Topology, geometry and arithmetic of hyperbolic 3-manifolds

双曲3流形的拓扑、几何和算术

• 批准号: 0906155
• 负责人: Marc Culler
• 金额: $ 27.67万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-01 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0906155&HistoricalAwards=false
• 关键词: Topology; geometry; arithmetic; hyperbolic; manifolds

Culler and Shalen will continue their research on hyperbolic 3-manifolds. One of the main themes of their work is the connection between topologically defined invariants of such manifolds and their quantitative geometric invariants such as volume. This has involved interactions between very classical techniques in 3-manifold topology, some of which go back to Papakyriakopoulos's work in the 1950's, and more geometric methods such as the log(2k-1)-theorem of Anderson, Canary, Culler and Shalen, the work of Kojima and Miyamoto on hyperbolic manifolds with totally geodesic boundary, and the work of Agol, Dunfield, Storm and Thurston based on properties of the Ricci flow with surgeries. A second theme, which recently has grown out of the first, is the connection between the number-theoretic invariants of a manifold such as its trace field and quantitative geometric invariants such as its Margulis number. This aspect of the work depends on combining the earlier work with new group-theoretic observations, and has already brought into play such deep number-theoretic ingredients as the work of Siegel and Mahler on the unit equation in algebraic number fields.Hyperbolic manifolds are geometric objects that arise in many branches of mathematics and in many applications of mathematics. The first hyperbolic manifold, called hyperbolic space, was discovered in the 19th century and settled---in the negative---the ancient problem of whether Euclid's fifth postulate could be deduced from his other postulates. Hyperbolic manifolds may be thought of as geometric objects which at small scales are indistinguishable from hyperbolic space, but whose large-scale behavior is more complicated. A major theme in modern geometry is the interaction between the quantitative properties of a geometric object, for example those defined in terms of distances, lengths, areas and volumes, and their "topological"properties which are more qualitative and are unchanged when the object is deformed. In the case of hyperbolic manifolds, so much progress has been made in recent years in relating the quantitative and topological theories that they may be said to be completely unified at an abstract level. The present project involves making our understanding the connection in a more concrete way. Doing this turns out to involve deep ideas from many branches of mathematics.

Culler和Shalen将继续他们对双曲三维流形的研究。他们的工作的主要主题之一是拓扑定义的不变量之间的连接等流形和定量几何不变量，如体积。这涉及到3-流形拓扑中非常经典的技术之间的相互作用，其中一些可以追溯到Papakyriakopoulos在20世纪50年代的工作，以及更多的几何方法，如安德森，金丝雀，Culler和Shalen的log（2k-1）-定理，Kojima和Miyamoto关于具有全测地边界的双曲流形的工作，以及Agol，Dunfield，Storm和Thurston基于手术的Ricci流的性质。第二个主题，最近已经成长出的第一个，是数论不变量之间的连接流形，如其跟踪领域和定量几何不变量，如其马古利斯数。这方面的工作取决于结合早期的工作与新的群论的意见，并已发挥这种深刻的数论成分的工作西格尔和马勒的单位方程在代数数域。双曲流形是几何对象，出现在许多分支的数学和许多应用的数学。第一个双曲流形，称为双曲空间，是在19世纪世纪发现的，并以否定的方式解决了欧几里得的第五公设是否可以从他的其他公设推导出来的古老问题。双曲流形可以被认为是在小尺度下与双曲空间无法区分的几何对象，但其大尺度行为更为复杂。现代几何学的一个主要主题是几何对象的定量属性之间的相互作用，例如距离，长度，面积和体积，以及它们的“拓扑”属性，这些属性更具定性，并且在对象变形时不变。在双曲流形的情况下，近年来在定量理论和拓扑理论的联系上取得了很大的进展，可以说它们在抽象的层次上是完全统一的。本项目涉及以更具体的方式使我们理解这种联系。这样做涉及到数学许多分支的深刻思想。