Low-dimensional manifolds and computation

本项目关注低维拓扑和几何问题，重点关注计算方面的问题。二维和三维流形为许多物理现象提供了几何模型。计算问题在这些领域的数学研究中扮演着越来越重要的角色。低维流形的研究反过来又对计算几何和拓扑学做出了贡献。它还影响了计算复杂性理论，该理论研究计算机解决问题需要多长时间等问题。计算拓扑技术已经导致了计算机图形学、可视化、医学和分子建模以及图像识别算法的改进。本计画将研究计算拓扑主题的几个问题。许多重要的拓扑和几何问题的计算复杂性是未知的，即使有显式算法存在。最近的研究结果揭示了复杂性理论、最小曲面理论、法向曲面理论和等周问题之间的有趣联系。本项目将在此方向进行研究，研究结识别、结属、同源属和相关拓扑问题的复杂性。该项目还包括研究法向曲面的广义理论的计划。法向曲面是相对于3流形的固定三角剖分来说特别简单的曲面。它们是PL类中最小曲面的类似物。这些曲面的集合是离散的，非常适合于计算，导致在分类，复杂性，枚举和算法识别方面的应用。索引- 1，或几乎法向曲面，在最近的流形识别工作中获得了惊人的成功。该项目的第三个重点是多区域等周问题，它询问包围给定大小的多个区域的最短边界是什么。目的是给出一个数学证明，证明肥皂泡所假设的结构是最优的。诸如找出包围平面上三个给定区域的最短曲线这样的问题仍然没有解决。同样缺少的是对稳定的类肥皂泡结构的一般性质的理解，比如最小化的结构是否总是有连通的区域。最后，该项目使用基于法向曲面的思想来研究曲线和曲面演变的问题。这种演化方法在许多领域具有实际应用的潜力，类似于平均曲率演化的许多应用。

Abstract for NSF proposalLow Dimensional Manifolds and ComputationThis project concerns problems in low-dimensional topology and geometry, with an emphasis on computational aspects. Manifolds in two and three dimensions provide the geometric models for many physical phenomena. Computational issues are playing an increasing role in mathematical investigations in these areas. Research in low-dimensional manifolds in turn is making contributions to computational geometry and topology. It alsoimpacts computational complexity theory, which studies questions such as how long it takes a computer to solve a problem. Techniques of computational topology have already led to improvements in algorithmsin computer graphics, visualization, medical and molecular modeling and image recognition. Several problems on the theme of computational topology will be investigated in this project. The computational complexity of many important problems in topology and geometry is unknown, even when explicit algorithms exist. Recent results have revealed intriguing connections between complexity theory, minimal surface theory, normal surface theory and isoperimetric problems. The project will pursue research in this direction, examining the complexity of Knot Recognition, Knot Genus, Homology Genusand related topological problems. The project also includes plans to investigate the generalized theory of normal surfaces. Normal surfacesare particularly simple surfaces relative to a fixed triangulation ofa 3-manifold. They are the analogs of minimal surfaces in the PL Category. The collection of such surfaces is discrete and well suited to computation, leading to applications in classification, complexity, enumeration and algorithmic recognition. Index-one, or almost normal surfaces, were applied with spectacular success in recent work on manifold recognition. A third focus of the project concerns multi-region isoperimetric problems, which ask what are the shortest boundaries enclosing multiple regions of a given size. The goal is to give a mathematical proof that the configurations assumed by soap bubbles are optimal. Questions such as finding the shortest curve enclosing three given areas in the plane remain open. Also missing is an understanding of general properties of stable soap-bubble-like configurations, such as whether the minimizing configurations always have connected regions. Finally, the project examines questions of curve and surface evolution, using ideas based on normal surfaces. Such evolution methods have the potential for practical applications in many areas, similar to the many applications of mean curvature evolution.