Optimal Geometry: Theory and Computation

研究人员Sullivan0071520和他的合作者研究几何优化问题，比如为空间中的表面和节点寻找最小能量形状。他们将其最近的三端嵌入常平均曲率曲面的分类扩展到具有任意数目的共面端的曲面的更一般情况，并且还研究了具有截断端的曲面的细节。此外，调查者还对这些表面进行数值计算，以便例如创建交互式计算机图形。该项目利用Willmore的弹性弯曲能及其梯度流，在欧几里得和球面空间中发现新的极小曲面。威尔莫尔流最近被证明有短期解决方案，但调查人员考虑它是否会没有长期解决方案。这个项目还研究了最小化长度的纽结的构型，给出了小结的长度的新的下界，以及关于长度随交叉数增长的新的渐近界。最后，研究人员利用他在曲线和曲面的数值建模方面的经验，对与曲率相关的量的几何自然离散化给予了新的理解。许多现实世界的问题可以转化为形状的某些特征的优化形式；从数学上讲，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，因此是恒定平均曲率曲面。细胞膜是更复杂的双层表面，它最大限度地减少了弹性弯曲能量，在数学上称为威尔莫尔能量。当绳索被拉紧时，或者如果带电的打结的电线自我静电排斥，打结的曲线就会达到最佳形状；了解这种配置有助于解释DNA等生物分子的行为。这个项目探索了这样的物理自然问题，从理论和计算的角度来看，这些问题仍然具有挑战性。

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.