Colour by numbers

COMMERCIAL PROFILE:Applied mathematics is a key discipline to solving real world issues, and a network of mathematicians and analysts centred at University of Limerick is engaged in the critical task of demystifying and promoting it in Ireland.

WHAT IS the link between a traffic jam and a settling pint of Guinness; or the price of options on a futures market and paint dripping; or cooking a frozen burger and fuel spillages polluting a water table? The answer is applied mathematics formulas. We're getting rather good at applied maths in Ireland, thanks to a Science Foundation Ireland-funded network of mathematical modellers and scientific computational analysts, centred at the University of Limerick.

Macsi (Mathematics Applications Consortium for Science and Industry) is supported by the largest single grant ever awarded to the field of mathematics in Ireland - €4.34 million from the Science Foundation Ireland Mathematics Initiative - and aims to develop a coherent strategy for the solution of problems which arise in science, engineering and industry in Ireland. It promotes co-operation between universities by operating as a network, having full members at UL, UCC, NUIG, DIT, UCD, DCU and TCD.

This industrial mathematics network is the first of its kind in Ireland. Its members envisage it as the initiation of a new era in Irish mathematics, where mathematicians are viewed as scientists with key skills needed for understanding and investigating different phenomena in the physical, biological and social sciences.

This is where the practicalities of settling pints and burgers come in, as project leader Prof Stephen O'Brien of University of Limerick explains: "Most people don't realise that you can do so much with mathematics. You can solve real world problems with mathematical modelling at a fraction of the cost."

An example of this is work done by Macsi with microchip maker Analog Devices. Polysilicon fuses, which are used in microprocessors, are blown during manufacturing as a means of programming chips. There are occasions when these fuses mysteriously "heal" themselves, thus impairing the performance of the chip.

"There is a way of investigating and solving the problem experimentally, but this is hugely expensive," O'Brien points out. "By using mathematical modelling we can solve the problem very cheaply."

A far more accessible example, O'Brien notes, is that of a farmer whose field forms a right-angled triangle - the two shorter edges of which are 30 and 40 metres. The farmer can use a simple mathematical model to deduce that the longest edge is 50 metres long without having to physically measure it.

So, uses for applied maths can be very down to earth - or as esoteric as you like either. "It's quite surprising in applied maths how problems can turn out to be very similar," O'Brien notes. "You can take two apparently completely different problems and they can turn out to be the same in mathematical terms."

He cites what has become a famous mathematical equation for production stock or commodity option prices as an example of this. "The celebrated Black- Scholes equation of economics, which predicts option prices, is technically a backward diffusion equation," he points out. "This is the same equation which is used for predicting the movement of heat through a substance. Both are known as 'random walks', with the molecules in the substance moving in different directions randomly, and the options market works similarly."

Technically, the same model can be used to predict the cooking speed of a burger, or how droplets will form and fall from a freshly painted ceiling.

Mathematical modelling does not specify a novel kind of mathematics, but a philosophy of asking how things work and where emphasis is placed on the application of mathematics in non-mathematical disciplines such as finance, economics, biology, physics, chemistry or industry.

"An applied mathematician is a kind of lapsed pure mathematician in the sense that he/she would like to prove every result formally, but is sometimes unable to do so, and must make intuitive leaps in the search for understanding," O'Brien explains. "Otherwise the flow of air over airplane wings would not be understood.

"At the more advanced level, mathematical models can be developed to predict the wavelength of the waves in a settling pint of stout or the pricing of credit risk derivatives. Mathematical modellers perceive themselves as being scientists as well as mathematicians and are interested in other disciplines apart from mathematics. Without this philosophy, most modern technology would not exist: airplanes would not fly, man would not have reached the moon, there would be no scientific weather forecasts. There are even mathematical models for the dynamics of marriage!"

O'Brien points to the chicken and egg nature of applied mathematics and how it was developed to solve real world problems. "An obvious example of this is Newton and the development of the calculus. This arose from his practical investigations in mechanics and geometry, and the need to measure areas under curves. Similarly, differential geometry was invented in the context of making maps and surveying a region of the Earth."

Macsi formally came into existence in 2007. Since then it has engaged in more than 20 separate research projects with a variety of leading firms such as Diageo, Analog Devices Limerick, Dell Computers, Waterford Crystal, Boston Scientific, Kostal and Transitions Optical.

Macsi is also about demystifying applied mathematics and promoting it in Ireland. "Most of the multinational industries in Ireland are heavily dependent on solving problems through applied mathematics," O'Brien points out. "Developing a coherent Irish strategy for the solution of problems which arise in science, engineering and industry is therefore paramount to the continuing attractiveness of Ireland as a business location for international companies. Many countries, such as the United Kingdom, have for years recognised the importance of applied mathematical modelling in most areas of modern life. Ireland has for too long lagged behind, with no mathematical modelling centre present in the country. Thankfully, this is now changing."

A sign of this is the appointment of Prof Andrew Fowler as SFI-funded Stokes Professor of Industrial and Applied Mathematics at the University of Limerick. Prof Fowler delivered his inaugural lecture last month, in which he covered topics from Chaos Theory to ice caps on Mars, drumlin formation and the mathematical modelling of leukaemia.

As part of its ongoing work, Macsi is holding a summer school at UL from June 21st to 26th where school students spend a week at UL learning about mathematical modelling and visiting some of our industrial partners. (macsi.ul.ie). Also, from June 28th Macsi is hosting a meeting of ESGI-70 (European Study Group with Industry) at UL. This sees a group of industrialists, enterprise partners and international mathematicians come together for a week to investigate problems of interest to industrial/ enterprise partners.

In the new academic year, UL is introducing an MSc in mathematical modelling aimed at introducing skills needed for advanced mathematical modelling.

With such a range of exciting applications, it's clear that applied mathematics is anything but dry - especially when it's being applied to a pint of the black stuff.

Layman's guide

A farmer whose field forms a right-angled triangle - the two shorter edges of which are 30 and 40 metres long - can use a simple mathematical model to deduce that the longest edges is 50 metres long without haveing to physically measure it

A mathematical model for the real world

Macsi is collaborating with industry on a range of projects, including:

Developing more accurate forecasts of energy output for wind farms with Airtricity;

Helping South African Breweries to solve a problem of labels wrinkling on beer bottles;

Helping Kerry Group to cool cheeses to the required temperature before packing without tarnishing the products;

Working with technology company Point Topic to find more cost-efficient ways of upgrading internet infrastructure to allow subscribers to connect to exchanges, thereby allowing huge increases in consumer upload and download speeds;

And, very importantly, given the current international financial meltdown, the development of new mathematical models for pricing credit risk derivatives - in particular using network percolation models to price collateralised debt obligations. This research is being conducted in collaboration with IKB Deutsche Industriebank (Dusseldorf) and Depfa Bank (Dublin).