We are living in a golden age of numbers, with a new world of opportunities created by theoretical and applied maths. Chloe Stothart figures out why being a mathematician today really counts.
"There has never been a more exciting time to be a mathematician," according to Peter Grindrod, president of the Institute of Mathematics and its Applications and a visiting professor of mathematics at Oxford University.
"In the next few years, there will be really important developments across a range of sectors - in services, energy, the environment, advanced manufacturing, counterterrorism and the detection and prevention of crime and extremism - all of which have a mathematical element.
"Growth in data and computing power has enabled us to do things that could only be dreamt about a few years ago," he says.
It seems that we are in something of a golden age of numbers. Mathematicians are at the forefront of modelling climate change, developing technologies to reveal weapons hidden in travellers' luggage, predicting the spread of disease, shortening hospital waiting times and easing traffic congestion.
Professor Grindrod has been leading from the front. He was one of the brains who helped make possible the customer loyalty cards that are used by two major supermarket chains to extract and process information on the shopping habits and lifestyles of customers in 20 million households.
Mathematicians are moving into political consciousness, too, especially with the urgent new interest in counterterrorism. John Reid, the Home Secretary, recently called for a new "Enigma group" - similar to the mathematicians at Bletchley Park who helped break German codes in the Second World War - to lead technological efforts to stay ahead of violent extremists.
Even in popular culture, maths and mathematicians have been enjoying a higher profile of late. Number-based games such as sudoku have had a phenomenal reception. The 2001 Hollywood film A Beautiful Mind won four Oscars for its depiction of the life of Nobel prizewinning mathematician John Nash. This year's televised Royal Institution Christmas lectures will be only the third set of lectures to focus on maths since the series began in 1825 (see panel opposite).
The new world of opportunities for mathematicians has been opened up by the explosive increase in computing power that can be accessed relatively cheaply, the huge growth in the amount of electronic data collected by organisations of all kinds and the need for security in electronic communications and financial transactions.
Professor Grindrod's work on customer loyalty cards would not have been possible without databases holding information on millions of shoppers, which supermarkets can now collect at their tills. In the past, the large data sets needed to make accurate forecasts of behaviour were rare. Today, myriad areas from biology to retail are "data rich", Professor Grindrod says, which allows researchers to predict the actions of everything, from shoppers to micro-organisms, with greater accuracy.
As computers have gained in power and speed, mathematical models that require many laborious calculations can now be run fast enough for real-world applications. It is possible to build a model that shows, in just a few minutes, how thousands of patients move through hospitals. This allows health policymakers to see how changes in resources can affect provision.
"The availability of computing power makes maths much more useful," says Chris Budd of Bath University. "Computers are doing for maths what the printing press did for literature."
The rise of digital communications has opened yet more doors. Maths drives web search engines such as Google, which use matrices to rank search results, and provides the encryption needed to secure transactions such as online banking.
It might appear that all the excitement is in applied maths. Not so. Pure theoretical maths has a big role to play. A buzz was generated in the mathematical world when several theories that had gone unsolved for centuries were cracked recently. Fermat's last theorem was proved in 1993, and Grigori Perelman claimed to have solved the Poincare conjecture in 2003. Fields that were once regarded as pure maths can sometimes find a practical application years later. Number theory now helps to make secure online transactions possible.
"Things that look quite applied depend on theoretical areas of maths," says Marcus du Sautoy, professor of maths at Oxford University. "There's now much more fluidity between the subjects, and that's one of the exciting modern developments."
Mathematicians' collaborations with other disciplines and with industry are also pushing maths into new territories. Southampton University's Paul Harper - who models hospital waiting queues, the spread of diseases and the impacts of screening policies - says he works with colleagues in a wide range of disciplines. "I am working with people in social sciences working on the impact of ageing on healthcare and with people in geography on things such as the spread of diseases. Working with people in different disciplines has opened up new horizons."
Commercial projects have changed how mathematicians work in subtle ways.
Bill Lionheart at Manchester University says that mathematicians prefer to work on long-term problems, but industry needs answers quickly. So mathematicians have developed a process of refinement in which they provide the best solution they can in the time they have available and work with the client later to fine-tune their efforts according to need.
"As mutual trust develops, the company becomes willing to fund more theoretical work with longer term benefits," he adds.
But the shift in maths can lead mathematicians away from academe. There are opportunities for mathematicians to earn big sums in the City, to take up new roles in industry or to work with firms on emerging technologies.
Several maths departments say their masters students get headhunted during their courses.
The shortage of mathematicians being trained is something that David Hand, of Imperial College London, refers to as an "international crisis".
Maths still lags well behind big arts subjects as an A-level choice, with almost 30,000 fewer students taking maths than English in 2006. But the numbers are slowly beginning to increase, with some 3,000 more people taking maths at A level in 2006 than in 2005. Some 1,200 more students applied for maths degree courses in 2005 than in 2003. The number of maths academics has also gone up slightly from 2,910 in 2003-04 to 3,355 the next year. The question is whether numbers of students will continue to rise in line with demand for mathematicians.
The demand for mathematicians looks set to continue as yet more stimulating areas emerge. The successful decoding of the human genome is likely to lead to more work to unlock further genetic secrets. The Clay Mathematics Institute in Massachusetts offered $1 million prizes to mathematicians who could solve seven theories that have baffled scholars. And most mathematicians can name at least three fields that are ripe for new breakthroughs.
But, as Professor Budd says, maths is changing so fast that it is hard to say which areas will be big in a decade. "It all depends on the young generation. A lot of discoveries in maths are made by relatively young people," he adds.
MODELS FOR A BETTER WORLD
Paul Harper Senior lecturer, operational research group, School of Mathematics, Southampton University
Hospital waiting lists and the rationing of medical treatment are political hot potatoes, but Paul Harper is making sure that maths informs the policies.
He models the spread of diseases, the impact of different screening and treatment policies and the effects that changes in resources, staffing and scheduling can have on hospital queues.
Dr Harper says hospitals are well suited to research into stochastic systems, a branch of maths and operational research in which he specialises. These involve complexity and variability, which in the case of hospitals could be patients in one department needing different combinations of treatments and variations in length of stay and resource consumption.
His model of the effectiveness of breast cancer screening allows managers in the National Health Service to see how the number of cancers detected and the years of life saved can vary depending on a screening strategy, such as more frequent screening or widening the age of women checked.
"There are financial constraints, so people have to make a decision about what is the best use of public money," Dr Harper says. "From their computers, managers can change some parameters and examine the consequences: what would happen if they had a different number of beds, operating theatres or staff levels."
Bill Lionheart School of Mathematics, Manchester University
Manchester University's maths department is helping to combat terrorism by developing the maths needed to make airport baggage scanners better at detecting hidden weapons.
Bill Lionheart's work is an example of how maths can combine theories created decades ago with cutting-edge research. He is working out the maths needed to make a machine that will scan in 3-D, such as a medical scanner, run at the higher speeds of a 2-D airport X-ray machine; 3-D images will make it easier to spot weapons concealed in luggage.
In his work, Professor Lionheart draws on discoveries made by Fritz John, a German-Jewish mathematician who fled the Nazis. Professor John created 3-D pictures using a theory built on the geometry used to make 2-D images.
Professor Lionheart is adapting this for use in high-speed scanning.
"John was interested in this problem, but not for airport security. It is a nice illustration of how maths can work. It can be a long time before an element finds a useful application," he says. "It is a two-way street to some extent; technology drives maths development, but sometimes the maths problems have been solved if we know where to look."
David Hand Head of statistics, department of mathematics, Imperial College London
Anyone who has ever been phoned by a bank to query a suspicious credit card transaction has been at the receiving end of sophisticated maths of the sort that David Hand performs.
He develops the statistical models that predict the behaviour of individuals so that banks can decide whether to grant a loan or a mortgage or if a transaction might be fraudulent.
The simplest model might weight the age, income and credit history of a customer to produce a score that can be compared to a threshold.
But much more sophisticated models are being developed. For example, models can predict if someone might default on repayments or if a customer might like an alternative financial product.
Professor Hand says: "Investment banking underwent a mathematical revolution 30 years ago; personal banking is now going through a similar revolution."
The need for statistics in personal banking has been driven by a huge demand for loans, credit cards and mortgages, combined with growing competition between banks.
There are millions of credit cards in circulation in the UK and thousands of mortgage products on the market - human interviewers could not investigate every transaction or assess every detail of each application.
DO PRIME NUMBERS MAKE YOU A BETTER FOOTBALLER AND OTHER MATHEMATICAL MYSTERIES
Marcus du Sautoy is on a mission to show the public how exciting maths can be. The Oxford University mathematician is giving this year's Royal Institution Christmas lectures, which sold out in record time.
"It is only the third time since 1825 the Royal Institution lectures have been on maths. That this year's lectures were the fastest selling is a great indication for maths," he says.
Professor du Sautoy has particular affection for the Christmas lectures because they helped to pique his interest in maths. As a schoolboy, he watched Sir Christopher Zeeman give his lecture in 1978. "He wanted to show kids something exciting about maths. I saw it on TV and thought 'I want to do that'," he recalls.
Now it is Professor du Sautoy's turn. He aims to enthuse his audience with discussions of the shape of the universe, of how to create secret messages and of why David Beckham wears the number 23 shirt at Real Madrid.
Before Beckham's arrival, he says, all the top players at the Spanish club wore shirts with prime numbers on - Brazilian aces Ronaldo and Carlos wore 11 and 3; Zidane had the number 5 and Raul number 7. Beckham also chose to pull on a shirt with a prime number on it.
"It has to be more than a coincidence that the building blocks of the team wore prime-numbered shirts, and prime numbers are the building blocks of maths," he says.
He tested the theory on his own Sunday league side by getting them to put on prime-numbered shirts after a run of bad performances. It worked for a while. The team were promoted, but they slid back after a season. Now, he jokes, he needs to make another mathematical discovery that could help them to regain their glory.
The public have an appetite for maths, Professor du Sautoy says, and mathematicians must feed it by finding ways to show the fun, wonder and mystery of numbers.
Professor du Sautoy points to the enthusiastic response he gets when presenting maths programmes and the popularity of sudoku puzzles as evidence of the public's love of problem-solving. The key is not to kill this interest by demanding that people understand every detail or overemphasising arithmetic - something that our school system is sometimes guilty of doing, he says. Mathematicians, who must obsess over detail in their professional lives, have to learn to "go for the big, broad sweep" when communicating maths to the public.
But Professor du Sautoy believes that maths communication is in a good state in this country. "Being a scientist is about discovery but also communication. Discovery does not really exist without communication," he says.