Computational Science and the Ratio of Perimeter to Area

Abstract

Computational science and Engineering is a mixture of scientific computing and applied mathematics (and more). I will describe specific topics from a basic course in this quickly growing area. I hope there will be discussion of what is essential and whether a unity of presentation is possible - or if every subject has to go its own way.

I will also present a problem in plane geometry, to find the maximum area with fixed perimeter. With no other constraints, the Greeks knew that a circle wins. If the set must lie in a square, the solution changes. this leads to a continuous form of the "Maximum Flow-Minimum Cut Theorem" and there are applications of many kind: medical imaging, landslides, traffic flow,... There are also unsolved problems: What set inside a cube gives the least ratio of surface area to volume? (We also hope for a new approach to the original isoperimetric problem).