Discrete Conformal Geometry of Surfaces and Applications

曲面的离散共形几何及其应用

• 批准号: 1811878
• 负责人: Feng Luo
• 金额: $ 10.56万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-07-01 至 2022-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1811878&HistoricalAwards=false
• 关键词: Discrete; Conformal; Geometry; Surfaces; Applications

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.

如今，数字表面正在由扫描仪、成像系统、传感器和许多其他设备大规模生产。 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。 这个国家科学基金会资助的项目旨在开发数学理论，以有效和快速地对数字表面进行分类。 这个国家科学基金会资助的项目的成功完成将在医学成像，游戏行业，工程和许多领域的应用。PI在这一领域的一些早期工作已经用于健康和游戏行业。 该项目工作将进一步推进这一进程，并在经典黎曼曲面理论和共形几何的基础上发展深层数学。 黎曼曲面的经典理论是将曲面分类为共形反同态的有力工具。该理论中最重要的定理是Poincare-Koebe单值化定理。该定理在数学内外都有广泛的应用。然而，非平坦表面上的均匀化映射和度量的计算通常非常困难。该提案的目标是发展一个理论的离散共形几何多面体表面，建立对应的单值化定理在离散设置，并证明收敛的离散共形映射到共形映射。PI将通过合作开发有效的算法来计算均匀化指标。 发展离散共形几何的主要内容是定义离散共形性的概念。在最近与合作者的联合工作中，PI引入了曲面上多面体度量的离散共形等价的概念。PI和他的合作者证明了紧凑多面体表面的一致化定理的离散版本，并表明离散的共形映射收敛到环面和圆盘上的共形映射。 建立所有曲面的离散共形映射的收敛性和证明所有非紧单连通曲面的离散一致化定理仍然是几个主要的公开问题。我们引入的离散共形性与Alexandrov、Pogorelov和Thurston关于三维双曲空间中凸曲面的工作、经典Wyle问题和Koebe圆域猜想密切相关。 该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

An effective Lie–Kolchin Theorem for quasi-unipotent matrices

拟单能矩阵的有效李科尔钦定理

• DOI: 10.1016/j.laa.2019.07.023
• 发表时间: 2019
• 期刊: Linear Algebra and its Applications
• 影响因子: 1.1
• 作者: Koberda, Thomas;Luo, Feng;Sun, Hongbin
• 通讯作者: Sun, Hongbin

Computational Conformal Geometry Behind Modern Technologies

现代技术背后的计算共形几何

• DOI: 10.1090/noti2164
• 发表时间: 2020
• 期刊: Notices of the American Mathematical Society
• 作者: Xianfeng GU, Feng LUO

Co-evolution of Opinion and Social Tie Dynamics Towards Structural Balance

舆论和社会关系动态的共同演化走向结构平衡

• 发表时间: 2022
• 期刊: Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA
• 作者: Haotian Wang, Feng Luo

Discrete conformal geometry of polyhedral surfaces and its convergence

• DOI: 10.2140/gt.2022.26.937
• 发表时间: 2020-09
• 期刊: Geometry & Topology
• 作者: F. Luo;Jian Sun;Tianqi Wu