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

对复变函数的研究是现代数学的核心，可以粗略地描述为对曲面之间保角映射的研究。这些地图的最近应用是它们在三维中的二维表面上的数据处理中的使用，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方法来获得这些表面上的标准化坐标，以帮助数据比较。这个项目的目的是通过证明黎曼映射定理，一个基础定理，在研究复杂的功能，开发这些表面映射的一个新的有限近似。本研究将与清华大学著名的计算和低维几何与拓扑专家孙健、大卫和罗锋博士合作进行。给定紧致曲面上的分段平坦度量，我们可以将其构造为欧氏三角形的胶合，从而得到具有有限个锥点的几何三角剖分。本研究考虑了此类指标的离散保形性概念，允许通过锥点处的保形因子来缩放边长，直至达到Delaunay条件;并且边翻转以在Delaunay三角剖分之间过渡。在最近的工作中，一个一致化结果被证明，这表明任何这样的度量是离散共形的度量与等角亏损（曲率）在每个锥点。此外，一个离散的里奇流实现了这种一致化，大量的计算表明，这种流的分段线性映射收敛到全纯映射的三角剖分变得更精细。对于圆盘的情形，将试图证明这种收敛性，以得到经典的黎曼映射定理。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.