EAPSI: Developing a new theory of discrete angle-preserving surface maps

复函数的研究是现代数学的核心，可以粗略地描述为研究曲面之间的保角映射。这些地图最近的一个应用是它们用于处理三维二维表面上的数据，例如来自医学成像或3D面部扫描的数据。这些地图提供了一种在这样的表面上获得标准化坐标的方法，以帮助进行数据比较。这个项目的目的是通过证明黎曼映射定理来开发这些曲面映射的新的有限逼近，黎曼映射定理是复函数研究中的一个基本定理。这项研究将与北京清华大学著名的计算和低维几何与拓扑学专家孙健、顾大卫和罗峰博士合作进行。给定紧凑曲面上的分段平坦度量，我们可以将其构造为欧几里得三角形的粘合，从而产生具有有限数量圆锥点的几何三角剖分。这项研究考虑了这种度量的离散一致性的概念，允许通过锥点处的保形因子缩放边长度，直到Delaunay条件；以及边翻转到Delaunay三角剖分之间的过渡。在最近的工作中，证明了一个均匀化结果，它证明了任何这样的度量都是离散共形的，在每个锥点处具有相等的角亏(曲率)的度量。此外，离散的Ricci流实现了这种均衡化，并且对这种流的大量计算表明，随着三角剖分变得更精细，得到的分段线性映射收敛到全纯映射。对于圆盘的情况，将尝试证明这种收敛，以获得经典的Riemann映射定理。该NSF EAPSI奖是与中国科技部合作资助的。

The study of complex functions is central to modern mathematics, and may be roughly described as the study of angle-preserving maps between surfaces. A recent application of these maps is their use in the processing of data on two-dimensional surfaces in three dimensions, such as data from medical imaging or 3D facial scans. The maps provide a way to obtain standardized coordinates on such surfaces, to aid data comparison. This project will aim to develop a new finite approximation of these surface maps by proving the Riemann mapping theorem, a foundational theorem in the study of complex functions. This research will be conducted in collaboration with Drs. Jian Sun, David Gu, and Feng Luo, noted experts on computational and low-dimensional geometry and topology, at Tsinghua University in Beijing, China.Given a piecewise flat metric on a compact surface, we may construct it as a gluing of Euclidean triangles, resulting in a geometric triangulation with a finite number of cone points. This research considers a notion of discrete conformality for such metrics that allows for scaling of edge lengths by conformal factors at the cone points, up to Delaunay conditions; and edge flips to transition between Delaunay triangulations. In recent work, a uniformization result was proven, which demonstrated that any such metric is discrete conformal to a metric with equal angle defect (curvature) at each cone point. Furthermore, a discrete Ricci flow realizes this uniformization, and numerous computations of this flow suggest that the resulting piecewise linear maps converge to holomorphic maps as the triangulations become finer. For the case of the disc, attempts will be made to prove this convergence to obtain the classic Riemann mapping theorem. This NSF EAPSI award is funded in collaboration with Chinese Ministry of Science and Technology.