Sitting in a pub in Leicester Square, talking to one of the most brilliant mathematicians of our generation, is not the way one would normally expect to spend a sultry evening in early June. But it turns out that Artur Avila, winner of the 2014 Fields Medal, takes very spontaneous holidays, and is a big fan of the pub.
In the UK, Artur certainly does not conform to the stereotype of a mathematician. He is good-looking, stylishly dressed in a white T-shirt and designer jeans and asking about the best London nightclubs. However, Avila was born and bred in Rio de Janeiro, famous for its spectacular parties and beautiful beaches. He still spends half his time there, based at the National Institute of Pure and Applied Mathematics (IMPA), and spends the other half at the French National Centre for Scientific Research (CNRS) in Paris, where the nightlife is terrible, according to Avila.
It’s perhaps unusual for an exceptional mathematician not to spend all their time in the USA or Europe, where there are higher numbers of world class research institutions, but Avila believes ‘it’s significant that I studied at IMPA because it shows that Brazil has institutions that can prepare someone to do maths at a high level, and it’s not necessarily true that you always have to go to the United States or to Europe to advance.’
And, of course, Rio has many appealing features: ‘I have several times brought collaborators to the beach with me and we would just sit and share ideas with each other, with the sound of the sea in the background.’
There are clearly advantages of this arrangement to Brazil too. In 2014, Avila won the Fields Medal – a prize often referred to as the equivalent of the Nobel Prize for mathematicians, awarded to up to four mathematicians every four years. Whilst the Nobel Prize and Fields Medal may be similar in terms of prestige attached to them, in reality they have very different aims. The Nobel Prize is often awarded years after a discovery has been made, when its full impact has become evident. The Fields Medal on the other hand is only awarded to mathematicians under 40 years of age, believing that this will stimulate further work. Often, mathematicians are awarded the prize on their ‘last chance’; that is, in the last awarding year before they are 40. This was true of all four winners in 2010, and the other three winners in 2014. However Avila was just 35, and the work that won him the medal had been completed years earlier.
We would just sit and share ideas with each other, with the sound of the sea in the background.
The Fields Medal announcements in 2014 received particular attention because it was the first time in its 78-year history that it was awarded to a woman, Maryam Mirzakhani. Less publicised was the fact that it was also the first time a Latin American had won a Fields Medal. Indeed, no Brazilian scientist has ever won a Nobel Prize either, and Avila believes that `there is a tradition in Brazil to think that no science that comes from there has any quality, and it’s good to give examples that that’s not the case’. Indeed, Avila is convinced that the saturated nature of the academic job market in the West and the constant battle for research positions and funding will make universities in ambitious, rapidly developing countries such as Brazil much more attractive. Which, ‘once you have the type of people who have good plans and lots of energy, and who do high quality work’, will further help to make the world sit up and take notice – provided that institutions ‘don’t make it more complicated than it needs to be.’
Strangely enough, the 2014 Fields Medal was not the first time that Artur and Maryam have shared success. They both won gold medals at the International Mathematical Olympiad (IMO) in 1995 in Toronto and Avila recalls ‘she sat next to me during the prize ceremony, and then of course again in Seoul in 2014.’
And of course, [the IMO] makes you realise that maths can be fun.
The IMO is an annual mathematics competition for pre-university students, of extreme difficulty. A quarter of all Fields Medallists have won medals at the IMO, which is even more incredible given that the IMO began over twenty years after the Fields Medal was first awarded. Countries have rigorous training regimes, typically consisting of intensive camps where students work on problems similar to those that they will face in the competition.
Like many others, Avila believes that training for and competing in the IMO was a major formative experience in the process that led him to become a mathematician: ‘It was the first structured thing that came along that was really challenging. When you start, you struggle with these unfamiliar problems, but that gives you focus to work very hard on them. And of course, it makes you realise that maths can be fun.’
The training was held at IMPA, which opened up the world of mathematics research to Avila. Although he enjoyed the Olympiad, he decided not to participate in further competitions after returning with the gold medal from Toronto at the age of 16, but to immediately begin his undergraduate education at IMPA. At 21, he completed his doctorate, having already proved some outstanding results in his chosen area of dynamical systems.
Dynamical systems is a branch of mathematics that considers how a point in space moves over time if you apply a fixed rule to it repeatedly. A classic example of this is the motion of the bob of a pendulum. The central aim is to determine the behaviour of the point after a long period of time, and there are three main possibilities for this. The first is periodic behaviour, such as the usual back and forth oscillations of a pendulum. However, if the air resistance is drastically increased, the pendulum would eventually come to a halt. In dynamical systems, the system reaches a fixed point. Both fixed points and periodic behaviour are fairly intuitive, but the third possibility – chaos – is much more surprising.
If it were not that hard, then somebody would have solved it already.
If a second pendulum is attached to the end of the first pendulum, its behaviour becomes very complex. Sometimes the bottom pendulum may flip over completely rather than oscillating from side to side. Whether this flip happens, and how long it takes, depends on the position from which the pendulum is first released. This dependence on the initial conditions is enormously sensitive and very counterintuitive: raising the bottom pendulum ever so slightly may cause it pendulum to take a thousand times longer to flip over. This behaviour is known as chaos, and it leads to trajectories that look completely irregular, and entirely different to one where the pendulum has been released from a slightly different position.
Although the behaviour is entirely deterministic, in practice it is so complicated that it is best understood in a probabilistic manner. In Avila’s words, ‘the interesting part is that you start with a deterministic system that apparently doesn’t have have any randomness inside it, and due to the complexity of the system, it is better modelled with something that is very random.’
Communicating is a bit difficult because ideally you don’t want to lie. The mathematical reality is complicated.
Understanding which systems display chaos was a huge strand of research in the 1970s. However, several important questions remained open. One of these concerned a general class of dynamical systems known as unimodal maps. Mathematicians believed that these could be categorised according to their long term behaviour as either regular or stochastic. Regular systems eventually show periodic behaviour or converge towards a fixed point, but stochastic systems have chaotic orbits that appear random, and hence are best understood using probabilistic tools. In 2003, Avila and his collaborators showed this to be true, proving that a randomly chosen unimodal map will either be regular or stochastic. This exceptional result provided an overarching understanding of these systems, and was the culmination of a long line of research.
This is only one of Avila’s groundbreaking results, but he is generally unwilling to reduce the details of his work to a neat, easily understood analogy. `Communicating is a bit difficult because ideally you don’t want to lie. The mathematical reality is complicated.’ For the general public, he believes that the increased visibility of mathematics and mathematicians after the announcement of his Fields Medal is more important than the exact specifics of his work.
The hope is that some young people may be inspired to follow more in his footsteps than those of Messi!
When talking to Avila about his research, he projects a certain sense of ease. He is driven to do mathematics because he is ‘just kind of curious’ and he tries to satisfy that curiosity wherever he is. Astonishingly, he often does mathematics without writing anything down, and has made breakthroughs on the train into work and on flights from Rio to Paris. `The advantage of making the computation without paper is that your memory restricts the complexity of the problem, so you have to structure the question in a smarter way. When you finally succeed you have a better understanding than if you used brute force.’
This is not the only way in which Avila seems relaxed and clear-headed about his research. Stories of mathematicians dedicating years to solving one problem, and the emotional highs and lows that come with it, are commonplace. That’s not for Avila. ‘I don’t find it very smart to fix your ideas on one famous problem. They are often essentially technical challenges that are extremely hard, because if it was not that hard, then somebody would have solved it already.’
Instead, he prefers to work on many different problems and dive into areas that he knows little about. ‘I come as an outsider and look at the traditional problems in the field, but without knowing the usual methods that people use. In the past I have made conjectures which were initially completely wrong because I was so unfamiliar with the topic. But in the end the new approach solved the problem.’ This wide ranging approach has been very fruitful, and Avila has worked with over 30 collaborators worldwide.
These collaborations show the more social side to Avila, and bring us back to the pub. Conversation over, he finishes the rest of his whisky and sets off towards the bright lights of Leicester Square and its clubs, ready to party the night away.
Lead photograph by Tânia Rêgo/Agência Brasil. Licensed under Creative Commons CC BY–NC 2.0.
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