Geometry and Topology of Polyhedral Surfaces

如今，从3D扫描到数字世界中的医学成像，多面体表面以惊人的速度产生。几乎所有出现在计算机屏幕上的曲面都是多面体曲面。一个紧迫的任务是产生一个粗略的分类和分类这些表面。数学中处理曲面的一个里程碑式的结果是经典的庞加莱-柯比单值化定理。 它是一个非常强大的工具，可以对3-空间中的光滑曲面进行分类，直到保持角度的同态（即，保形性）。然而，经典的一致化定理的算法实现是非常困难的。PI和他的合作者最近引入了一个离散对应的共形性，并证明了一个离散版本的多面体曲面的单值化定理。该方案从理论上研究了当三角剖分网格越来越细时，离散共形几何是否收敛到光滑共形几何。 收敛的数值证据是非常强大的。 离散一致化定理的收敛性问题的成功解决将影响离散一致化定理在许多领域的应用，经典的Poincare和Koebe一致化定理是对具有黎曼度量的光滑曲面进行共形同态分类的最有力的工具。 该定理是20世纪世纪数学的支柱之一，在数学内外有着广泛的应用。然而，它是非常困难的算法使用它来分类多面体表面。最近，Luo和他的合作者引入了多面体曲面的离散共形性的概念，并证明了一个离散单值化定理。离散保形性的两个主要特征如下。首先，离散保形性是算法的，其次，存在一个有限维（凸）变分原理来找到离散一致化度量。该建议的目标是调查几个主要的剩余问题，在多面体表面的离散保形几何。例如，有强有力的数值证据表明，离散保形收敛到光滑（经典）保形时，三角剖分是适当的选择。然而，它的理论证明仍然缺乏。这是建议中要解决的主要问题。罗计划使用他在过去10年中开发的有限维变分原理来解决收敛问题。

Polyhedral surfaces are produced at an alarming rate ranging from 3-D scanning to medical imaging in the digital world these days. Almost all surfaces appeared in any computer screen are polyhedral surfaces. An urgent task is to produce a coarse categorization and classification of these surfaces. A mile stone result in mathematics dealing with surfaces is the classical uniformization theorem of Poincare-Koebe. It is an extremely powerful tool to classify smooth surfaces in the 3-space up to angle preserving diffeomorphisms (i.e., conformality). However, to algorithmically implement the classical uniformization theorem is very difficult. The PI and his collaborators introduced a discrete counterpart of conformality and proved a discrete version of uniformization theorem for polyhedral surfaces recently. This proposal investigates theoretically whether the discrete conformal geometry converges to the smooth conformal geometry as triangulation meshes become finer and finer. The numerical evidences for the convergence are very strong. The successful resolution of the convergence issue will have impacts on the applications of the discrete uniformization theorem in many fields.The classical uniformization theorem of Poincare and Koebe is the most powerful tool to classify smooth surfaces with Riemannian metrics according to conformal diffeomorphisms. This theorem is one of the pillars of the 20th century mathematics and has a wide range of applications within and outside mathematics. However, it is very difficult to use it algorithmically for categorizing polyhedral surfaces. Luo and his collaborators introduced a notion of discrete conformality for polyhedral surfaces and proved a discrete uniformization theorem recently. Two main features of the discrete conformality are the following. First, the discrete conformality is algorithmic and second, there exists a finite dimensional (convex) variational principle to find the discrete uniformization metric. The goal of the proposal is to investigate several major remaining issues in the discrete conformal geometry of polyhedral surfaces. For instance, there are strong numerical evidences suggesting that the discrete conformality converges to smooth (classical) conformality when the triangulations are suitably chosen. However, a theoretical proof of it is still lacking. This is the main problem to be resolved in the proposal. Luo plans to use the finite dimensional variational principles that he developed in the past 10 years to approach the convergence problem.