Veering Triangulations and Visualization

这个项目将研究三维（3D）空间的拓扑结构：宇宙可能具有的可能形状。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重新粘在一起来描述。这种结构称为三角测量，对于使用计算机研究拓扑空间特别有用。转向三角剖分是一种非常特殊的三角剖分，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。该项目的核心目标之一是正式证明这种关系，这将使这些流的研究开放给计算实验和探索。除了回答关于转向三角剖分的其他问题外，该项目还将研究关于高维空间三角剖分的基本问题。该项目的另一个目标是数学可视化，使用3D打印和虚拟/增强现实来帮助研究，教学和推广。PI开发了一门本科课程，学生在其中学习3D设计技能，并应用他们的数学知识来生产3D打印物体，并为3D打印和虚拟现实的学生项目提供建议。他计划通过新技术和数学可视化项目来扩展这些活动。PI计划与CIMAT（墨西哥）的合作者一起设计廉价的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。在这个项目中，PI的目标是与合作者一起证明转向三角剖分对应于伪Anosov流，它们可以用来产生新的Cannon Thurston映射（与双曲几何相关的空间填充曲线），以及某些类型的流形有许多转向三角剖分。这项工作还将扩大现有的转向三角测量普查。在另一个方向，该项目旨在建立连接属性的三角测量在任意数量的维度，在拓扑不变量和人口普查建设的潜在应用。PI和合作者计划进一步为Thurston开发虚拟现实体验（和其他）几何形状，以及一个增强现实应用程序来研究球体外翻，并与研究软件“SnapPy”的开发人员合作，该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响进行评估来支持审查标准。

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.