Maths for the 21st century

June 25, 1999

In anticipation of the fourth world maths conference that is taking place in Edinburgh next month, Julian Hunt and Jack Carr

survey future applications of mathematics and computing Mathematicians like to work on very definite problems and are always searching for good new ones to solve. This is why they can be as famous for stating a problem or a conjecture as for solving it. Lewis Fry Richardson, the British Quaker mathematician and meteorologist, asked the curious question, in the 1920s, on the top of an open London bus: "Does the wind have a velocity?" In other words, does the measured wind always vary smoothly or could it suddenly jump from one value to another?

This is a serious practical problem, which the Californian Fields medallist Stephen Smale - winner of the Fields Medal, the top prize for mathematicians - recently included in his list of great unsolved problems for the millennium. Politicians are also well known for their questions. Gladstone's questions about the Balkans and devolution appear to remain unsolved to this day.

But how do mathematicians' "problems" relate to the difficult questions of industrial engineers, medical and commercial researchers, and environmental scientists? These also need to be analysed and understood conceptually and, where possible, numerically and accurately. Increasingly, industry finds that mathematicians can help.

For example, the aerospace and automotive industries have to calculate swirling flows somewhat like the vortex you see in the plug hole of a bath. In mathematicians' eyes these observations lead to intriguing problems about long thin vortices and how they can be tangled or knotted, which also have wider relevance for applied physics in the study of "fault lines" within the atomic structure of crystals, and in the vortices found in very low-temperature "super fluids".

Perhaps surprisingly, this approach has proved to be practical. The mathematical classifications of these different types of swirling flow are now routinely applied by the aircraft industry to describe succinctly the complex flow patterns over wings. This may well lead to designs of low-drag aircraft and quieter jet engines whose turbulence is suppressed by thousands of controlled moving micro-panels on the surfaces of wings or the walls of engines.

The basic mathematics of this problem, as with so many others, was first considered by the 19thcentury mathematician and physicist James Clerk Maxwell, whose unifying theory for electromagnetic and light waves led to the invention of radio. He first suggested the connection between knots and vortices.

In this case mathematics helps solve a particular problem, but more often its main use is in providing methods of analysis, concepts, computation and measurement that specialists in other disciplines can use.

The techniques now available have changed enormously since 1900 when the German mathematician Hilbert addressed a mathematical congress in Paris. He listed 23 mathematical problems for the coming century, but did not include any mention of approximate methods or statistics. But in one problem on the "calculus of variations" he pointed to the important general technique, now widely used, of studying the overall properties of mathematical systems. At that time mathematicians were just beginning to develop methods to solve approximately the kinds of differential equations that describe natural phenomena and industrial processes, whose use Richardson pioneered for numerical weather prediction.

We are at an equally exciting point in time today. Many new developments in the applications of mathematics and the new mathematical problems that arise will be presented at ICIAM'99, the Fourth International Congress on Industrial and Applied Mathematics.

This joint effort of over 20 scientific societies around the world

is to be held in Edinburgh on

5-9 July. Its organisation, led by Sir Michael Atiyah, is supported by the mathematical community in the UK, involving chiefly Edinburgh and Heriot-Watt universities, the Institute of Mathematics and its Applications, and the International Centre for Mathematical Sciences in Edinburgh, Maxwell's birthplace.

Sponsorship has come from scientific and commercial organisations. Among the 1,200 or more presentations at the congress there will be many examples of mathematicians dealing with new problems, ranging from tumour growth in medicine to superalloys in metallurgy, as well as many more where the use of mathematics is well established - engineering, biotechnology and various aspects of environment modelling, such as air pollution dispersion.

Following on from the successful session on women in mathematics held at the European Congress of Mathematics in Paris in 1992, there will be a mini-symposium of presentations by postdoctoral women mathematicians, organised by the Association for Women in Mathematics and European Women in Mathematics. Distinguished women mathematicians are also giving some of the invited lectures, including Leah Edlestein Keshet of Canada on mathematical models in biological problems, in particular the mechanical activity within living cells; and Margaret Wright of Bell Labs in the US on the mathematical problems of optimisation related to industrial and computer systems, a theme which harks back to Hilbert.

A new aspect of the congress will be papers on the latest mathematical models and computer simulations for the insurance and financial markets. One example is statistical modelling relating medical data and risk; another is the use of differential equations, often generalising those used in physical sciences, applied to the valuation of financial derivatives.

There are many other examples of how mathematicians' overview of analytical and modelling techniques in different fields enables them to provide practical advice in a wide range of scientific and commercial situations. Many universities, led by Oxford, now have a broad-based expertise in providing such "synoptic" advice to industry. They have joined together to form a European Consortium for Mathematics in Industry, with support from the European Community.

One of their mini-symposia concerns the mathematical basis for the design of conductor chips. Chips are now smaller than the diameter of a hair, a result of advances in solid-state physics and also new approaches to calculating how electrical currents and electromagnetic waves pass through the various materials in these devices. New statistical methods are being presented in Edinburgh with applications both to basic physics research and to industrial problems.

This is one of many situations where those who are faced with practical computational problems, such as the modelling of water quality in rivers or the erratic price movements in financial markets, have to decide whether the calculations should be made by assuming that the problem is inherently deterministic or whether it is better treated statistically, like working out the chances of a ball ending in a winning position in a pinball machine. This is the question for a special session at ICIAM'99 run by Noel Barton of the Commonwealth Scientific & Industrial Research Organisation of Sydney.

For some problems, this question becomes less relevant as computers become increasingly able to calculate every individual random step in a process, even at the detailed level at which molecules collide and interact with each other. This is the basis of modern computational chemistry and drug design, and has enabled traffic flow on the M25 to be modelled by tracking the movement of every car; an improvement on earlier deterministic models of the "waves" of traffic jams.

Greater availability of computer power has forced a deeper analysis of the underlying mathematics in practical models, particularly to answer the question as to whether an accurate calculation is even possible, or conversely whether several possible solutions exist, the topic of a lecture by Olga Oleinik of Moscow.

How does research in applied mathematics actually get used in practice, and how long does it take? The presentations by participants from industry and environmental organisations provide some answers, including the Numerical Algorithms Group of Oxford, which exports its mathematical software around the world, and the Meteorological Office, whose mathematically based numerical weather forecasting codes are continually under development to meet the challenges for improved accuracy set year on year for it by Parliament.

Other answers come from successful collaborations between academics on the one hand and industry and commerce on the other. In the metal-making industry, cheaper and stronger materials have resulted from recently improved processes, which were developed with the aid of new mathematical models, for example to describe how magnetic fields control molten metal as it is poured and solidifies. This example of the timely application of mathematical research contrasts with many other cases where researchers and industries take many years to complete the research and apply it in practice.

Nowadays almost all mathematics is far more likely to be applied than in the past. Recent work on number theory is being used to design better systems and codes for secure communications protocols. These are essential to ensure your money is safe at the "hole in the wall" or when you do your shopping on the internet.

Julian Hunt is professor of mathematics, University of Cambridge. Jack Carr is professor of mathematics,

Heriot-Watt University. For further

details of the Congress in Edinburgh, see

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