曲流形的几何行为。

9612317 Fenley Foliations are a fundamental tool in the study of the topology of 3-manifolds. A Reebless two dimensional foliation F in a closed 3-manifold M lifts to a foliation (F1) by topological planes in the universal cover. This project proposes to study the topological structure of F1, the geometric properties of F1, and how these relate to the topology and geometry of M. Of special interest is the study of the large scale geometry and asymptotic behavior (when M is hyperbolic) of leaves of F1. A second topic to be considered consists of flows in 3-manifolds and the large scale geometry of flow lines when lifted to the universal cover. The principal investigator has constructed many examples of metrically efficient (called quasigeodesic) pseudo-Anosov flows in hyperbolic 3-manifolds and now proposes to study the consequences of the quasigeodesic property for the geometry of the stable and unstable foliations associated to these flows and the finite depth foliations to which these quasigeodesic flows are almost transverse. A foliation is a decomposition of an object of dimension n (called a manifold) into a disjoint union of objects of dimension less than n (called the leaves of the foliation), much the same as a book (dimension 3) is the union of its pages (of dimension 2). In mathematics these objects are quite common; for instance, a flow without fixed orbits is a foliation of dimension 1 (the leaves are the flow lines and they have dimension 1). A general principle of manifold theory is the following: the smaller the dimension, the easier to understand. Hence foliations are very useful; understand the leaves and how they are put together to form the manifold and one partially understands the manifold. This project attempts to understand how the geometry of the leaves interacts with the geometry of the manifold; in the case of a book, the leaves sit flat in the manifold, but in general their geometric behavior is much more complex. The principal investigator hopes to describe this geometric behavior for a special class of manifolds called hyperbolic manifolds.