Topics in low-dimensional topology

Long计划从事广泛的项目，探索一系列几何、拓扑和数论问题。要探索的大多数问题都受到低维拓扑的启发，尽管其中一些问题适合更广泛的背景。这些内容包括覆盖空间的谱理论和同调的行为，通约器的结构，迹域的算法，实射影结构的变形和极限，以及相似区间交换映射在几何无限曲面群中的作用。统一的主题是将双曲流形在低维和高维中的许多不同方面联系在一起。尽管佩雷尔曼的工作将拓扑学和几何学结合在一起，但我们对这种几何学的理解还有很长的路要走。在本项目提出的方向上取得的进展将大大增进这种理解。如果一个空间是由小块组成的，那么它就被称为3流形，这些小块都“像”我们生活的普通三维空间。数学家知道如何非常精确地解释“like”，在这种情况下，有两个非常不同的概念非常重要：“拓扑类”和“几何类”。Perelmann最近的研究证实了Thurston在70年代提出的一个长期存在的猜想，即拓扑图和几何图密切相关。这是全球共识的重要组成部分。在几何流形中，到目前为止最重要的一类是双曲流形。这门课在数学的许多领域都很普遍，从低维拓扑，到动力系统，再到数论。事实上，这门课在物理学中也是至关重要的。然而，即使有了Perlemann的工作，人们对双曲流形本身的了解仍然相当少，尽管它们的重要性使它们成为了30多年来研究的磁石。这一建议直接针对双曲流形的结构研究方面和许多与之相关的问题。例如，他打算继续研究一个著名的古老猜想，即哪些二维物体可以存在于这些三维物体中。

Long plans to work on a wide spectrum of projects which explore a range of geometrical, topological and number-theoretical problems. Most of the issues to be explored are inspired by low dimensional topology, although several of the problems fit into a much wider context. These include the behavior of the spectral theory and homology of covering spaces, the structure of commensurators, the arithmetic of trace fields, the deformations and limits of real projective structures, and the role of similarity interval exchange maps in geometrically infinite surface groups. The unifying theme is the tying together of the many disparate aspects of hyperbolic manifolds in both low and higher dimensions. For while Perelman's work has drawn together the topological and geometric, our understanding of this geometry still has a long way to go. Progress in the directions proposed in this project would add significantly to this understanding.A space is called a 3-manifold if it is made of small chunks all of which are ``like'' the ordinary 3-dimensional space that we live in. Mathematicians understand how to interpret "like" very precisely and there are two very different notions which are of great importance in this setting: "topologically-like" and "geometrically-like". The recent work of Perelmann has verified a long-standing conjecture which emerged in the seventies with the work of Thurston, namely that the topological and geometrical pictures are intimately related. This is an important piece of global understanding. Amongst geometrical manifolds, by far the most important are the class called hyperbolic manifolds. This class is ubiquitous in many areas of mathematics, ranging from low-dimensional topology, to dynamical systems, to number theory. Indeed, this class is also crucially important in physics. However, even with Perlemann's work, hyperbolic manifolds themselves are still fairly poorly understood, although their importance have made them a magnet for research for well over thirty years. This proposal directs itself towards aspects of the structural study of hyperbolic manifolds and many issues related to them. For example, he intends to continue work on a famous old conjecture about which sorts of two-dimensional objects can live inside these 3-dimensional objects.