Discretely Structured

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What do discrete mathematicians do? When does 7 + 7 = 2? What sort of geometry makes sense to a computer? What is a matroid? Geoff Whittle has some answers to these questions. Anna Meyer spoke with him.

The mathematics that was successful over the last 300 years is what physicists used for modelling things like movement in space and time, or effects under a gravitational field, says Professor Whittle. “These are situations where you have a continuous change.” With the development of computers, which operate using only discrete numbers – those that jump from one to the next with no graduations in between - a completely new type of mathematics was needed. Discrete number systems are all around us - we are all familiar with the 12 hour clock, for example, where 7 hours after 7 o’clock is 2 o’clock, so that 7+7=2. The field of discrete mathematics
is now a major area of interest for mathematicians.

Professor Whittle’s research concentrates on matroid theory, a part of discrete mathematics that deals with the sometimes highly unusual geometric structures that can be constructed from sets of discrete numbers.

Familiar Euclidean geometry involves drawing shapes based on the infinite number of points that lie on a line of continuous numbers. Examples include a sphere or a cube, or the spatial positions on a map,which have a smooth, continuous connection between points.

Geometry using discrete numbers, however, is a little different; structures drawn from them have only a finite number of points. The points ‘jump’ from one to the next, and anything in between simply does not exist. “It’s not the familiar world of geometry,” said Professor Whittle. “But it’s a world that makes a lot of sense to a computer.” Matroids are a specific type of discrete geometry. The illustration is an example of a basic matroid. The points represent co-ordinates specified by numbers from a discrete number set. These are just like the x,y,z points on a continuous graph, or points of latitude and longitude on a map, except there are only a finite number of points in a matroid. The lines show the relationships between each point, such as whether several points lie on a straight line.

For any particular discrete number system, an enormous number of different matroids can be constructed. “You can decide to make points with only three co-ordinates, or even ones with 10 co-ordinates,” said Professor Whittle. “The ones we can draw are the ones that exist in two and three dimensions, but they could exist in a million dimensions.
Tragically, we can’t visualise those ones. That’s where the mathematics becomes important, as a way of rigorously testing that you’re not kidding yourself!”

Professor Whittle is investigating the properties of different matroids. He is trying to answer a key question about infinite sets. An infinite number of matroids can be obtained from each discrete number system. However, like a set of Russian dolls, some may be contained in others. It seems that if we avoid having matroids contained in others, then it is only possible to construct a finite set of matroids from a given discrete number system. If this could be proven, it would make matroids much simpler and easier to work with.

“Seeing that something is finite, when you would have expected it to be infinite, is very, very powerful,” he said. “I spend my time dreaming about this sort of stuff, much to my wife’s annoyance.”

Professor Whittle describes himself as “an old-fashioned type of pure mathematician,” preferring the intrinsic interest value of his work to any potential uses. “Personally, I don’t care about whether the stuff I do has any use or not – although a lot of it does turn out to be useful,” he said. “I view mathematics as being like music or poetry.”

“If you were talking to a musician, you wouldn’t dream of asking for applications, but somehow or other, mathematics has to sell itself. Because it is so useful, people think that it only exists for its use, but actually that’s not true. The development of mathematics is one of the rich streams of intellectual history.”

Every year, the NZIMA awards a Maclaurin Fellowship to enable a New Zealand mathematician to take time out for full-time research. The 2006 recipient was Geoff Whittle, a Professor of Mathematics at Victoria University.