Mathematical Sciences: Geometry, Topology and Arithmetic of Hyperbolic 3-Manifolds

小行星9625958 低维拓扑中的一个基本问题是 理解三维流形的几何和拓扑，允许有限体积的完全双曲结构。 特别感兴趣的是双曲3流形中不可压缩曲面的性质，以及数论和代数之间的相互作用，以及双曲3流形。 前者涉及到对存在和建构的研究， 双曲三维流形中的不可压缩曲面，后者 集中在如何某些不变量产生的代数数据 与双曲3流形相关的是与几何和 3-流形的拓扑 三维流形中曲面的研究在三维拓扑学中起着重要的作用。 事实上，三维流形拓扑学中许多最深刻的结果都是从曲面如何映射到三维流形的研究中产生的。 除了在三维流形拓扑学中的应用外，三维流形中曲面的研究在物理学中也有许多有趣的应用。 上面提到的数论联系在近年来证明双曲3-流形定理中很重要。 最简单的数论联系之一是通过双曲三维流形的体积。 这与双对数函数密切相关，双对数函数以多种形式出现 数学和物理学的其他分支。 ***

9625958 Reid One of the fundamental problems in low-dimensional topology is to understand the geometry and topology of 3-dimensional manifolds admitting a complete hyperbolic structure of finite volume. Of particular interest are properties of incompressible surfaces in hyperbolic 3-manifolds, and interactions between number theory and algebra on the one hand and hyperbolic 3-manifolds on the other. The former involves the study of the existence and construction of incompressible surfaces in hyperbolic 3-manifolds, and the latter concentrates on how certain invariants arising from algebraic data associated to a hyperbolic 3-manifold are related to the geometry and topology of the the 3-manifold. The study of surfaces in 3-manifolds plays an important role in 3-dimensional topology. Indeed, many of the deepest results in 3-manifold topology have arisen from the study of how surfaces map into 3-manifolds. Apart from their application in 3-manifold topology, the study of surfaces in 3-manifolds has had many interesting applications in physics. The number theoretic connections alluded to above have proved important in recent years in proving theorems about hyperbolic 3-manifolds. One of the simplest number theoretic connections is through the volume of a hyperbolic 3-manifold. This is closely connected to the Dilogarithm function, which appears in many guises in other branches of mathematics and physics. ***