Topology, geometry and arithmetic of hyperbolic 3-manifolds

Culler和Shalen将继续他们对双曲3流形的研究。他们工作的主要主题之一是这种流形的拓扑定义不变量与它们的定量几何不变量（如体积）之间的联系。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Anderson， Canary， Culler和Shalen的log（k-1）定理，Kojima和Miyamoto关于完全测地边界的双曲流形的工作，以及Agol， Dunfield， Storm和Thurston基于里奇流的性质与手术的工作。第二个主题，最近从第一个主题发展而来，是流形的数论不变量（如其迹场）与定量几何不变量（如马古利斯数）之间的联系。这方面的工作依赖于将早期的工作与新的群论观察相结合，并且已经发挥了诸如西格尔和马勒关于代数数域单位方程的工作等深刻的数论成分。双曲流形是在数学的许多分支和许多数学应用中出现的几何对象。第一个被称为双曲空间的双曲流形是在19世纪发现的，它否定地解决了欧几里得第五公设能否从他的其他公设中推导出来的古老问题。双曲流形可以被认为是几何对象，在小尺度上与双曲空间难以区分，但其大尺度行为更为复杂。现代几何的一个主要主题是几何物体的定量特性（例如用距离、长度、面积和体积来定义的那些特性）与它们的“拓扑”特性之间的相互作用。拓扑特性更定性，并且在物体变形时不会改变。就双曲流形而言，近年来在定量理论和拓扑理论的联系方面取得了很大进展，可以说它们在抽象层面上是完全统一的。目前的项目包括让我们以更具体的方式理解这种联系。事实证明，这样做涉及到数学许多分支的深奥思想。

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.