Mathematicians, the unsung heroes of research

Jan 13, 2014 by

You don’t often see the news headline ‘maths breakthrough’ but they do frequently happen behind the scenes. The symbolic language of mathematics seems intangible yet it underpins the tools of almost every scientific discipline.

When there is a scientific announcement of a new a device or a drug, scientists acknowledge their local, national and international collaborators in discovery. The accolades can extend back in time, as scientists remind us that they ‘stand on the shoulders of giants’, who paved the way to their new discoveries. But at the heart of new discoveries in physics, chemistry, and biology, often one finds key contributions from mathematicians who have studied fundamental aspects of the problem from a completely different point of view.

Unknown to most of us, the University of Melbourne Department of Mathematics and Statistics is world famous for work on an ongoing problem referred to as ‘self-avoiding walks’ or rather ‘the model for understanding the number of shapes a long chain molecule or polymers can make’. It was this fame that led to Nathan Clisby’s decision to join the ongoing investigation of this mathematical model at the University of Melbourne.

“It is an important model of polymers, and over the past 60 years it has resisted all attempts to find an exact answer, but it is such a ‘rich’ problem that new and interesting discoveries about self-avoiding walks are being made all of the time,” says Dr Clisby, who has recently published major progress on the problem.

“We need to understand the shapes long polymers can form in say liquids, as this may be a key to their ‘performance’ in chemical or biological systems.”

So what is a ‘self-avoiding walk’ and how does it model long chain polymers?

Imagine you go for a walk across the CBD of a city every morning. You mark out a path by unrolling a ball of red wool you string along behind you, and the destination doesn’t matter but you have to keep to two rules: you can’t retrace your steps and you can’t cross you own path at an intersection. Think of it as a loopless path along a city grid. The next day you do another walk with a ball of blue wool; it doesn’t matter if you travel the same streets as yesterday as long as you stick to the loopless rules. How many different pathways or 2D shapes can the balls of wool make? It sounds easy, but is in fact a fiendishly difficult and complex problem.

Now imagine this scenario in 3D: you are an ant crawling around a 100-storey Rubik’s cube going north, south, east, west, but also up and down. The path you take could be the same shape as a string of molecules that are free to move in space. The shape that is traced out is fixed in its length, but at any moment can take another path. How many shapes can you make? This is part of the problem that Dr Nathan Clisby tackles.

“The number of walks grows exponentially with length, which makes this a tremendously challenging computational problem. I developed a new algorithm which used a divide-and-conquer approach to dramatically improve the efficiency with which long walks could be counted,” Dr Clisby says.

“My work builds on the pioneering efforts of Tony Guttmann and others in the mathematical physics group at the University of Melbourne, who have made key contributions to the enumeration of self-avoiding walks and related models.”

The algorithms developed to study self-avoiding walks promise to be useful in studying real problems, such as attempting to understand how proteins fold, or predicting whether a candidate drug molecule is likely to be effective.

“While mathematicians often study problems for their inherent interest and beauty, surprisingly frequently one finds that the problems, which are mathematically interesting, are later discovered to be physically relevant,” Dr Clisby says.

via Mathematicians, the unsung heroes of research.

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