Low-dimensional Geometry and Topology

在70年代后期，PI发起了一项几何结构的研究，作为破译三维拓扑结构的工具。中心原理是几何化猜想，它断言任何3-流形都有正则拓扑分解成具有八种可能的局部齐次几何结构之一的块。在过去的两三年里，这个项目取得了巨大的进展。最显著的进步是帕尔曼用里奇流建立几何化的证明（已被接受）。此外，对非紧双曲流形的几何性质也有了很大的进展，证明了双曲3-流形不存在紧叶理、基本片理和拟椭球形流，并取得了一些显著的否定结果。3-流形的几何分解使我们对单个3-流形有了牢固的掌握。随着几何化猜想的被接受，这门学科进入了探索三流形之间丰富的相互联系以及流形上各种结构的新阶段。一个重要的研究课题是进一步阐明紧叶理、紧接触结构、不同性质流动和双曲结构之间的关系。另一个有趣的领域是研究主三流形，一个层合集，它的叶子都是可能的双曲三流形，紧的和非紧的。第三个重要的主题是理解3-流形的有限层盖的格，特别是双曲3-流形是否为虚Haken、虚正betti和虚纤维的问题。最后，也许也是最重要的主题是发展一个更大的图景来解释几何化，使用结构，如三流形上的叶状束来构建它们的几何形状。3流形是一个有3个自由度的空间，所以它可以用三个变量局部描述。通过现在被普遍接受的几何化猜想，三流形的拓扑结构在三维几何的8种风格的晶体群中得到了漂亮的描述。事实上，在非欧几里德几何或双曲几何中，大多数3流形都是晶体群的识别空间。与标准（欧几里得）晶体学不同，双曲几何中有无限多的晶体群，关于它们的结构、分类和相互关系仍有许多未解之谜。这个项目将试图揭示和解释其中的一些相互关系。

During the late 70's the PI initiated a study of geometric structures as a tool for deciphering three-dimensional topology. The central principle has been the geometrization conjecture, asserting that any 3-manifold has a canonical topological decomposition into pieces that have one of eight possible kinds of locally homogeneous geometric structures. The last two or three years have seen dramatic progress in this program. The most notable progress is Perlman's proof (which has been gaining in acceptance) using Ricci flow to establish geometrization. In addition,there has also been great progress on understanding the geometry of noncompact hyperbolic manifolds through the proof of the tame end and ending lamination conjectures, and there have been some notable negative results establishing that certain hyperbolic 3-manifolds don't have taut foliations, essential laminations, or quasigeodesic flows. Geometric decompositions of 3-manifolds have given us a firm grip on individual 3-manifolds. With the acceptance of the geometrization conjecture, the subject is entering a new phase of exploration of the rich interconnections amongthree-manifolds and various structures on manifolds. One important topic for investigation is to further clarify the relationships among taut foliations, tight contact structures, flows with varying properties, and hyperbolic structures. Another interesting area is to investigate is the master three-manifold, a laminated set whose leaves are all possible hyperbolic 3-manifolds, compact and noncompact. A third important topic is to understand the lattices of finite-sheeted covers of 3-manifolds and in particular, the questions of whether hyperbolic 3-manifolds are virtually Haken, virtually positive betti, and virtually fibered. The final and perhaps most significant topic is to develop a bigger picture to explain geometrization, using structures such as a foliated bundles over three-manifolds to build their geometry.A 3-manifold is a space which has 3 degrees of freedom, so it can be locally described by three variables. Through the geometrization conjecture, which is now generally accepted, the topology of three-manifolds have a beautiful description in terms of crystallographic groups in 8 flavors of 3-dimensional geometry. In fact most 3-manifolds are the identification spaces of a crystallographic group in non-Euclidean or hyperbolic geometry. Unlike for standard (Euclidean) crystallography, there are infinitely many crystallographic groups in hyperbolic geometry, and many mysteries remain about their structure, their classification, and their interrelationships. This project will seek to uncover and explain some of these interrelationships.