Hyperbolic 3-manifolds

双曲 3 流形

• 批准号: 1207720
• 负责人: Marc Culler
• 金额: $ 37.5万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-08-15 至 2016-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1207720&HistoricalAwards=false
• 关键词: Hyperbolic; manifolds

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.

这一建议的动机是三维流形的拓扑结构和几何结构的统一。主要研究双曲型3流形的定量几何，特别是用拓扑数据估计体积、马古利数和直径等定量不变量。这是一个与拓扑学和其他几个数学分支的联系发挥着意想不到的作用的领域。3流形拓扑中的经典技术，其中一些可以追溯到Papakyriakopoulos在20世纪50年代的工作，在双曲几何的背景下应用时变得特别强大。这些拓扑学思想与更多的几何和解析方法相互作用，例如Anderson、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关于具有完全测地线边界的双曲流形的工作；以及Agol， Dunfield， Storm和Thurston的工作，他们将里奇流的特性与手术应用到Haken流形和Dehn填充的研究中。此外，通过拓扑，双曲3流形的定量几何与它们的数论方面，特别是它们的迹域之间出现了令人惊讶的相互作用；这使得数论的深奥结果得以应用于这一学科。非欧几里得几何是纯数学中的一个经典课题，近几十年来有了显著的发展。这个主题的起源是在古代开始的尝试，证明欧几里得的第五个公理可以从他的其他公理推导出来。19世纪的研究表明，这是不可能做到的：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），它满足欧几里得公理，除了第五公理，三角形的内角之和总是小于180度。值得注意的是，双曲几何比欧几里得几何丰富得多。这解释了自20世纪60年代以来，在双曲几何和其他数学和科学分支之间发展起来的令人惊讶的各种相互作用。这些相互作用大多涉及对双曲流形的研究，这些几何对象具有双曲空间的小尺度几何，但在大尺度上具有更复杂的结构。例如，一条直线在双曲空间，如在欧几里得空间，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“周期性”行为。一些主要研究人员对双曲流形的早期研究产生了关于结的结果，该结果已被应用于研究重组DNA的结构。他们目前正在研究双曲流形几何的各个方面以及与上面提到的一些其他主题的联系。