Theory and Computation of Optimal Geometries

最佳几何理论与计算

• 批准号: 9727859
• 负责人: John Sullivan
• 金额: $ 8万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-09-01 至 2000-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9727859&HistoricalAwards=false
• 关键词: Theory; Computation; Optimal; Geometries

Sullivan 9727859 The investigator, with his collaborators, studies geometric optimization problems like finding minimum-energy shapes for surfaces and knots in space. This project completes the classification of embedded constant-mean-curvature surfaces with three ends and extends these results to greater numbers of ends. The elastic bending energy of Willmore is used computationally to drive several different sphere eversions, and to discover new minimal surfaces in euclidean and spherical space. The investigator also studies the singularities found in minimizers like soap films in higher dimensions; the foams likely to provide solutions to Kelvin's partioning problem; and configurations for knots that minimize energy or ropelength. Many real-world problems can be cast in the form of optimizing some feature of a shape; mathematically, these become variational problems for geometric energies. For instance, the films between cells in a foam minimize their area and thus are constant-mean curvature surfaces. Understanding these geometries will lead to better knowledge of important structural properties of foam materials. Cell membranes are more complicated bilayer surfaces, which seem to minimize an elastic bending energy known mathematically as the Willmore energy. Knotted curves achieve an optimal shape when a rope is pulled tight, or if a charged knotted wire repels itself electrostatically; understanding such configurations helps explain the behavior of biological molecules like DNA. This project explores such phsically natural problems that are still challenging from both theoretical and computational standpoints.

研究人员沙利文9727859和他的合作者研究几何优化问题，比如为空间中的表面和结点寻找最小能量形状。这个项目完成了三端嵌入的常平均曲率曲面的分类，并将这些结果扩展到更多的端。Willmore的弹性弯曲能在计算上被用来驱动几种不同的球面外倾，并在欧氏空间和球面空间中发现新的极小曲面。研究人员还研究了在高维肥皂膜等极小器中发现的奇点；可能为开尔文分区问题提供解决方案的泡沫；以及将能量或长度最小化的结的配置。许多现实世界的问题可以转化为形状的某些特征的优化形式；从数学上讲，这些问题变成了几何能量的变分问题。例如，泡沫中细胞之间的薄膜使其面积最小，因此是恒定平均曲率曲面。了解这些几何结构将有助于更好地了解泡沫材料的重要结构特性。细胞膜是更复杂的双层表面，它似乎将弹性弯曲能量最小化，在数学上被称为威尔莫尔能量。当绳索被拉紧时，或者如果带电的打结的电线自我静电排斥，打结的曲线就会达到最佳形状；了解这种配置有助于解释DNA等生物分子的行为。这个项目探索了这样的物理自然问题，无论是从理论上还是从计算角度来看，这些问题都仍然具有挑战性。