Geometry and Topology of Convex Projective Manifolds

射影几何是透视几何，其实践者包括研究平面上线的关联属性的希腊哲学家/数学家，试图渲染更逼真壁画的文艺复兴时期艺术家，以及开拓计算机图形学和视觉技术的计算机科学家。这种几何图形来自于将高维空间中的点沿直线投影到低维投影空间。与欧几里得几何不同，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，线的相交和线中的点的包含）。原则上，这种无法测量距离的特性似乎是一种缺陷；然而，在实践中，它为同时研究看似不同和不协调的几何图形提供了一个统一的框架。例如，投影空间有一些部分可以作为我们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，比如德西特空间和反德西特空间，这些都是现代物理学感兴趣的。最近，人们对适当凸域的兴趣增加了，适当凸域是射影空间中有趣的部分，它与双曲空间具有许多相同的性质，但具有双曲环境中没有的有趣的变形性质。这个项目的主要焦点是产生更多的这些适当的凸例子，并以系统的方式理解它们的几何、动态和代数性质。由于与透视和计算机视觉的内在联系，本项目所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，并将许多重要结果的精神传达给更广泛的非数学社区。适当凸域是射影空间的子集，它们与射影超平面和仿射空间中的凸不相交。双曲空间通过克莱因模型作为适当凸域的主要例子。适当凸域及其离散群商具有双曲空间和双曲轨道的许多性质。这个建议的一个要点是了解双曲几何中熟悉的概念如何在适当的凸几何中表现出来。该项目的三个主要目标是：1)发展Dehn手术的适当凸理论，该理论可用于从非紧化流形中产生闭合适当凸流形的例子；2)研究射影流形基本群的动态特性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具有有趣的代数性质（如薄性）的特殊线性群的子群。除了更好地理解适当凸流形的几何和动力学方面的明显潜力之外，该项目还应该通过阐明双曲几何的哪些几何特征是均匀负曲率的结果，哪些是更一般的几何结构的结果，从而加深对双曲几何的理解。

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.