Low-dimensional Geometry and Topology

在70年代后期，PI开始研究几何结构作为解读三维拓扑结构的工具。中心原理是几何化猜想，断言任何三维流形都有一个典型的拓扑分解，分解成八种可能的局部齐性几何结构之一。在过去的两三年里，这个项目取得了巨大的进展。最显着的进展是珀尔曼的证明（已获得接受）使用里奇流建立几何化。此外，通过证明非紧双曲流形的驯服和末端层合结构，对非紧双曲流形几何的理解也有了很大的进展，并且有一些显著的负面结果证明某些双曲3-流形不具有拉紧叶理、本质层合或拟短程流。三维流形的几何分解给了我们一个坚实的把握个别三维流形。随着几何化猜想的被接受，这门学科进入了一个新的阶段，即探索三维流形和流形上各种结构之间丰富的联系。研究的一个重要课题是进一步阐明拉紧叶理、紧密接触结构、变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维流形，一个层合集，其叶子都是可能的双曲三维流形，紧的和非紧的。第三个重要的课题是理解3-流形的有限片覆盖格，特别是双曲3-流形是否是虚哈肯、虚正贝蒂和虚纤维的问题。最后一个也可能是最重要的主题是发展一个更大的图景来解释几何化，使用诸如三流形上的叶状丛这样的结构来构建它们的几何。三流形是一个有三个自由度的空间，所以它可以用三个变量局部描述。通过几何化猜想，三维流形的拓扑结构在8种三维几何中得到了一个漂亮的晶体群描述。事实上，大多数3-流形是非欧几何或双曲几何中晶体群的识别空间。与标准的（欧几里得）晶体学不同，双曲几何中有无限多个晶体群，它们的结构、分类和相互关系仍有许多未解之谜。本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.