I study these geometric optimization problems both theoretically and also numerically, using Brakke's Evolver and other software tools. The Evolver was originally designed for computing minimal surfaces, but now it can also minimize other energies like Willmore's. Minimization of this bending energy can, for instance, be used to turn a sphere inside out in an optimal way. To create such a numerical sphere eversion we start with a sphere contorted so as to be halfway inside-out and let the energy simplify it in either direction. I plan to study ways to use this process to create other regular homotopies; this work should also lead to a better understanding of possible singularities in the Willmore flow.
Foams are infinite clusters of soap bubbles, built from cells separated by films which meet in certain singularities. Voronoi diagrams from crystal structures have a similar combinatorics, and give good starting points for relaxation into foams. An interesting question asks for the most efficient foam with equal-volume cells; surprisingly the best known answer combines cells of different shapes and pressures. The Evolver is an excellent tool for attacking this question and others about the elastic properties of foams. I plan to compute foams based on all the TCP structures (crystals of certain transition metals) to look for better equal-volume candidates and to understand their elastic behavior.
My research in optimal geometries involves a combination of mathematical theory and numerical experiments. Many physically natural problems are still challenging from both standpoints, and the interplay between computer experiments and rigorous proofs is what allows progress on both fronts. In addition to the work described above on foams and Willmore surfaces, I have looked at energies and thickness measures for knots, singularities in higher-dimensional soap films, complete surfaces of constant mean curvature, and configurations of points on a sphere.
The most popular knot energies are based on the idea of spreading electric charge along the rope of a knot to make different strands repel each other. This seems to be a promising way to simplify tangled curves; experiments have failed to find any unknotted curve which does not simplify to a round circle, although we have shown that some other knot types could get stuck at a suboptimal configuration. The energies for Hopf links reduce to Coulomb energies for finite sets of points on the sphere; this leads to interesting topological questions about these configuration spaces. A thickness measure for knots is a way to calculate the length of rope needed to tie a particular knot; I have recently been able to describe the geometry of such ropelength minimizers.
I have recently collaborated on a project to explicitly understand complete constant mean curvature (CMC) surfaces with three ends; these turn out to be described by triples of points on a sphere. This work should form a basis for further progress on understanding more complicated CMC surfaces. In a foam, CMC surfaces come together with particular singularities, known for over a century. But in higher dimensions, the analogous singularities have not been classified. I have listed the candidates (related to semi-regular polyhedra) and computations will be necessary to determine which ones actually exist.