权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Geometry and Topology of Convex Projective Manifolds

凸射影流形的几何和拓扑

基本信息

批准号：
1709097
负责人：
Samuel Ballas
金额：
$ 13.93万
依托单位：
Florida State University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-09-01 至 2023-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1709097&HistoricalAwards=false
关键词：
Geometry Topology Convex Projective Manifolds

项目摘要

Projective geometry is the geometry of perspective, whose practitioners over time have included Greek philosopher/mathematicians studying incidence properties of lines in the plane, Renaissance artists attempting to render more realistic frescoes, and computer scientists pioneering computer graphics and vision techniques. This geometry comes from projecting points in a higher dimensional space along lines to a lower dimensional projective space. Unlike Euclidean geometry, this geometry has no well-defined notions of distance or angle. Its only meaningful geometric notion is incidence (for example, intersections of lines and inclusion of points in lines). In principle, this inability to measure distance initially seems like a drawback; however, in practice it provides a unified framework for studying seemingly disparate and incongruous geometries simultaneously. For example, projective space has pieces that serve as models for the familiar Euclidean geometry, the non-Euclidean spherical and hyperbolic geometries, and other exotic geometries, such as de Sitter and anti de Sitter space, that are of interest in modern physics. Recently, there has been increased interest in properly convex domains, which are interesting pieces of projective space that share many properties with hyperbolic space but enjoy interesting deformation properties absent in the hyperbolic setting. A primary focus of this project is to produce more of these properly convex examples and to understand their geometric, dynamic, and algebraic properties in a systematic fashion. Due to built-in connections with perspective and computer vision, many of the low dimensional examples under study in this project can be effectively visualized and rendered with the aid of a computer to produce vibrant dynamic graphics. This feature will allow the involvement of students with limited mathematical background in portions of the research as well as conveying the spirit of many of the important results to the broader non-mathematical community. Properly convex domains are subsets of projective space that are disjoint from a projective hyperplane and convex in the affine space produced by removing such a hyperplane from projective space. Hyperbolic space serves as the prime example of a properly convex domain via the Klein model. Properly convex domains and their quotients by discrete groups share many properties with hyperbolic space and hyperbolic orbifolds. A main point of this proposal is to understand how familiar concepts in hyperbolic geometry manifest themselves in properly convex geometry. Three main aims of the project are 1) developing a properly convex theory of Dehn surgery that can be used to produce examples of closed properly convex manifolds from non-compact ones, 2) investigating how dynamic properties of the fundamental group of a projective manifold manifest themselves geometrically, in analogy with geometric finiteness for hyperbolic manifolds, and 3) using properly convex structures to produce subgroups of the special linear group with interesting algebraic properties (such as thinness). In addition to the obvious potential to better understand geometric and dynamical aspects of properly convex manifolds, this project should also yield a deeper understanding of hyperbolic geometry by elucidating which geometric features of hyperbolic geometry are consequences of uniform negative curvature and which are consequences of more general geometric structure.

射影几何是透视几何，其实践者随着时间的推移包括希腊哲学家/数学家研究平面中线条的入射性质，文艺复兴时期的艺术家试图渲染更逼真的壁画，以及计算机科学家开创计算机图形学和视觉技术。这种几何来自于将高维空间中的点沿沿着投影到低维的投影空间。与欧几里德几何不同，这种几何没有明确定义的距离或角度概念。它唯一有意义的几何概念是关联（例如，线的交点和线中包含的点）。原则上，这种无法测量距离的能力最初似乎是一个缺点;然而，在实践中，它提供了一个统一的框架，可以同时研究看似不同和不协调的几何形状。例如，射影空间有一些片段可以作为熟悉的欧几里得几何、非欧几里得球面几何和双曲几何以及其他奇异几何的模型，如现代物理学中感兴趣的德西特空间和反德西特空间。最近，人们对真凸域越来越感兴趣，真凸域是射影空间的有趣片段，与双曲空间共享许多属性，但在双曲设置中没有享受有趣的变形属性。这个项目的主要重点是产生更多的这些正确的凸的例子，并了解他们的几何，动力学和代数性质在一个系统的方式。由于与透视和计算机视觉的内在联系，该项目中研究的许多低维示例可以在计算机的帮助下有效地可视化和渲染，以产生充满活力的动态图形。这一功能将允许有限的数学背景的学生参与部分研究，以及传达的精神，许多重要的成果，以更广泛的非数学界。真凸域是射影空间的子集，它们与射影超平面不相交，并且在仿射空间中是凸的，通过从射影空间中移除这样的超平面来产生仿射空间。双曲空间作为通过克莱因模型的适当凸域的主要例子。真凸域及其由离散群生成的真凸域与双曲空间和双曲轨道有许多共同的性质。这个建议的一个要点是了解如何熟悉的概念在双曲几何表现自己在适当的凸几何。该项目的三个主要目标是：1）发展Dehn手术的真凸理论，该理论可用于从非紧流形产生闭真凸流形的例子，2）研究射影流形的基本群的动力学性质如何在几何上表现出来，类似于双曲流形的几何有限性，3）利用适当的凸结构产生特殊线性群的具有有趣代数性质（如稀疏性）的子群。除了明显的潜力，以更好地了解几何和动力学方面的适当凸流形，该项目还应产生更深入的了解双曲几何，阐明哪些几何特征的双曲几何是一致的负曲率的后果，哪些是更一般的几何结构的后果。