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.

Sullivan 9727859调查员及其合作者研究了几何优化问题，例如在太空中寻找最小能量形状。 该项目完成了嵌入式恒定均值曲面的分类，并将这些结果扩展到更多的末端。 Willmore的弹性弯曲能在计算上用于驱动几个不同的球体，并在欧几里得和球形空间中发现新的最小表面。 研究者还研究了在更高尺寸的肥皂膜（例如肥皂膜）中发现的奇异性。这些泡沫可能为开尔文的分离问题提供解决方案；和结构的配置，以最大程度地减少能量或ropelgength。 可以以优化形状的某些特征的形式施加许多现实世界的问题。从数学上讲，这些变成了几何能量的各种问题。 例如，泡沫中细胞之间的膜最小化其面积，因此是恒定均值的曲率表面。 了解这些几何形状将使人们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，它似乎可以最大程度地减少弹性弯曲能在数学上称为Willmore Energy。 打结的曲线在拉紧绳索时达到最佳形状，或者如果带电的打结线静电驱除；了解这种构型有助于解释生物分子等DNA的行为。 该项目探讨了从理论和计算的角度来看仍然具有挑战性的自然问题。