Efficient Triangulations and Normal Surface Theory

提案：DMS-9971719PI: William jacob本项目涉及有效三角剖分法和法向曲面理论在3流形研究和理解中的发展和应用。3流形的三角剖分确定了3流形中的一类曲面，称为法线曲面。一个基本原理是，如果流形包含一个有趣的曲面，那么对于任何三角化，流形都包含一个有趣的法线曲面。有效三角剖分的指导原则是减少不必要的法线曲面，提高基于三角剖分算法的计算效率。他们在这方面取得了显著的成功，此外，还揭示了与3流形拓扑密切相关的新的组合信息。这些方法产生了新的、有效的算法来识别3球、确定3流形是否为Haken流形以及解决解结问题。低维流形提供了大多数物理现象的几何模型。研究和理解所有可能的模型是很自然的。三角剖分法和法向曲面理论为这种研究提供了环境。这些领域的研究为计算拓扑和复杂性理论的发展做出了重要贡献。这些技术似乎有可能有助于更好的计算机可视化，并可能有助于医学建模的进步。

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.