Low-dimensional manifolds and computation

低维流形和计算

• 批准号: 0072348
• 负责人: Joel Hass
• 金额: $ 16.49万
• 依托单位: University of California-Davis
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-15 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0072348&HistoricalAwards=false
• 关键词: Low; dimensional; manifolds; computation

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.

NSF提示尺寸歧管和计算的摘要涉及低维拓扑和几何形状中的问题，重点是计算方面。两个维度和三个维度的歧管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起着越来越多的作用。低维流形的研究反过来为计算几何和拓扑做出了贡献。它也破坏了计算复杂性理论，该理论研究了诸如计算机解决问题所需的时间。计算拓扑的技术已经导致了计算机图形，可视化，医学和分子建模以及图像识别的改进。 该项目将研究有关计算拓扑主题的几个问题。即使存在明确的算法，拓扑和几何学上许多重要问题的计算复杂性也未知。最近的结果揭示了复杂性理论，最小表面理论，正常表面理论和等值问题之间的有趣联系。该项目将朝这个方向进行研究，研究结识别，结属，同源性属和相关拓扑问题的复杂性。该项目还包括研究正常表面的广义理论的计划。正常表面特别简单的表面相对于3个manifold的固定三角剖分。它们是PL类别中最小表面的类似物。此类表面的收集​​是离散的，并且非常适合计算，从而导致分类，复杂性，枚举和算法识别的应用。在最近在多种识别方面的工作中，索引一个或几乎正常的表面在最近的工作中取得了惊人的成功。该项目的第三个重点涉及多区域等级问题，该问题询问包含给定尺寸多个区域的最短边界是什么。目的是给出数学证明，表明肥皂气泡假设的配置是最佳的。诸如找到封闭飞机中三个给定区域的最短曲线之类的问题仍然开放。还缺少对稳定的类似肥皂泡的配置的一般属性的理解，例如最小化配置是否始终具有连接的区域。 最后，该项目使用基于正常表面的想法来检查曲线和表面演变的问题。这种进化方法具有许多领域的实际应用，类似于平均曲率进化的许多应用。