Geometry and Topology of Convex Projective Manifolds

射影几何是透视的几何学，随着时间的推移，它的实践者包括研究平面中线的入射性质的希腊哲学家/数学家，试图绘制更逼真壁画的文艺复兴时期的艺术家，以及开创计算机图形和视觉技术的计算机科学家。这种几何来自于将高维空间中的点沿线投射到低维射影空间。与欧几里德几何不同，该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，直线的交点和直线中的点的包含)。原则上，无法测量距离最初看起来像是一个缺点；然而，在实践中，它为同时研究看似完全不同和不协调的几何图形提供了一个统一的框架。例如，射影空间有一些片段用作熟悉的欧几里德几何、非欧几里德球面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，真凸域是一种有趣的射影空间，它与双曲空间有许多相同的性质，但具有双曲背景下所没有的有趣的变形性质。这个项目的一个主要焦点是产生更多这样的适当的凸示例，并以系统的方式了解它们的几何、动态和代数性质。由于与透视和计算机视觉的内在联系，本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，以生成充满活力的动态图形。这一特点将允许数学背景有限的学生参与部分研究，并将许多重要成果的精神传达给更广泛的非数学社区。真凸域是射影空间的子集，它们与射影超平面不相交，并且在仿射空间中由射影空间中去掉这样的超平面而产生凸集。通过Klein模型，双曲空间是真凸域的主要例子。真凸域及其由离散群构成的商与双曲空间和双曲奥布洛尔德有许多共同的性质。这一建议的一个主要观点是理解双曲几何中熟悉的概念如何在真凸几何中表现出来。该项目的三个主要目标是：1)发展Dehn运算的适当凸理论，该理论可用于从非紧流形产生封闭的适当凸流形的例子；2)研究射影流形的基本群的动态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)利用适当的凸结构来产生具有有趣的代数性质(如稀疏性)的特殊线性群的子群。除了更好地理解真凸流形的几何和动力学方面的明显潜力外，这个项目还应该通过阐明双曲几何的哪些几何特征是一致负曲率的结果，以及哪些是更一般的几何结构的结果，来更深入地理解双曲几何。

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.