Discrete Conformal Geometry of Surfaces and Applications

如今，扫描仪、成像系统、传感器和许多其他设备正在大量生产数字表面。如何对这些表面进行比较和分类是一个迫切而重要的问题。这项由美国国家科学基金会资助的项目旨在开发数学理论，对数字表面进行高效、快速的分类。这项国家科学基金资助项目的圆满完成，将在医学影像、游戏产业、工程等多个领域得到应用。PI在这一领域的一些早期工作已经被用于健康和游戏行业。该项目工作将进一步推动这一进展，并发展基于经典黎曼曲面理论和保形几何的深度数学。经典的黎曼曲面理论是将曲面分类到共形微分同胚的有力工具。该理论中最重要的定理是Poincare-Koebe的均匀化定理。该定理在数学内外有着广泛的应用。然而，一般情况下，非平坦曲面上的均匀映射和度量的计算是非常困难的。该建议的目的是发展多面体曲面的离散共形几何理论，建立离散环境下的一致化定理的对应关系，并证明离散共形映射到共形映射的收敛。PI将通过协作开发高效的算法来计算统一化指标。发展离散保形几何的主要内容是定义离散保形的概念。在最近与合作者的合作中，PI为曲面上的多面体度量引入了离散共形等价的概念。PI和他的合作者证明了紧多面体曲面的一个离散化定理，并证明了离散的共形映射收敛到环面和圆盘上的共形映射。在建立所有曲面的离散保角映射的收敛和证明所有非紧单连通曲面的离散一致化定理方面，仍有几个主要的公开问题。我们引入的离散构象与Alexandrov，Pogorelov和瑟斯顿在双曲空间中凸面上的工作，经典的Wyle问题和Koebe圆域猜想密切相关。在经典数学领域开发的工具将被用来解决建议的问题。这个奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Digital surfaces are being massively produced from scanners, imaging systems, sensors and many other devices these days. How to compare and categorize these surfaces is an urgent and important problem. This National Science Foundation funded project aims to develop mathematical theories for an efficient and speedy classification of digital surfaces. A successful completion of this National Science Foundation funded project will have applications in medical imaging, game industry, engineering and many fields. Some of the earlier work of the PI in this area have already been used in health and game industries. The project work will further advance this progress and develop deep mathematics based on the classical Riemann surface theory and conformal geometry. The classical theory of Riemann surfaces is a powerful tool for classifying surfaces up to conformal diffeomorphisms. The most important theorem in the theory is the Poincare-Koebe's uniformization theorem. The theorem has a wide range of applications within and outside mathematics. However, computation of uniformization maps and metrics on non-flat surfaces are very difficult in general. The goal of the proposal is to develop a theory of discrete conformal geometry for polyhedral surfaces, to establish the counterpart of the uniformization theorem in the discrete setting, and to prove convergence of discrete conformal maps to conformal maps. The PI will develop efficient algorithms to compute uniformization metrics through collaboration. The main ingredient in developing a discrete conformal geometry is to define the notion of discrete conformality. In a recent joint work with collaborators, the PI introduced a notion of discrete conformal equivalence for polyhedral metrics on surfaces. The PI and his collaborators proved a discrete version of uniformization theorem for compact polyhedral surfaces and showed that discrete conformal maps converge to conformal maps on tori and disks. There remain several major open problems of establishing the convergence of discrete conformal maps for all surfaces and proving the discrete uniformization theorem for all non-compact simply connected surfaces. The discrete conformality introduced by us is closely related to the work of Alexandrov, Pogorelov and Thurston on convex surfaces in hyperbolic three-space, the classical Wyle problem and the Koebe circle domain conjecture. Tools developed in the classical area of mathematics will be used to solve the proposed problems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.