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.

该项目将研究三维（3D）空间的拓扑：宇宙可能具有的形状。可以通过将其切入四面体并记录如何将它们粘合在一起的形状来描述这种形状。这种称为三角剖分的结构对于使用计算机研究拓扑空间特别有用。弯曲的三角剖分是一种非常特殊的三角剖分，似乎与可能的流过空间的流动密切相关，即流体循环的方式。该项目的核心目标之一是正式证明这种关系，该关系将向计算实验和探索开放这些流程的研究。除了回答有关转向三角形的其他问题外，该项目还将调查有关高维空间三角的基本问题。该项目的另一个目标是使用3D打印和虚拟/增强现实来帮助研究，教学法和外展。 PI开发了一门本科课程，在该课程中，学生学习3D设计技能并运用其数学知识来生产3D打印对象，并在3D打印和虚拟现实中为学生项目提供了建议。他计划通过新技术和数学可视化项目扩展这些活动。 PI与CIMAT（墨西哥）的合作者一起设计廉价的3D印刷滑动瓷砖难题和相关的教育材料，将分发给拉丁美洲代表性不足的社区。其他外展活动将包括外部论文，公开谈判和YouTube视频。在此项目中，PI旨在证明，弯曲的三角剖分与伪 - anosov流相对应，它们可用于生成新的Cannon-Thurston地图（与超级填充曲线相关的新型填充曲线（与超质量几何形式相关的空间曲线），并且这些类型的类型都如此。这项工作还将扩展现有的转向三角群的普查。在另一个方向上，该项目试图在任意数量的维度中建立三角剖分的连通性属性，并在拓扑不变和人口普查中使用潜在的应用。 PI和合作者计划进一步为Thurston（和其他）几何形状开发虚拟现实经验，以及增强现实应用程序，以调查领域，并与研究软件“ Snappy”的开发人员合作，以扩展其可视化能力。该奖项通过评估了众多的依据，反映了NSF的奖励，以反映出NSF的审查，这是一项伟大的支持者的精彩范围。