Beyond the Thurston Geometries

在这个项目中，我们将研究几何结构和结构的代表品种。 这些主题是以前的研究计划，专注于低维双曲流形的自然产物。 在过去的十年里，克莱因群和三维流形的几何领域取得了惊人的进展。因此，这些领域能够重新调整其工作重点。 这种情况已经变得更像表面，人们感兴趣的是家庭和空间的结构。 尝试理解三维流形的谱系是很重要的，它们是如何通过各种拓扑和几何运算联系起来的。这导致了对许多其他类型的几何结构的研究，例如投影和洛伦兹结构，这些结构没有不变的度量。 这些结构本身就很有趣。 它们还提供了一个上下文，在其中查看不同的8个三维度量几何之间的关系，导致过渡几何的概念。 此外，还有关于其他维度的几何结构的有趣问题，涉及一系列不同的李群，如高维中的双曲结构，曲面上的复和真实的投影结构，以及曲面群到更高秩李群的表示。研究空间上各种类型度量的想法至少可以追溯到世纪后期，庞加莱，克莱因，等人 他们的动机很大程度上来自于对理解物理现象的渴望。 现代物理学，从爱因斯坦开始，已经导致了对复杂数学的更大需求，以理解物理宇宙，特别是来自度量几何的数学。 虽然物理世界并不是一个完全均匀的结构，如本项目中研究的结构类型，双曲和洛伦兹几何被认为是理解物理现象的有用模型。 三维几何特别有吸引力，因为它对许多人来说都是视觉上可访问的，包括数学初学者和那些技术背景较低的人。 它还导致创建了许多已被广泛使用的图形界面。

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.