Efficient Triangulations and Normal Surface Theory

高效的三角测量和法线表面理论

• 批准号: 9971719
• 负责人: William Jaco
• 金额: $ 11.54万
• 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: 1999
• 资助国家: 美国
• 起止时间: 1999-07-15 至 2003-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9971719&HistoricalAwards=false
• 关键词: Efficient; Triangulations; Normal; Surface; Theory

Proposal: DMS-9971719PI: William JacoAbstract. This project involves the development and use of efficienttriangulations and normal surface theory in the study and understanding of3-manifolds. A triangulation of a 3-manifold determines a class ofsurfaces in the 3-manifold called normal surfaces. A fundamental principalis that if a manifold contains an interesting surface, then for anytriangulation the manifold contains an interesting normal surface. Theguiding principle of efficient triangulations is reducing unnecessarynormal surfaces and improving the efficiency of computations fortriangulation based algorithms. They have exhibited remarkable success indoing this and, in addition, have revealed new combinatorial informationintimately related to the topology of 3-manifolds. These methods have ledto new, effective algorithms for recognizing the 3-sphere, determining ifa 3-manifold is a Haken manifold, and solving the unknotting problem.Low-dimensional manifolds provide the geometric models of most physicalphenomena. It is natural to study and understand all possible such models.Triangulations and normal surface theory provide an environment for such astudy. Research in these areas is making significant contribution tocomputational topology and complexity theory. It seems possible that thesetechniques will contribute to better computer visualization and possiblyto advances in medical modeling.

提案：DMS-9971719 PI：William J. Abstract。这个项目涉及到在三维流形的研究和理解中有效的三角剖分和法向曲面理论的发展和应用。三维流形的三角剖分确定了三维流形中的一类曲面，称为法向曲面。一个基本原理是，如果一个流形包含一个有趣的表面，那么对于任何三角形的流形包含一个有趣的正常表面。高效三角剖分的指导原则是减少不必要的法向曲面，提高三角剖分算法的计算效率。他们已经表现出显着的成功indoing这一点，此外，揭示了新的组合信息密切相关的拓扑结构的3流形。这些方法为三维球面的识别、三维流形是否为Haken流形的判定以及解纽结问题的求解提供了新的有效算法，低维流形为大多数物理现象提供了几何模型。研究和理解所有可能的这类模型是很自然的，三角剖分和法向曲面理论为这类研究提供了环境。这些领域的研究为计算拓扑学和复杂性理论做出了重要贡献。这些技术可能有助于更好的计算机可视化，并可能促进医学建模的发展。