Interview Artur Ávila

The man who calculates

Marcos Pivetta Published in September 2014

rtur Ávila, a Rio de Janeiro-based specialist in a field called dynamical systems in which the objective is to develop a theory capable of predicting the long- A term evolution of natural and human phenomena, received the Fields Medal, the most important international mathematics award, on August 13, 2014. At 35, he became the first Brazilian and South American to be granted such an honor, which is given every four years by the International Mathematics Union (IMU) to researchers under 40. In addi- tion to Ávila, who works at the National Institute of Pure and AGE 35 Applied Mathematics (IMPA) in Rio de Janeiro and is head of research at the National Center for Scientific Research SPECIALTY (CNRS) in Paris, the medal was also awarded to the Austrian Dynamic systems Marnin Hairer, the Canadian Manjul Bhargava, and the Ira- EDUCATION nian Maryam Mirzakhani, the first woman to win the award. Federal University Of the four winners, who also received €10,000 in cash, Ávila of Rio de Janeiro was the youngest. “For the other candidates, this was the (UFRJ) and last chance to win the medal because of the age limit. Due National Institute to circumstances at the time, I thought I had little chance of of Pure and being recognized this year,” Ávila said in an interview in Rio Applied Mathematics de Janeiro, after traveling to Seoul to receive the award at the (IMPA) 27th International Congress of Mathematicians. The excellence of this Brazilian, who also gained French INSTITUTION citizenship last year, manifested itself early. The only child IMPA and of divorced parents (he has a half-sister through his father), National Center for he had a middle-class upbringing and attended good schools. Scientific Research In 1995, at the age of 16, Ávila won the gold medal at the In- (CNRS), Paris ternational Mathematical Olympiad. Two years later, while

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pESQUISA FAPESP | 15 still a student at the traditional Santo the existence of another strong candi- strated its ability to organize major events Agostinho High School and without hav- date in a similar subarea and the fact that helped its candidacy. Many countries ing enrolled in a university, he completed I was younger than the other candidates. that have hosted the event have never a Master’s degree at IMPA. In 2001, at the I still had one more chance and could won a medal, such as South Korea, In- age of 21, he finished his Doctorate, also win the medal in 2018. For the others, dia and Spain. The medal is for recogni- at IMPA, along with an undergraduate this was the last chance to win because tion of mathematical research, a purely degree at the Federal University of Rio of the age limit. Due to circumstances at scientific issue. It is the first time a prize de Janeiro (UFRJ). “Smart students usu- the time, I thought I had little chance of winner has completed all of his education ally like to show off and ask many ques- being recognized this year. through the doctoral level in a develop- tions,” says researcher Welington Celso ing country, rather than in Japan, parts de Melo, Ávila’s PhD advisor. “Arthur Who was the other candidate in your of Europe, the United States or Israel. I was different. He did not talk much, but subarea? studied only in Brazil, and this did not when he asked questions I was unable to The Iranian Maryam Mirzakhani, who hold me back. The quality of the PhD I answer them immediately. I would have also won the award, was the other candi- did at IMPA was the same as whatever to go home and think about the answer.” date. It was exceptional that they gave the I could have done abroad. It is a clear Married to a researcher in econom- medal to both of us. We work in neigh- demonstration of the quality of what can ics, with no children, Ávila lives in both boring subareas, and that made it unlikely be done here in Brazil. This, of course, is Paris and Rio de Janeiro, the two cit- that both of us would be honored in the due to the teaching and research IMPA ies that let him do what he likes best: same year. For this reason, and because has been carrying out for decades. solve big mathematical problems. Among it was the last chance for her but not for his achievements are solutions for the me, I thought that I would not win. How did you see yourself at 21, finishing Schrödinger operators, mathematical your PhD? Did you feel you were special tools that help describe the evolution of Have you and Maryam worked to- because you were a prodigy? vector states in quantum systems. Even gether? I was younger than most PhD students. before winning the Fields Medal, Ávila, No. However, I have worked with peo- However, I knew that I could finish my who has over 50 published papers, en- ple who have worked with her. She has PhD early but still not become a great joyed enormous prestige in mathematical already used results from my work and researcher. You can be an excellent math circles. His way of conducting research is vice-versa. She works in a subarea that student and have excellent grades but not unusual. He does not read much, he does intersects with mine, and we have com- have the ability to do research. Even if, not teach, and he can work at home, in mon interests. So, we could work togeth- in this context, you are able to do PhD his office, or even on the beach if he is in er in this subarea, and we are certainly research, you could have difficulty con- Rio. He prefers to learn a new research working in the same direction with com- tinuing in a research-oriented career. topic by establishing partnerships with mon co-authors. For some reason, we Sometimes, you cannot continue to pro- colleagues who are specialists in the field have never even met. duce the same quality of work. You could in question. “You are talking, and the also not demonstrate any exceptional person explains what is most important. Normally, do the organizers of the award abilities at the start of your career, and You do not necessarily need to read all avoid giving the medal to mathemati- then at some point things could click. of the literature on a problem,” he said. cians in similar subareas? That is how I saw things, and my goals Averse to interviews, Ávila says he has no There are no rules. However, I under- were very basic. During my PhD, my goal vocation for communicating mathemat- stand that if there is a situation in which was to perform research to obtain basic ics to the general public, a duty that will one candidate could wait four years un- results. I intended to follow the normal be difficult to avoid after receiving the til the next award, they might prefer to path of a researcher when he wants a greatest international award ever granted award the medal to people from a greater career, without lofty future goals. I had to a Brazilian researcher. variety of subareas. It was what I would reasonable ambitions because I knew that think and what could have happened to obstacles could arise and probably would. It is true that, unlike the Nobel Prize, me. I could certainly have been a can- the Fields Medal winners are notified didate in 2018, too. I was not in a hurry. You started studying at IMPA very ear- in advance that they were chosen to re- ly. How did that happen? ceive the award? In 2018, the International Congress of That was because IMPA sometimes ac- We knew before the announcement. I Mathematicians will be in Rio de Ja- cepts younger students who are still in knew five months in advance and had neiro. Do you think the choice of Bra- high school. They do this if they perceive to keep it secret. It was a long time, but zil to host the meeting influenced your that the student is able to do the work. I I contained myself. candidacy for the medal? knew about this, and this aroused my in- The decision on where to hold the con- terest in doing the same thing. This wish Your name was already under consid- gress is separate from the award commit- was granted when I returned from the eration to receive the medal four years tee decision. They are quite different is- 1995 International Mathematical Olym- ago. Did you expect to win the award sues. The meeting involves mathematical piad, at which I won the gold medal. this time? development questions and organization IMPA suggested that I take one of the I didn’t expect to win this time due to issues. The fact that Brazil has demon- level 1 courses shortly before starting

16 | special issue November 2015 h a that the system follows, new laws that

ssan can be used by the system. These laws are e

o p o no longer deterministic but rather statis- tical and probabilistic. We then have to nard

leo ask questions and try to give answers in terms of probabilities and system behav- iors instead of having absolute certainty. We try to model the system stochastically [using a probabilistic description of the processes]. We try to treat the system in whatever way possible.

It is correct to think of the Sun and its planets as an example of a chaotic dy- namical system? In the planetary system, it is difficult to describe the emergence of chaos. This Fields Medal awards ceremony during the International Congress of Mathematicians in Seoul is still very complex and is not very well understood. However, an event in which chaotic phenomena appear could be due the Master’s degree. If everything went systems was due to the historical chance to the interaction of quadratic functions well, I would enroll. In fact, that is what of my being at IMPA. [second-order polynomials], which every I did while still in the last year of high child learns about in school. After a long school. I started at the Master’s level, How would you explain the area of dy- time, what would be the effect of the re- and after a while I continued on to the namical systems to a layperson? peated application of the same quadratic Doctorate more or less normally, taking In general, dynamical systems is the study law? It could result in the emergence of courses at IMPA. At one point, I began of systems that change over time, with a chaos. This is a very simple example of to talk with researchers, with Welington, rule that describes the transition from what happens. and that was how I got my start in the one moment in time to the next; between field of dynamical systems. today and tomorrow, for example. This Some people say that you’re a great rule could be very simple. But, over a long problem solver, perhaps more so than Why did you become interested in math- period of time, you see complicated be- a formulator of theories. Do you agree ematics and not science? havior emerging. We call some systems with this statement? I don’t know. I always liked math, even chaotic. The study of this chaotic behav- Many times in my career I have sought before I understood the difference be- ior that emerges in the long term is one of out known difficult problems and worked tween the fields, since I was 5, for no spe- the principal concerns in the area of dy- hard to solve them. Because I did this sev- cial reason. I also liked other fields that namical processes. [The results and meth- eral times, it is certainly true that I solved I thought were science. But, in mathe- ods originating from the area of dynami- many problems. But, to a lesser extent, I matics, you can advance much faster on cal systems are used to explain complex also worked on building and developing your own and I had this contact with the phenomena in fields such as chemistry these theories, which sometimes involve mathematics Olympiads, which gave me (reactions, industrial processes), physics not only solving but also formulating the encouragement and focus and also served (turbulence, phase transitions, optics), problem. In the beginning, I resolved a to transition me to IMPA. biology (species competition, neurobi- problem related to Schrödinger operators, ology) and economics (growth models, but later, I also constructed a theory and How did you become interested in dy- financial market behavior).] solved problems related to it. However, namical systems? certainly, the most visible aspect of my I have certain characteristics as a re- Superficially, people associate chaos work is my many solutions of dynamical searcher that adapt well to research in with disorganization, but there are rules systems problems in different contexts. dynamical systems and that would also within the chaos, right? adapt to other areas. I am an analyst. I We have become able to better describe Your PhD advisor, Professor de Melo, work with analyses, statistics, and geom- good-quality chaotic systems, which have said that you have always been very se- etry. In my case, I was more exposed to certain characteristics. They are sensi- lective in your choice of projects and the area of dynamical systems because I tive to initial conditions, and in them, became interested in the great problems was at IMPA and in direct contact with small changes create large effects. At first of mathematics, trying to avoid being Welington. That is why I chose dynami- glance, on the one hand, it seems to be sidetracked by smaller questions. Was cal systems, where these characteristics something that prevents us from saying that your strategy? are very important. You can treat this anything useful about the system that de- I work on things I like, with problems topic using these techniques or others. I stroys the possibility of forecasts. But, on that particularly interest me, that I con- like the area, but the choice of dynamical the other hand, it introduces new rules sider beautiful. Often, the problems

pESQUISA FAPESP | 17 considered difficult are fundamental How does intuition help the mathema- What is your work schedule like, in Rio because some aspect of them is of great tician? and Paris? interest. Theories also develop around The most difficult parts of mathematical I go back and forth. I do not spend six these problems. The mathematician is work are those that involve creativity, whole months here or there, it is much usually attracted by the richness of the which lead to discovering things that are, more broken up. I spend some months theory around these objects. Working obviously, different from the basic rules. there, others here. I try to avoid winter with these problems allows us to explore Every top mathematician has excellent in Paris, but there are some exceptions. more pleasurable things. However, I do technical skills and can follow known The details of the trips are decided at the not discard a problem because others paths without difficulty. This is certainly last minute and, depending on the cir- think it is not important. I also worked true in my case. What trips up research is cumstances, I choose the specific dates. on questions that I knew would not have having to go outside established methods, I have a great deal of flexibility because I a monumental impact. I solve these sim- discover something and try to identify a only research and do not teach. It is one pler problems faster. I do not spend most way to attack the problem. Faced with of my characteristics. I prefer to spend of my time working on them because they the unknown, there is no rule as to how my time on research, and I do not think can be solved quickly. They are simpler. to approach something. You must rely on I have great teaching skills. I have PhD your intuition to decide how to attack a students, but basically, I do not teach. So, You did various projects with colleagues. problem. This involves a bit of experi- I do not have to follow a calendar, which Do you like to work as part of a team? ence, which helps a lot in developing in- I would have to do if I had to teach. I mainly like it when I want to learn tuition on a topic. You go in one direction something. I do not tend to read. because you hope it will work, but you It is true that you wake up late and usu- cannot formalize it mathematically yet. ally work more at night? What do you mean? I certainly continue waking up late, at I read very few mathematics books and Why did you become a French citizen? 11:00 or even at 13:00. However, this var- papers. I completed my PhD in Brazil and went ies a lot. It depends on the previous day to France in 2001. My first positions and on things that could be more en- How can you do research like that? were in France, and I spent five years grossing. At night, I work before going In mathematics, you can advance without there before returning to Brazil. After to bed or, if I wake in the middle of the having a deep knowledge of the literature; that, I spent three years in Brazil, and night, I can think about mathematics. it is more important to have a very pre- then started spending half of my time However, I try to work in the afternoons cise understanding of the fundamentals. here and half there. The time I spent in too. I often work with colleagues and I I learn these important things more eas- France complemented my training as a am not going to work with them after ily by talking to other researchers. That is mathematician, and I extended my ar- midnight. I have worked in various sit- when collaboration is useful. You are talk- eas of research. I finished my PhD with uations. Most recently on the beach or ing, and the person explains what is most the ability to conduct research at a high walking along the beach, for example. important. You do not necessarily need level. My results were recognized as im- Not so much at night. to read all of the literature on a problem. portant, but I had a restricted view of the area and its position within the whole Are your routines in Paris and Rio Is this unique to you or do many math- of mathematics. In Paris, I had contact similar? ematicians work this way? with the largest community of math- Not at all. To begin with, I do not have It is not a completely unique character- ematicians in the world and unparal- a set routine. There, I tend to go to my istic. Mathematicians work in differ- leled activity. This forced me beyond office more often. There is no beach ent ways. Some mathematicians like to my area of expertise at the time, one- there, of course. So I meet co-authors, read a lot. I do not. I know a lot because dimensional dynamics, and made me colleagues and students. I try to have I have already solved many problems. look for other things to be able to in- several meetings during the day, not nec- I often begin to work in an area, doing teract with these people who were not essarily in my office. Some days I stay at research, even before studying the area. necessarily interested in the same things home, too. In France, this occurs more Before studying, I try to solve a prob- that I was. In this search with such good frequently. In Brazil, I usually invite peo- lem. However, it is very difficult to start professionals, with so many possible co- ple to meet me near my home. When I from nothing, without knowing anything. authors, I started working in other areas, work alone, I tend to stay at home or go So, I begin a collaboration, and before I and my work was lauded due to what I to the beach. I am not always working even learn a topic in depth, I have already did in these areas. The mathematician intensely. When I am not working on a solved an important problem; that moti- that I am today is a result of my time in very specific line of research or when I vates me more. I have changed subareas France and my training in Brazil. Thus, am a bit lost, unsure of how to approach several times, and each time, I solved an I thought it was more accurate to con- a problem—which happens most of the important problem right away and only sider myself a French-Brazilian math- time—working intensely doesn’t help later understood what the theory said ematician. If I am a French-Brazilian much, and there is no way I can work about that problem. It involves a bit of mathematician, it makes sense to have for many hours at a time. I don’t work the technical characteristic of the person French nationality too, which also leads many hours a week then. It is different and their intuition, too. For me, it works. to practical benefits for my life there. from some situations, which occur now

18 | special issue November 2015 and then, in which I hope, know or imag- to winning the medal. What do you ready well-known. So, it is not so impor- ine that something is going to work and think of this possibility? tant that I play this role. I think that to a involve a lot of technical but directed I think that Brazil and IMPA have been slight extent, I have the role of helping work. In these cases, I work very in- producing excellent mathematicians promote mathematics to people who tensely, for many hours a day. for some time. I am a bit reluctant to are not mathematicians and who do not put pressure on any one person. It is not know that we do high-level mathemat- Brazil’s math and science teaching is supernatural to win an award. What was ics in Brazil. poor. Do you consider yourself an ex- considered impossible occurred. It oc- ception, given this? curred within a continuum of improve- Do you intend to give talks in schools? I think that I am more a natural conse- ment in Brazilian mathematics. It is not I will probably do something along those quence of the evolution of science, espe- a unique event that cannot be repeat- lines, but the objective is to play this role cially mathematics, which is perhaps the ed. However, it really is a rare award. alongside people who have more of a most developed field in Brazil in terms of Not receiving it does not mean your re- calling for it. I am very limited in vari- international impact. This is due to the search is of poor quality. I like to remind ous aspects related to explaining math inherent properties of mathematics but people that Germany—which has 100 to a more general public. It is not one of also to the people who work in the field. Nobel prizes to Brazil’s none—has only my skills. I already find it difficult to talk Mathematics depends more on human one Fields Medal winner. You can see to non-PhD students in mathematics, resources than on material resources. how rare this award is. This does not even in my subarea. On the other hand, Having focused people can be enough diminish the quality of the research car- I have greater visibility. We still have not to go far. When one depends decided how we are going to on many material resources, reconcile this. However, in such as laboratories, the will practice, due to my limita- of the researchers will not tions, I will not be commu- be enough, no matter how nicating directly with people. competent they are. [Brazil’s prestige is evidenced by the But wouldn’t it be natural fact that there are currently for people to expect you to four Brazilian voting repre- have greater contact with sentatives at the IMU gen- The Fields Medal is non-specialists? eral assembly, one less than They can ask, but I have a powers like the United States choice. I think I can do some- and France. At the Interna- a clear demonstration thing positive indirectly. tional Conference in Seoul, There are many competent four mathematicians from of the quality of the people who communicate IMPA gave talks.] much better than I do. I do not need to be the person What could be done in the research that can be speaking. I can be next to schools to awaken interest him or her. in mathematics? conducted in Brazil Given my background, I did What is your life like outside not have contact with teach- of mathematics? ing in schools. I went to very select ried out by German mathematicians and In Brazil, I try to go to the gym often schools, I went directly to IMPA and their contributions in any way. You can- and, when possible, to the beach. I live did not spend much time at a university. not measure things by these awards; it on Leblon beach. I like to walk in the I formally attended a university, but I would create immense distortions. The neighborhood and do typical things, studied at IMPA. I did not have contact analysis is much more complicated. With like going to juice cafés. In Rio, I have with the reality of teaching in Brazil. If the medal, it is easy to show people that childhood friends with whom I main- I talk about education, it is more about Brazil did something at the highest in- tain contact, and I organize get-togeth- what I imagine it is like. I do not teach ternational level. Before, Brazil already ers. Everything I do is very normal. I at a university, so I do not have daily did so, but it was harder to demonstrate don’t do anything very odd, no high- contact with that reality. I prefer to let it. People could even say “so, where is risk sports, no trips unrelated to math- other people who have more contact the award?” However, now they can’t. ematics. In Paris, I meet with a group with and ideas on this topic discuss it. However, things should not be measured of math colleagues after work and we There are high-level mathematicians in that way because that is not the focus. go to bars and such. who have much better ideas about this that I do. Do you think that, from now on, you Paris is famous for its cultural life, its will be a sort of ambassador for Brazil- museums. Do you go to these places Another Brazilian at IMPA, Fernando ian science and mathematics abroad? often? Codás, has been mentioned in relation Among mathematicians, IMPA was al- No. n

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