Topology, geometry and arithmetic of hyperbolic 3-manifolds

卡勒和沙伦将继续他们对双曲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的不变量与它们的定量几何不变量(如体积)之间的联系。这涉及到三维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1950‘S的工作，更多的几何方法，如Anderson，Canary，Culler和Shalen的LOG(2k-1)定理，小岛和宫本在具有完全测地线边界的双曲流形上的工作，以及Agol，Dunfield，Storm和瑟斯顿基于外科手术的Ricci流的性质的工作。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(如它的迹场)和定量几何不变量(如它的Marguis数)之间的联系。这方面的工作依赖于将前人的工作与新的群论观察相结合，并已经发挥了像Siegel和Mahler关于代数数域上的单位方程的工作这样深刻的数论成分。双曲流形是出现在数学的许多分支和许多数学应用中的几何对象。第一个双曲流形，被称为双曲空间，在19世纪被发现，并解决了-否定-欧几里得的第五公设是否可以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。现代几何学中的一个主要主题是几何对象的定量属性(例如，根据距离、长度、面积和体积定义的属性)与其更具定性且在对象变形时不变的“拓扑”属性之间的相互作用。在双曲流形的情况下，近年来在将定量理论和拓扑理论联系起来方面取得了如此大的进展，以至于它们可以说在抽象水平上是完全统一的。目前的项目涉及以更具体的方式使我们对这种联系的理解。事实证明，这样做涉及到许多数学分支的深刻想法。

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.