Theory and Computation of Optimal Geometries

点击翻译按钮获取中文摘要

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.