Minimal surfaces are mathematicians' idealization of soap films. They are of interest to artists and architects, to designers of strong, lightweight structures, and to materials scientists who want to understand the microstructure of compound materials. The lecture will touch on these applications and will also try to explain how computers have become extremely useful in the hunt for minimal surfaces and in their application to physical problems. Computer graphics animations of interest to both scientists and artists will be shown.
Abstract: A recent issue of the Philosophical Transactions of the Royal
Society of London was devoted to ``Curved surfaces and chemical
structures.'' Differential geometry is playing an important role in the
experimental physics of materials, but in ways that are surprising---and
sometimes troubling for mathematicians. I will discuss this by considering
the paper of Viet Elser in this issue, which purports to construct its
title surface: A cubic Archimedian screw.
We know Archimedes' screw by the name of the helicoid, an embedded minimal surface. The ways in which a properly embedded minimal surface of finite topology can diverge is a fundamental question, whose relevance is underscored by some strong theorems and by some recently-discovered examples. The helicoid stands as the prototype. There is hope that the cases yet to be classified consist entirely of surfaces asymptotic to the helicoid. In the second, less applied, part of the lecture, I will present some of the evidence as to why this may be the case.