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

数学科学：双曲3流形的几何、拓扑和算术

• 批准号: 9625958
• 负责人: Alan Reid
• 金额: $ 5.94万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-08-01 至 2000-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9625958&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Geometry; Topology; Arithmetic

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. ***

低维拓扑学的基本问题之一是理解具有有限体积的完全双曲结构的三维流形的几何和拓扑。特别感兴趣的是双曲3流形中不可压缩曲面的性质，以及数论和代数与双曲3流形之间的相互作用。前者研究了双曲3-流形中不可压缩曲面的存在和构造，后者研究了与双曲3-流形相关的代数数据中产生的某些不变量如何与该3-流形的几何和拓扑相关。三维流形曲面的研究在三维拓扑学中占有重要的地位。事实上，许多关于3流形拓扑的最深刻的结果都来自于对曲面如何映射成3流形的研究。除了在3流形拓扑中的应用外，3流形表面的研究在物理学中也有许多有趣的应用。上述数论联系近年来在证明双曲3-流形的定理中被证明是重要的。最简单的数论联系之一是通过双曲3流形的体积。这与二重对数函数密切相关，二重对数函数以多种形式出现在数学和物理的其他分支中。* * *