Three-Manifolds and Geometry

三维流形是与我们生活的宇宙密切相关的空间。为了更好地理解这种空间，人们需要几个研究领域的方法，如几何拓扑学、代数、数论和理论物理。流形的不变量是仅依赖于流形类型并携带有用信息的关联对象。这个由美国国家科学基金会资助的项目解决了与三维流形的分类和数论流形不变量的实现有关的几个重要的、尚未解决的问题。双Lipschitz几何是研究紧致流形泛覆盖的一个自然框架，它将几何拓扑学与几何群论联系起来。它也是研究代数集局部几何的自然框架。在这两种情况下，三维流形都扮演着重要的角色。该项目的一个目标是完成3-流形群的准等距分类，这是一个许多研究人员已经工作了20年的项目，PI和Behrstock在过去几年中取得了非常重要的进展。另一种是将复奇点的Lipschitz几何应用于重要的开放问题，如高维Zariski等值性的几何意义(Pi和Anne Pichon最近解决了2维情况)、Zariski多重性猜想、复杂映射芽的分类等。在光谱的另一端，对于双曲流形，流形的几何、算术、表示理论和基于量子的不变量之间也存在假定的联系，PI和他的学生在这方面取得了进展，而且还在继续。

Three-manifolds are spaces closely related to the universe we live in. To develop a better understanding of such spaces one calls upon methods from several research areas, such as geometric topology, algebra, number theory, and theoretical physics. An invariant of a manifold is an associated object that depends only on the manifold type, and carries useful information. This NSF funded project addresses several important, unsolved problems related to classification of three-manifolds and realization of number-theoretic manifold invariants. Bi-Lipschitz geometry is a natural framework for the geometric study of universal covers of compact manifolds, where it connects geometric topology with geometric group theory. It is also the natural framework for the study of the local geometry of algebraic sets. In both situations, three-dimensional manifolds play an important role. One aim of the project is to complete the quasi-isometric classification of 3-manifold groups, a project on which a large number of researchers have been working for two decades, and on which the PI and Behrstock made very significant progress over the last several years. Another is to apply Lipschitz geometry of complex singularities to significant open questions, such as the geometric meaning of Zariski equisingularity in higher dimensions (the PI and Anne Pichon recently resolved the 2-dimensional case), the Zariski multiplicity conjecture, classification of complex map germs and more. At the other end of the spectrum, for hyperbolic manifolds there are also postulated connections between geometric, arithmetic, representation-theoretic and quantum based invariants of manifolds, on which the PI and his students have made progress, and which is continuing.