Veering Triangulations and Visualization

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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.