Optimal Geometry: Theory and Computation

最佳几何形状：理论与计算

• 批准号: 0071520
• 负责人: John Sullivan
• 金额: $ 9.75万
• 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: 2000
• 资助国家: 美国
• 起止时间: 2000-09-15 至 2004-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0071520&HistoricalAwards=false
• 关键词: Optimal; Geometry; Theory; Computation

Sullivan0071520 The investigator, with his collaborators, studies geometric optimization problems like finding minimum-energy shapes for surfaces and knots in space. They extend their recent classification of embedded constant-mean-curvature surfaces with three ends to the more general case of surfaces with any number of coplanar ends, and also investigate in detail surfaces with truncated ends. In addition, the investigator computes these surfaces numerically, in order, for instance, to create interactive computer graphics. This project uses Willmore's elastic bending energy, and its gradient flow, to discover new minimal surfaces in euclidean and spherical space. The Willmore flow has been recently shown to have short-time solutions, but the investigator considers whether it can fail to have long-time solutions. This project also studies configurations for knots which minimize ropelength, giving new lower bounds for the ropelength of small knots, and new asymptotic bounds on the growth of ropelength with crossing number. Finally, the investigator uses his experience with numerical modeling of curves and surfaces to give new understanding of geometrically natural discretizations for quantities related to curvature. 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, thin films, like those in foams, usually minimize their area and thus are constant-mean curvature surfaces. Cell membranes are more complicated bilayer surfaces which 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, which remain challenging from both theoretical and computational standpoints.

Sullivan0071520调查员及其合作者研究了几何优化问题，例如在太空中寻找表面和结的最小能量形状。 他们将最近的嵌入式均值屈肌表面的分类扩展到具有三个末端的三端，并以任何数量的共面末端进行更一般的表面案例，并详细研究具有截断端的末端。 此外，研究者以数字计算这些表面，例如，创建交互式计算机图形。 该项目使用Willmore的弹性弯曲能及其梯度流，以发现欧几里得和球形空间中新的最小表面。 最近已显示Willmore流量有短期解决方案，但是研究人员认为它是否无法使用长期解决方案。 该项目还研究了为结的结构，这些结构可最大程度地减少ropelength，从而为小结的ropelength提供了新的下限，以及对ropelength和crossing number的生长的新渐近界限。 最后，研究人员利用他在曲线和表面的数值模型中的经验，对与曲率相关的数量进行新的几何自然离散化提供了新的了解。 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，像泡沫中的薄膜一样，通常使其面积最小化，因此是恒定均值的曲率表面。 细胞膜是更复杂的双层表面，可最大程度地减少弹性弯曲能在数学上称为Willmore能量的能量。 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。 该项目探索了这种自然问题，从理论和计算的角度来看，这些问题仍然具有挑战性。