Raimundo was from the state of Amazonas and grew up tending a riverside plot of yuca. At age 15, he went to Manaus, where he got a job as a waiter at the governor's mansion. Balancing work and school, he graduated from high school. Raimundo decided to try his luck in Rio de Janeiro, was accepted for a civil service post and entered the Reinsurance Institute of Brazil (IRB), which helped to pay for his college - "I think he studied accounting," his son says. At IRB, he met Lenir. They had one son [Artur Ávila, the subject of this biography], who was still small when they separated [he was eight]. He was raised by his mother.

At school, I always wanted to read things that weren't being taught yet, texts from future years. My parents didn't have much of a way of guiding me to more specific texts, but I would read pretty much anything, ask them to buy books, etc. and, based on these books, I was learning.

For the first time, I felt I couldn't do anything.

People didn't study for the sake of knowledge, but to pass the test. In the schedule, after physics there came a Portuguese class, and then geography. In a system like that, what could I learn? I preferred mathematics; I chose to learn one thing well, for life.

IMPA sometimes accepts younger students who are still in high school. They do this if they perceive that the student is able to do the work. I knew about this and this aroused my interest in doing the same thing. This wish was granted when I returned from the 1995 International Mathematical Olympiad, at which I won the gold medal. IMPA suggested that I take one of the level 1 courses shortly before starting the Master's degree. If everything went well, I would enrol. In fact, that is what I did while still in the last year of high school.

[Welington de Melo] was an extremely rigorous mathematician, committed to the highest quality of the work carried out at IMPA. He wanted the work to be at the highest possible level, but, because of this rigour, his reputation scared students a little. I found it a bit risky to enrol, I think in 1997, in a course taught by Welington at IMPA. With great caution, I did not enrol in the course, but went to attend classes. He turned out to be, unlike his reputation, very nice to me. At the end of the course, he said that I had done well.

Just before welcoming the Brazilians to Stony Brook, Lyubich had written a series of articles in which he proved his most important results. "Very few people really understood what it was about", he commented, "and Welington was a notable exception. It was his proposal that Artur explore this line of research." ... The three spent a month tossing ideas back and forth, in a style of doing mathematics that only requires a blackboard, chalk, and space to walk back and forth. The daily conversations took place in the Institute's rooms, at Lyubich's house, in restaurants or during walks through the woods around the university. ... "Occasionally, amazed, I realised that Lyubich and I were a little behind Artur," recalled de Melo, "he was so young. .. I would forget about it but then be scared." ... After a month of intense discussions, the trio had a clear strategy for resolving the problem that consumed them, but the proof was still out of reach. There was an obstacle that refused to budge. Lyubich and de Melo decided to leave it in the boy's hands. "That was in March", remembers Artur. "I kept the problem in my head and a few months later, in September or October, I had a weird idea."

We prove that in any non-trivial real-analytic family of quasi-quadratic maps, almost any map is either regular, that is, hyperbolic (i.e., it has an attracting cycle), or stochastic (i.e., it has a probability absolutely continuous invariant measure). To this end we show that the space of analytic maps is foliated by codimension-one analytic submanifolds, 'hybrid classes'. This allows us to transfer the regular or stochastic property of the quadratic family to any nontrivial real analytic family.

I completed my PhD in Brazil and went to France in 2001. My first jobs were in France and I spent five years there before returning to Brazil. After that, I spent three years in Brazil, and then started spending half of my time here and half there. The time I spent in France complemented my training as a mathematician and I extended my areas of research. I finished my PhD with the ability to do research at a high level. My results were recognised as important, but I had a restricted view of the area and its position within the whole of mathematics. In Paris, I had contact with the largest community of mathematicians in the world and unparalleled activity. This forced me beyond my area of expertise at the time, one-dimensional dynamics, and to look for other things, in order to be able to interact with these people who were not necessarily interested in the same things that I was. In this search with such good professionals, with so many possible co-authors, I started working in other areas and my work was lauded due to what I did in these areas. The mathematician that I am today is a result of my time in France and my training in Brazil.

Ávila leads and shapes the field of dynamical systems. With his collaborators, he has made essential progress in many areas, including real and complex one-dimensional dynamics, spectral theory of the one-frequency Schrödinger operator, flat billiards and partially hyperbolic dynamics. Ávila's work on real one-dimensional dynamics brought completion to the subject, with full understanding of the probabilistic point of view, accompanied by a complete renormalization theory. His work in complex dynamics led to a thorough understanding of the fractal geometry of Feigenbaum Julia sets. In the spectral theory of one-frequency difference Schrödinger operators, Ávila came up with a global description of the phase transitions between discrete and absolutely continuous spectra, establishing surprising stratified analyticity of the Lyapunov exponent. In the theory of flat billiards, Ávila proved several long-standing conjectures on the ergodic behaviour of interval-exchange maps. He made deep advances in our understanding of the stable ergodicity of typical partially hyperbolic systems. Ávila's collaborative approach is an inspiration for a new generation of mathematicians.

Artur Ávila is awarded a Fields Medal for his profound contributions to dynamical systems theory, which have changed the face of the field, using the powerful idea of renormalization as a unifying principle.

Artur Ávila is clearly an exceptional talent in the world of mathematics. But he also is a symbol of the remarkable creativity that we can find among young researchers in the developing world. At TWAS, we are proud of our links to this scholar, and we are confident that he has many more years of important work ahead of him.

Given a system that evolves in time, what can be said about its behaviour over long time scales? Dynamicists have been occupied by this question since Newton, but since then our understanding of what we should aim at has changed a lot. This happened in large measure due to the discovery of chaos, and how it can arise even in some of the simplest situations. In this public lecture, Ávila will reflect on how his field has changed over time, and examine current questions at the forefront of dynamical systems theory.

Many times in my career I sought out known difficult problems and worked hard to solve them. Since I did this several times, it is certainly true that I solved many problems. But, to a lesser extent, I also worked on building and developing these theories, which sometimes involve not just solving, but also formulating the problem. In the beginning, I resolved a problem related to Schrödinger operators, but later I also constructed a theory and solved problems related to it. But, certainly, the most visible aspect of my work is my many solutions to dynamic systems problems in different contexts

Being both Brazilian and French, it is important to me that conditions are favourable for mathematical research in both countries. It is important not to let the researchers' careers deteriorate, not to put off talents. Some of these brilliant minds could indeed be discouraged by increased competition, difficult financial conditions and a lack of social recognition for researchers. As a consequence, they may turn away from this path. We must ensure that this does not happen. For me it is different. I started my career very early and I am unable to do anything but maths. But we must also think of a more egalitarian research system, not only conducive to internationally recognised researchers. Ideas are universal and research is based on the work of many people, including also those who will unfortunately never be distinguished.

Things tend to become more and more technical and thus people get further apart as they don't speak the same language. It has become clear that for an analyst it is more difficult to understand what they are now doing in algebra. My view on this is that people should just do what they want. If someone wants to think about prime numbers he should do that, if he wants to think about fractals then that is fine as well. Some researchers will make progress working on isolated topics and others might want to bridge together areas and examine the connections between discoveries. Not everyone has to do that. Some people might just want to play on their own. You cannot anticipate what is going to work out. It is also not necessary to decide which fields of mathematics have to be stimulated, since we don't know whether the next discovery in geometry will depend on a discovery in algebra and so on. This is also my point of view regarding focusing on pure or applied mathematics. What might be crucial in the solution of some very specific problem (with industrial applications or whatever) often happens to be just some beautiful object that someone studied for its own sake. The motivation of the latter (often similar to the motivation a child has for playing a game) is irrelevant to the fact that what has been discovered turns out to be useful to the former. And of course it works both ways: much of my work deals with theories that first arose in connection to physics, but for me what matters is that these theories are rich mathematically, and I study them from this perspective. So "pure" mathematicians have to recognise that many of the nice objects they have at their disposal with rich mathematical theorems that one works on just because they are beautiful, without caring about the motivation or the application, might have not been discovered if it was not for some physical model that someone was examining because they cared about the physics of it. Thus, by admitting this and letting people work on what they want, the whole community benefits from the eventual (unpredictable) interactions.