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Veering Triangulations and Visualization

转向三角测量和可视化

基本信息

批准号：
2203993
负责人：
Henry Segerman
金额：
$ 34.54万
依托单位：
Oklahoma State University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-08-01 至 2025-07-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2203993&HistoricalAwards=false
关键词：
Veering Triangulations Visualization

项目摘要

This project will study the topology of three-dimensional (3D) spaces: the possible shapes that a universe could have. Such a possible shape can be described by cutting it into tetrahedra and recording how to glue them back together. This structure, called a triangulation, is particularly useful for studying topological spaces using computers. Veering triangulations are a very special kind of triangulation, which seem to be closely connected to the possible flows through the space, that is, the ways in which a fluid could circulate. One of the central goals of the project is to formally prove this relationship, which will open the study of these flows to computational experimentation and exploration. In addition to answering other questions about veering triangulations, the project will investigate foundational questions about triangulations of higher-dimensional spaces. Another goal of the project is in mathematical visualization, using 3D printing and virtual/augmented reality to aid in research, pedagogy, and outreach. The PI has developed an undergraduate course in which students learn 3D design skills and apply their mathematical knowledge to produce 3D printed objects and has advised student projects in both 3D printing and virtual reality. He plans to extend these activities with new technologies and mathematical visualization projects. With collaborators at CIMAT (Mexico), the PI plans to design inexpensive 3D printed sliding tile puzzles and associated educational materials, to be distributed to underrepresented communities in Latin America. Other outreach activities will include expository papers, public talks, and YouTube videos.In this project the PI aims to prove, together with collaborators, that veering triangulations correspond to pseudo-Anosov flows, that they can be used to produce new Cannon-Thurston maps (space-filling curves associated to hyperbolic geometry), and that certain types of manifolds have finitely many veering triangulations. The work will also extend the existing census of veering triangulations. In another direction, the project seeks to establish connectivity properties for triangulations in an arbitrary number of dimensions, with potential applications in topological invariants and census building. The PI and collaborators plan to further develop virtual reality experiences for the Thurston (and other) geometries, as well as an augmented reality application to investigate sphere eversions, and to work with developers of the research software “SnapPy” to extend its visualization capabilities.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.

这个项目将研究三维（3D）空间的拓扑结构：宇宙可能具有的形状。这种可能的形状可以通过将其切割成四面体并记录如何将它们粘合在一起来描述。这种结构被称为三角剖分，对于使用计算机研究拓扑空间特别有用。转向三角测量是一种非常特殊的三角测量，它似乎与空间中可能的流动密切相关，也就是说，流体可以循环的方式。该项目的中心目标之一是正式证明这种关系，这将为这些流动的研究打开计算实验和探索的大门。除了回答有关转向三角测量的其他问题外，该项目还将调查有关高维空间三角测量的基本问题。该项目的另一个目标是数学可视化，利用3D打印和虚拟/增强现实来帮助研究、教学和推广。PI开设了一门本科课程，让学生学习3D设计技能，并应用他们的数学知识来生产3D打印对象，并为学生提供3D打印和虚拟现实方面的项目建议。他计划用新技术和数学可视化项目来扩展这些活动。与CIMAT（墨西哥）的合作者一起，PI计划设计廉价的3D打印滑动瓷砖拼图和相关的教育材料，分发给拉丁美洲代表性不足的社区。其他拓展活动将包括说明性论文、公开演讲和YouTube视频。在这个项目中，PI的目标是与合作者一起证明，转向三角形对应于伪anosov流，它们可以用来产生新的Cannon-Thurston图（与双曲几何相关的空间填充曲线），并且某些类型的流形具有有限多个转向三角形。这项工作还将扩展现有的转向三角测量普查。在另一个方向上，该项目寻求在任意数量的维度上建立三角测量的连通性属性，在拓扑不变量和人口普查建筑中具有潜在的应用。PI和合作者计划进一步开发Thurston（和其他）几何图形的虚拟现实体验，以及用于调查球体版本的增强现实应用程序，并与研究软件“SnapPy”的开发人员合作，扩展其可视化功能。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。