Theory and Computation of Optimal Geometries

沙利文9727859 这位研究者与他的合作者一起研究几何优化问题，比如寻找空间中曲面和节点的最小能量形状。 这个项目完成了三个端点的嵌入式常平均曲率曲面的分类，并将这些结果扩展到更多的端点。 Willmore的弹性弯曲能量用于计算驱动几个不同的球体外翻，并发现新的极小曲面在欧氏空间和球面空间。 研究人员还研究了在更高维度的肥皂膜等极小化器中发现的奇点;泡沫可能提供开尔文分割问题的解决方案;以及使能量或绳长最小化的结的配置。 许多现实世界中的问题可以转换为优化形状的某些特征的形式;在数学上，这些问题变成了几何能量的变分问题。 例如，泡沫中单元之间的膜使它们的面积最小化，因此是恒定平均曲率表面。 了解这些几何形状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，这似乎使数学上称为Willmore能量的弹性弯曲能量最小化。 当绳子被拉紧时，打结的曲线会达到最佳形状，或者如果带电的打结线会静电排斥自己;理解这样的配置有助于解释生物分子（如DNA）的行为。 这个项目探讨了从理论和计算的角度来看仍然具有挑战性的物理自然问题。

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.