Beyond the Thurston Geometries

超越瑟斯顿几何

• 批准号: 1308184
• 负责人: Steven Kerckhoff
• 金额: $ 17.87万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-15 至 2017-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1308184&HistoricalAwards=false
• 关键词: Beyond; Thurston; Geometries

In this project we will study geometric structures and the structure of representation varieties. These topics are a natural outgrowth of previous research programs that focused on low-dimensional hyperbolic manifolds. The areas of Kleinian groups and the geometry of 3-manifolds have seen an amazing amount of progress over the last decade. As a result these areas are in a position to refocus their efforts. The situation has become more like that of surfaces where one is interested in families and spaces of structures. It is important to try to understand the genealogy of 3-manifolds, how they are related by various topological and geometric operations. This has led to the study of many other types of geometric structures, such as projective and Lorentzian structures, that don't have an invariant metric. These structures are interesting in their own right. They also provide a context in which to view the relation between the different eight 3-dimensional metric geometries, leading to the concept of transitional geometry. Furthermore, there are interesting questions about geometric structures in other dimensions, involving an array of different Lie groups, such as hyperbolic structures in high dimensions, complex and real projective structures on surfaces, and representations of surface groups into higher rank Lie groups.The idea of studying various types of metrics on spaces dates back at least to the late 19th century and the work of Poincare, Klein, and others. Much of their motivation came from the desire to understand physical phenomena. Modern physics, beginning with Einstein, has led to an even greater need for sophisticated mathematics to understand the physical universe, particularly that coming from metric geometry. Although the physical world is not a completely homogeneous one like the type of structures studied in this project, hyperbolic and Lorentzian geometry are believed to represent useful models for understanding physical phenomena. Geometry in dimension 3 is particularly appealing because it is visually accessible to many people, including beginning mathematics students and those with less technical backgrounds. It also has led to the creation of a number of graphical interfaces that have been widely utilized.

在这个项目中，我们将研究几何结构和表示簇的结构。这些主题是以前专注于低维双曲流形的研究项目的自然产物。在过去的十年里，Kleian群的面积和三维流形的几何学取得了惊人的进展。因此，这些领域能够调整其努力的重点。这种情况变得更像是人们对结构族和空间感兴趣的表面。重要的是要试图理解3-流形的谱系，它们是如何通过各种拓扑和几何运算联系在一起的。这导致了对许多其他类型的几何结构的研究，例如射影结构和洛伦兹结构，它们没有不变的度规。这些结构本身就很有趣。它们还提供了一个上下文，可以在其中查看不同的8个三维公制几何之间的关系，从而产生了过渡几何的概念。此外，还有关于其他维的几何结构的有趣的问题，涉及到不同李群的阵列，如高维的双曲结构，曲面上的复射影结构，以及表面群到高阶李群的表示。研究空间上各种类型的度量的想法至少可以追溯到19世纪末，Poincare，Klein等人的工作。他们的大部分动机来自于了解物理现象的愿望。从爱因斯坦开始的现代物理学导致了对复杂数学的更大需求，以理解物理宇宙，特别是来自公制几何的宇宙。尽管物理世界并不像本项目中研究的结构类型那样完全同质，但双曲和洛伦兹几何被认为代表了理解物理现象的有用模型。三维几何特别吸引人，因为它对许多人来说都是可视的，包括初学数学的学生和那些技术背景较少的人。它还导致了一些已被广泛使用的图形界面的创建。