Optimal Geometry: Theory and Computation

Sullivan0071520 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。 他们最近的分类扩展到更一般的情况下，与任何数量的共面结束的表面，嵌入式常平均曲率曲面与三个端部，并详细调查与截断端部的表面。 此外，研究人员计算这些表面的数字，以便，例如，创建交互式计算机图形。 该项目使用Willmore的弹性弯曲能量及其梯度流，在欧氏空间和球面空间中发现新的极小曲面。 Willmore流最近已被证明有短期的解决方案，但研究人员认为，它是否可能无法有长期的解决方案。 本计画也研究使绳长最小化的绳结构形，给出小绳结绳长的新下界，以及绳长随交叉数增长的新渐近界。 最后，研究人员利用他的经验与曲线和曲面的数值建模，给新的理解几何自然离散量有关的曲率。 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，薄膜，如泡沫中的薄膜，通常使其面积最小化，因此是恒定平均曲率的表面。 细胞膜是更复杂的双层表面，其使数学上称为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。 这个项目探讨了这些物理上自然的问题，从理论和计算的角度来看，这些问题仍然具有挑战性。

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.