Mathematics in a geometric universe

DISTINGUISHED PROFESSOR ROBERT MCLACHLAN - Institute of Fundamental Sciences, College of Sciences

How do we predict the future behaviour of complex systems like the weather, or the orbits of planets? As anyone affected by an unexpected downpour will know, this is no easy task.

‘Getting from the laws of motion, which are understood, to what the weather is going to do is very difficult,’ says Distinguished Professor Robert McLachlan of Massey University’s Institute of Fundamental Sciences. ‘There’s a fundamental issue that mathematical errors build up in time when you’re trying to predict into the future behaviour of large, complex systems. Tiny errors to begin with will become larger.’

He has spent his career working on new mathematical methods that can be used in these difficult problems. ‘We found that in some areas, using certain kinds of new equations known as geometric integration gives more reliable results. The idea is that certain features of the equations are preserved exactly, and these are things that we know are conserved physically, such as mass and energy. Therefore, the numerical methods should also preserve these features. The normal numerical methods would introduce errors in those areas, but these methods avoid this issue.’

Distinguished Professor McLachlan’s methods are in widespread use in computational science, in areas as diverse as a possible celestial origin of the ice ages; the structure of liquids, polymers and biomolecules; quantum mechanics and nanodevices; biological models; chemical reaction-diffusion systems; the dynamics of flexible structures; and weather forecasting.

‘The solar system is an area where these methods have been used a lot, and in systems of planets around other stars as well,’ he explains. A classic problem in science is whether the solar system is stable, or whether the planets will eventually move out of their orbits. For example, it is known that over a very long period, the Earth does wobble up and down in an apparently random fashion. Will these small movements build up, or cancel out? ‘It’s very difficult to tell. That’s why, when running a simulation of the whole life of the solar system for example, the numerical methods have to be reliable.’

In some parts of pure maths, the objects you’re studying don’t really come up in the natural world, you just study them for their own sake. I try to combine the pure maths approach and the applied approach.


Another use for geometric integration is in predicting the weather, because among other things, it allows the shape of the Earth to be taken into account. Traditionally, weather forecasters worked on flat plane maps, which can be described using graphs with ordinary x and y coordinates. But, of course, the Earth is not flat, but is instead a sphere. ‘The famous thing about a sphere is you can’t have coordinates that cover it all. Latitude and longitude have trouble at the north pole, because longitude isn’t defined there.’

The course of Hurricane Sandy in 2012 is an example of the importance of taking the shape of the Earth into consideration in weather forecasting. The hurricane initially tracked in a north-easterly direction, before suddenly turning 90 degrees towards New York, causing a major disaster. Due to the use of better numerical methods, the European Centre for Medium Range Weather predicted this left turn six days in advance, and were the only ones worldwide to do so. ‘This shows the importance of having a good numerical method, used in the right place,’ says Distinguished Professor McLachlan. Since Hurricane Sandy, numerical methods have been improving, and new ideas mean that geometric integration may one day be of use here as well.

Distinguished Professor McLachlan also has an array of medals and awards nationally and internationally, and has been awarded visiting fellowships at several institutions, including Germany’s prestigious Oberwolfach Mathematics Research Centre. Although his work is widely used to tackle real-world problems, he sees himself as a pure mathematician at heart.

‘It’s got more that way as the years have gone by, either from my personality or from force of circumstances,’ he says. ‘In some parts of pure maths, the objects you’re studying don’t really come up in the natural world, you just study them for their own sake. I try to combine the pure maths approach and the applied approach. Pure mathematics is universal and definite, so once you’ve proved a theorem, everything is completely settled for all time. Making a definite theoretical contribution is extremely satisfying.’