The global theory of minimal surfaces in space is in a phase of explosive growth. Many new methods of constructing complete embedded minimal surfaces have recently been found; in place of a dearth of examples just a few years ago, we now have a quite varied collection of surfaces, including infinite families. A basic problem is to classify these examples, i.e. collect them into families with common properties and understood limits. Fruitful approaches have recently been developed that combine numerical simulation with methods from the theory of geometric structures on surfaces and classical complex analysis, notably Teichmuller theory. Some of the problems the team will attack are: Are there embedded minimal surfaces with one heliciodal end and arbitrary genus? Is the classical Scherk surface the unique desingularization of a pair of planes? Of what families is the Scherk surface the limit point? At the same time, the group hopes to make progress on simulation of minimal surfaces. For example, we hope to set up a library of Weierstrass representations of minimal surfaces which is reproducible, fully documented, and useful as a research tool.
A guiding philosophy in many areas of science, from physics to biochemistry to ecology, is that nature is maximally efficient; indeed, many explanations of natural phenomena have at their foundation the assumption that the phenomenon has optimized some or several of its features in the expression we witness. At its base, this philosophical principle is mathematical in nature: we search for principles in science that can be formulated as extremal problems. In mathematics, we can make this assumption of optimality very rigorous by expressing it as an equation. This leaves us with the problem of understanding all of the solutions of that equation. In this project, we aim to study one very rich type of optimization problem, the minimal surface problem, which is already known to have a number of quite subtle characteristics. (A minimal surface is one for which each small piece has less area than any other surface with the same boundary.) The study of these surfaces has its origins in physical problems studied first by Euler; then, a century later, the problem also arose in the studies of the behavior of rotating droplets and soap films by F. Plateau. Today the applications range from cosmology to the understanding of the structure of stable periodic structures in compound copolymers. As in many other optimization problems, for the minimal surface problem, we do not have much general information about solutions to the equation expressing extremality. At present though, we do have a wide variety of examples which help to guide our intuition, and which we are beginning to organize. It is thus a good model problem, enriching our understanding of all optimization problems.
