Hyperbolic 3-manifolds

这个建议的动机是三维流形的拓扑和几何之间的统一。 它主要集中在双曲3-流形的定量几何，特别是在估计这样的数量不变量，体积，马古利斯数和直径的拓扑数据。这是一个与拓扑学和其他几个数学分支的联系正在发挥意想不到的作用的领域。 经典技术在3流形拓扑，其中一些可以追溯到Papakyriakopoulos的工作在20世纪50年代，成为特别强大的应用时，在双曲几何的背景下。这些拓扑思想与更多的几何和分析方法相互作用，如安德森、Canary、Culler和Shalen的log（2k-1）定理，双曲空间的等周不等式，Kleinian群的代数和几何收敛理论，Kojima和Miyamoto关于具有全测地边界的双曲流形的工作，以及它们之间的相互作用。以及Agol，Dunfield，Storm和Thurston的工作，他们将Ricci流的性质与外科手术应用于Haken流形和Dehn填充的研究。此外，令人惊讶的相互作用正在出现，通过拓扑结构，定量几何的双曲3流形和他们的数论方面，特别是他们的跟踪领域，这使得应用程序的深入结果数论的主题。非欧几何是一个经典的主题，在纯数学已经看到了显着的发展，在最近几十年。这个问题有其起源的企图，开始在古代，以证明欧几里得的第五公理可以推导出他的其他公理。在19世纪的过程中证明了这是不可能的：有一种数学结构叫做双曲平面（二维）或双曲空间（三维或多维），它满足除了第五公理之外的所有欧几里得公理，并且三角形的角之和总是小于180度。值得注意的是，双曲几何比欧几里得几何丰富得多。这就解释了自20世纪60年代以来双曲几何与其他数学和科学分支之间发展起来的各种各样的相互作用。 这些相互作用中的大多数涉及双曲流形的研究，双曲流形是具有双曲空间的小尺度几何但在大尺度上具有更复杂结构的几何对象。例如，在双曲空间中的直线，就像在欧几里得空间中一样，总是延伸到无穷远;但在双曲流形中，局部为直线的路径（称为测地线）可能会表现出像圆一样的全局“周期”行为。一些主要的研究者在双曲流形上的最早的工作产生了一个关于结的结果，这个结果被应用于研究重组DNA的结构。他们目前正在调查的各种方面的几何双曲流形和连接与其他一些主题上述。

This proposal is motivated by the unification between the topology and the geometry of three-dimensional manifolds. It is primarily focused on the quantitative geometry of hyperbolic 3-manifolds, specifically on estimating such quantitative invariants as volume, Margulis number and diameter in terms of topological data. This is an area in which connections with topology and several other branches of mathematics are playing unexpected roles. Classical techniques in 3-manifold topology, some of which go back to Papakyriakopoulos's work in the 1950s, become particularly powerful when applied in the context of hyperbolic geometry. These topological ideas interact with more geometric and analytic methods, such as the log(2k-1) Theorem of Anderson, Canary, Culler and Shalen; the isoperimetric inequality for hyperbolic space; the theory of algebraic and geometric convergence of Kleinian groups; the work of Kojima and Miyamoto on hyperbolic manifolds with totally geodesic boundary; and the work of Agol, Dunfield, Storm and Thurston which applies properties of the Ricci flow with surgeries to the study of Haken manifolds and Dehn filling. Furthermore, surprising interactions are emerging, via topology, between quantitative geometry of hyperbolic 3-manifolds and their number-theoretic aspects, specifically their trace fields; this has allowed applications of deep results in number theory to the subject.Non-Euclidean geometry is a classical topic in pure mathematics which has seen remarkable developments in recent decades. The subject had its origin in the attempt, begun in ancient times, to prove that Euclid's fifth axiom could be deduced from his other axioms. It was shown in the course of the 19th century that this cannot be done: there is a mathematical structure called the hyperbolic plane (in two dimensions) or hyperbolic space (in three or more dimensions) which satisfies all of Euclid's axioms except the fifth, and in which the sum of the angles of a triangle is always less than 180 degrees. Remarkably, hyperbolic geometry turns out to be much richer than Euclidean geometry. This accounts for the astonishingly varied interactions that have developed since the 1960's between hyperbolic geometry and other branches of mathematics and science. Most of these interactions involve the study of hyperbolic manifolds, which are geometric objects that have the small-scale geometry of hyperbolic space but have a more complicated structure in the large. For example, a straight line in hyperbolic space, as in Euclidean space, always extends to infinity; but in a hyperbolic manifold, a path that is locally a straight line (called a geodesic) may exhibit globally "periodic" behavior like a circle. Some of the principal investigators' earliest work on hyperbolic manifolds produced a result about knots that has been applied to study the structure of recombinant DNA. They are at present investigating a variety of aspects of the geometry of hyperbolic manifolds and connections with some of the other topics mentioned above.