Theory and Computation of Optimal Geometries

这位研究者和他的合作者研究几何优化问题，比如寻找空间中曲面和结的最小能量形状。本项目完成了具有三个端点的嵌入式等平均曲率曲面的分类，并将这些结果扩展到更多的端点。利用Willmore的弹性弯曲能计算驱动了几种不同的球面，并在欧氏空间和球面空间中发现了新的最小曲面。研究者还研究了在最小化器中发现的奇点，比如高维的肥皂膜；泡沫有可能解决开尔文分割问题；以及最小化能量或绳长的绳结结构。许多现实世界的问题都可以通过优化形状的某些特征来解决；在数学上，这些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，这似乎最小化了弹性弯曲能量，数学上称为威尔摩能量。当绳子被拉紧，或者带电荷的打结线产生静电排斥时，打结曲线会达到最佳形状；理解这种结构有助于解释像DNA这样的生物分子的行为。该项目探索了从理论和计算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.