U.K. industry specialAt last: 140 year-old math problem solved by Imperial College researcher

Published 4 March 2008

Conformal mapping is a key theoretical tool used by mathematicians, engineers, and scientists to translate information from a complicated shape to a simpler circular shape so that it is easier to analyze; trouble is, until now it only worked for shapes which did not contain any holes or irregularities; attempts to solve these problems have defeated mathematicians for 140 years; a researcher at Imperial College London solves the problem

If first you don’t succeed, try, try again. Professor Darren Crowdy, Chair in Applied Mathematics at Imperial College, London, has made the breakthrough in an area of mathematics known as conformal mapping, a key theoretical tool used by mathematicians, engineers, and scientists to translate information from a complicated shape to a simpler circular shape so that it is easier to analyze. This theoretical tool has a long history and has uses in a large number of fields including modeling airflow patterns over intricate wing shapes in aeronautics. It is also currently being used in neuroscience to visualize the complicated structure of the grey matter in the human brain. A formula, now known as the Schwarz-Christoffel formula, was developed by two mathematicians in the mid-nineteenth century to allow them to carry out this kind of mapping. For 140 years, though, there has been a deficiency in this formula: It only worked for shapes which did not contain any holes or irregularities.

Now, Crowdy has made additions to the famous Schwarz-Christoffel formula which mean it can be used for these more complicated shapes. He explains the significance of his work, saying: “This formula is an essential piece of mathematical kit which is used the world over. Now, with my additions to it, it can be used in far more complex scenarios than before. In industry, for example, this mapping tool was previously inadequate if a piece of metal or other material was not uniform all over - for instance, if it contained parts of a different material, or had holes.” Crowdy’s work has overcome these obstacles and he says he hopes it will open up many new opportunities for this kind of conformal mapping to be used in diverse applications. “With my extensions to this formula, you can take account of these differences and map them onto a simple disk shape for analysis in the same way as you can with less complex shapes without any of the holes,” he added.

-read more in Darren Crowdy, “Schwarz-Christoffel Mappings to Unbounded Multiply Connected Polygonal Regions,” Mathematical Proceedings of the Cambridge Philosophical Society 142, no. 2 (March 2007): 319-39 (sub. req.)