PHOTO: EGBERTO NOGUEIRA_ÍMA FOTOGALERIA_2009

# Artur has a problem

## How a great mathematician is made

João Moreira Salles | Edição 95, Agosto 2014

*Translated by Flora Thomson-Deveaux*

Onboard a United Airlines flight to New York, mathematician Welington de Melo ordered a glass of wine. His traveling companion, Artur Avila, ordered another. The flight attendant looked askance: “How old are you?” Artur was 19, looked younger, and had to go without the wine. This was his first professional trip. He had been entrusted to the cares of his doctoral adviser, but, back on the ground,his mother hadn’t entirely come to terms with the decision to let him leave for the United States.

Under dry laws, Artur disembarked at JFK Airport and went on with Melo to Stony Brook University, on the north coast of Long Island, about 100 km away. The year was 1999. The two of them were going to meet Mikhail Lyubich, co-director of the Institute for Mathematical Studies, a leading center for mathematical research. Lyubich was born in Ukraine, where his reputation as a brilliant mathematician had not shielded him from the obstacles, large and small, faced by a Jew in the former Soviet Union. Kept far away from the country’s major academic centers, he had been discovered by an American colleague and then immigrated to the United States, where he became a citizen and now headed up the IMS. The meeting had been arranged months earlier, when Lyubich, at Melo’s invitation, had come to Rio to take part at a conference at the National Institute for Pure and Applied Mathematics (Impa).

When he met the Brazilian visitors at Stony Brook, Lyubich had just written a series of articles proving his most important discoveries. “Only a very few people truly understood what it was about,” he said recently, “and Welington was a notable exception. It was he who proposed that Artur explore that line of research.” Melo, then age 53, had gotten his doctorate in 1972, and Lyubich, then 40, had received his PhD in 1984. Artur, born in 1979, was a student still looking for a good problem for his doctoral thesis. Up until then, he’d been brought to Impa by his mother, Lenir, who found it more practical to wait around there rather than go home and fetch him later.

These representatives of three generations spent a month tossing ideas around, in a style of mathematics that only requires a blackboard, chalk, and space to pace up and down. Their conversations, which were daily, took place in classrooms at the institute, at Lyubich’s house, in restaurants, or during walks through the woods around the campus. Their collaboration was possible because mathematics stubbornly resists hierarchies. “The proof is the proof,” Artur says, referring to the irrefutable nature of truths. A kid who’s just arrived can trade ideas as an equal with established researchers. Or more than that: “Now and then I’d realize that Lyubich and I were a bit behind Artur,” Melo recalls. “He was so young… I’d forget that, and then I’d remember with a start.”

One day he and Artur went to New York to attend a talk. In the Village, not known for its dearth of restaurants, they went out to find somewhere to eat. Melo recalls the impossibility of reconciling their tastes: “I’d say, ‘What about this Korean place, Artur?’ and he’d say, ‘Never tried it.’ ‘And this Italian?’ ‘Don’t know what it’s like.’ Imagine never having eaten Italian. Artur wound up having lunch at a McDonald’s. He hardly knew anything about the world.”

After a month of intense discussions, the trio could make out a clear strategy for solving the problem that absorbed their attention, but the proof remained beyond their reach. There was one obstacle that refused to give. Lyubich and Melo decided to leave it in Artur’s hands. “That was in March,” Artur recalls. “The problem stuck in my head, and a few months later, in September or October, I had a weird idea.”

** I**n his short and magnificent 1940 book , the English mathematician G.H. Hardy draws attention to the fact that a theorem cannot be undone. Mathematics is the only science that deals with truth, something that can be demonstrated by popping into any library: mathematical literature is evergreen, while texts on other sciences become rapidly obsolete. Two thousand years have not written a wrinkle on the Pythagorean theorem. Nobody studies Ptolemy’s solar system, except out of historical interest, but Euclid is still standing. Mathematics works by accumulation, not substitution.

The permanent validity of mathematical truths is tied to the fact that the discipline is separate from the real world, outside of time and the circumstances of the universe. The French mathematician and philosopher Henri Poincaré wrote that mathematical discovery is the activity which least borrows elements from the outside world. The mind feeds off itself. The classic introduction to a geometry treatise goes: “Let us consider three distinct systems of things. The things composing the ﬁrst system, we will call points; those of the second, we will call straight lines; and those of the third system, we will call planes.” Things. Mathematics requires one to deal with the most remote and inhuman objects that the human mind has ever imagined, as Belgian mathematical physicist David Ruelle puts it.

Artur Avila, age 30, with a permanent five o’clock shadow, PhD in mathematics from Impa, lives between France and Brazil. In Paris, he works at the Centre National de la Recherche Scientifique (CNRS), a state-sponsored research organization. In Rio, he is a researcher at Impa. He has been accumulating increasingly important awards. The world’s great mathematical research centers press invitations on him, and many would like to hire him. When a layperson asks for him to explain what he does, Artur rubs his eyes, a gesture that is usually accompanied by a long silence. “My work is a bit hard to explain. I study the structure of operators. Does that make sense, *operators*? An operator is an infinite, symmetrical matrix. That operator has a spectrum…”

It goes on like that, but there’s no need to be embarrassed. It’s common for mathematicians to not understand what their colleagues are doing. One paper by a 37-year-old Vietnamese mathematician, Ngô Bâo Chau, has been sitting on the desk of the editor-in-chief of a prestigious mathematical journal. The article’s implications seem to be remarkable, but all the specialists who were asked to check the publication said that they weren’t qualified to judge whether it was correct or not.

Carlos Gustavo Tamm Moreira, who goes by Gugu, Artur’s colleague and collaborator, is an easygoing 36-year-old who splits his love between mathematics, football team Flamengo, and Karl Marx. He likes to tell an apocryphal story about the time he ran for city council as a candidate for the Brazilian Communist Party. The platform gave him 18 seconds to introduce himself on television. At top speed, he rattled off: “Hello, I’m Gugu, the Communist Party’s candidate for city council, number 21602. You all already know my work: I proved that the stable intersections of regular Cantor sets are thick where the sum of the Hausdorff dimensions is larger than one.” It reflects well the rarefied nature of the world mathematicians inhabit.

Physics studies the natural world; biology deals with living organisms. These are sciences whose object of study can be understood by a layperson. Math is different, although we imagine that we are familiar with it. We assume it is the stuff we learn in school – arithmetic, geometry, algebra, probabilities – but taken to an extreme. In theory, this isn’t wrong. In practice, the difference between school mathematics and that which springs from mathematical research institutes is not measured in degrees of complexity, but in leaps of quality, as if the school syllabus were a caterpillar and high mathematics a butterfly. It would be impossible to intuit the latter just by looking at the former. Imagine someone in that position. That’s all of us non-mathematicians.

Artur’s job is to think up butterflies. In his vocabulary, they are *objects* – infinite, complex, chaotic, unique, predictable, probable, elegant, beautiful, monstrous objects. These adjectives, all of them, are a part of mathematicians’ lexicon, some with precise, technical meanings, others employed to describe states of mind. These objects only exist mentally. Nobody knows where they reside. Mathematicians have still not decided whether what they do is *invent* or*discover* their objects. “Where does so much order lie?” is how Artur puts it, a question which he has no interest in answering because it’s not a mathematical one.

The common currency in mathematics is what some call the theorem-credit, which serves to measure the quantity and quality of the problems that each solves. By this yardstick, in Artur’s generation, few other mathematicians have racked up so many points. From January to November last year, he produced at least six major articles. In California, he decided to tackle a problem introduced in 1964 and popularized in 1980, after a physicist promised 10 martinis to whoever could solve it. Working alongside a Ukrainian colleague from Irvine, Artur managed to prove what was thereafter referred to in the literature as the “Ten Martini Problem.” The same week that he demonstrated that a conjecture that mathematicians had been working on for years was false, he had an insight that allowed him to make significant progress on one of his most ambitious projects: to construct, on his own, a general theory for a problem rooted in physics.

Artur, like a number of mathematicians educated at Impa, works with dynamical systems – a field that investigates the laws of processes that evolve over time. It emerged with Newton’s studies on the movement of the planets. Today, theorems in dynamical systems are tools to help describe the evolution of epidemics, prove that any weather forecast more than five or six days out is no better than a coin toss, or describe the demographical impact of a change in this or that variable. Take a population of wolves. If there are few specimens, food will be plentiful and the population will grow. Inversely, a large population of wolves will produce scarcity in the food supply, and a decrease in population. The dynamical system will describe the way in which this population progresses: it is a way of using the conditions of the present to find out what may be expected of the future.

Often, what one expects is regularity. A ball tossed into a bowl will come to rest in the bottom. A pendulum will oscillate between two points. Systems with a finite number of states, which conform to patterns, are called regular. But there are dynamical systems which behave more interestingly, and these are Impa’s specialty. At first, they evolve in a predictable fashion. Then, suddenly, and violently, they stop displaying any recognizable pattern and become irregular. These systems are extraordinarily sensitive to tiny initial discrepancies. As the saying goes: “For want of a nail the shoe was lost; for want of a shoe the horse was lost; for want of a horse the rider was lost; for want of a rider the battle was lost; for want of a battle the kingdom was lost.” Dynamical systems predict the impact of the nail on the instability of the kingdom.

When the behavior of a system no longer conforms to any pattern, it is called chaotic*. **Chaos *can mean many things. Here, it is a concept that expresses everything that *cannot *be known about the future. In the absence of certainties, one describes, in never-ending detail, how the system will change: up until *this *point it will evolve regularly, oscillating between *such* and *such* states; from *this *point on it will be chaotic, manifesting *these* particular characteristics.

The smoke from a cigarette will rise in a thin column until, for reasons independent of the breeze or the movement of the hand, it will unravel and form arabesques with unpredictable trajectories. This is a good image for a complex system that evolves from regularity into chaos. Taking the first molecule of smoke to leave the cigarette, one can easily predict its future position within one second. In 10 seconds, however, the molecule will have strayed, and it will be impossible to anticipate where it will be.

Lyubich, Melo, and Avila are dynamicists of non-regularity, specialists in chaos. They had met in Stony Brook to study a specific class of systems with chaotic characteristics. They were not worried about wolves, pendulums, or cigarettes. They were working exclusively with mathematical models, but, to keep with the analogy, it was as if they wanted to understand the area just above where the smoke begins to dissipate. What was happening there?

They used a mathematical technique that makes it possible to dive like a bathyscape into the tiniest structures of this space. They would take a small interval from the region of the arabesques and put it under a purely logical microscope. The space would widen, like a zoom on Google Earth. When they analyzed the enlargement, they saw that, within the disorder, surrounded by chaos on all sides, there were small patches of order – little regular columns of smoke, so to speak. They then put this tiny space occupied by the regular column under the microscope, and, once again, a new enlargement revealed formless smoke everywhere, shot through with miniscule incidences of regular threads.

The trio kept on this way, on this vertiginous dive into ever-smaller intervals. It was no novelty that, given any particular point in a chaotic space, a window of order could be found nearby. But regular chaotic spaces – or stochastic spaces, as mathematicians prefer to call them – appear interwoven in a complex fashion, and what the three of them did was to show the universality of this organization. They discovered the law that governs the behavior of an entire class of systems that evolve towards chaos, as if the description of the unfurling smoke also explained the changing shapes of clouds, the path of a branch down a waterfall, or the whirling of leaves in a windstorm.

** I**n January 2009, ten years after Stony Brook, Artur woke up in the middle of the night in his apartment in Leblon, Rio de Janeiro, purchased with his wife, economist Susan Schommer, a native of Brazil’s South doing postdoctoral work at Impa. “Well, what now? Do I try to go back to sleep, or do I think for a while?” He decided to think. He lay there in the dark, looking at the ceiling. Outside, the last revelers from some pre-Carnaval party lurched along the street, singing and falling over. Inside, there was nothing but a man lying still in bed, eyes open, next to his sleeping wife.

But there was movement here. Without stirring, Artur began to turn mathematical objects over in his head, like someone circling a statue in order to see it from every possible angle. He was returning to a problem that he had set aside six years before, when he couldn’t see how to move forward with it. “I started thinking *gently*,” he says. His thoughts were wandering, adrift: “I had two objects, but I didn’t know how one related to the other. I’d hit a wall.” Until that night, he’d only seen the object as two separate parts, not fitting together. Suddenly, it came: “But if I change perspective, it shows itself as *this*. It *is* this. I can move forward.” The sensation was that of a person who squints at the shattered forms of a cubist painting and, upon taking one step back and maybe another to the side, finally manages to reconstruct the form – there is the woman, the guitar, and the score. It is all a single whole.

Still in the dark, Artur began calculating the consequences of his new point of view, and saw that he could produce much more information this way. “The narrative had thickened,” he explains. His object, which up until then had not revealed much of itself, began to spin increasingly fantastic stories, as if he had found the secret to one of those hermetically sealed mystery boxes found in penny arcades – think Fred Astaire in *The Band Wagon* – that, when hit just the right way, open into a carnival’s worth of flags, spring-loaded puppets and circus music. Artur was excited, but went back to sleep. “I didn’t even write it down. I’m not afraid of forgetting my intuitions.”

The following day, he decided to “attack the object from all sides” (mathematicians’ vocabulary is brimming with bellicose metaphors). “Ten days, eighteen hours a day. Technically it was very difficult, but the idea was there.” He spent all day walking in circles around the apartment. Now and then he’d stop, stare at the ceiling, and scribble something on the paper to help guide his reasoning. “Most of the work happens in your head. It’s a feeling of total absorption. I remember opening a bottle of champagne that was in the fridge. The cork exploded, the wine started pouring out, and I didn’t move, I just stood there and thought, ‘It’s not supposed to be pouring out, normally that doesn’t happen…”

At every turn, increasingly improbable things happened to the object – exactly what Artur wanted. He was looking for a proof by contradiction: if he was wrong, then the object was monstrous, and “horrible things happened to it.” Mathematical objects can be easy to visualize (a circle) or very complex (the kind that Artur always works with), but, in order to exist, all of them need to possess one characteristic: they must be logical. A horrible object is one that displays characteristics that eventually nullify it, as if it bore a genetic mutation so serious that it made its existence impossible.

“I kept going like that until I found a contradiction. After a week of work, the proof ad absurdum was done. My conjecture was true.” Artur had just taken a major step towards solving a problem that had arisen in physics: quasiperiodic Schrödinger operators. Up until then, the problem had only been partially understood. Artur had envisioned the possibility of employing dynamical systems to understand it globally.

** A**rtur usually gets up around noon. He often works in bed, and treasures idle time. He thinks that public transit is good for doing math, which is one of the reasons he dislikes cars. He has had great ideas on long rides on the Paris metro. In 2008, during a flight from Rio to Paris, he decided to have a go at a problem he’d been fighting with for two years. “I think it was between one film and another on that little screen,” he says. He began turning things over in his head and, astonished, saw that their complexity could be reduced to a simple expression. When the plane landed at Charles de Gaulle Airport, he had solved the problem – and discovered another piece of Schrödinger’s puzzle.

Artur prefers to “do sums in his head” – and by “sums” one shouldn’t think of times tables, but rather of ideas being put together, mental geographies full of valleys, peaks, folds, abysses, and discontinuities. “Paper is brute force. In your head you can’t manipulate huge objects, and that forces me to make simpler calculations,” he explains. He isolates the characteristics that interest him most and discards the rest – “I make a cartoon of the object.”

His style of thinking alternates formal terms with day-to-day expressions. “In a paper that I wrote with Gugu, we classified objects as good, very good, excellent, and when the excellent ones had extra characteristics that made them the best objects possible, they were *cool* objects.” There’s also “the dark side,” a place where “you find horrible things, particularly detestable ones, that violate your ability to understand them.” A problem morphs into a geography divided into boring zones, paradises, and hells. In the boring zones, all behaviors are predictable. Switzerland. In the paradises, interesting and unexpected things happen. In hell, all proofs fail and you must show that everything there vanishes for lack of logic. It’s an un-place.

Jean-Christophe Yoccoz, 52, is French. He lived in Brazil from 1981 to 1985, a period he spent teaching at Impa. In 1994, thanks to his work on dynamical systems, he was awarded the greatest honor in the profession, the Fields Medal, given every four years to at least two and as many as four mathematicians no older than 40 (since the medal’s creation in 1936, only 48 people have won it). Yoccoz resembles a leprechaun – chubby, charming, and quite red (sunburn, in this case). He is married to a Brazilian and spends stretches of time in the country. “Artur is certainly the most talented mathematician I’ve ever met, at least among those I’ve had considerable contact with,” he says in an office at Impa.

In order to explain Artur, Yoccoz returns to an old taxonomy of mathematicians: there are those who construct theories and those who solve problems. Artur is a solver, according to Yoccoz. “He has the talent of a Jean Bourgain or a Terence Tao” – two of the greatest mathematicians currently working – “the kind who sees things that others don’t. You have an impassable problem, they look at it, and *bam!* you get to the other side.”

Mathematical discovery is a mysterious process. The ability to *see* differently is one of its characteristics. Intuition is another. A great mathematician once defined intuition as the ability to “know without knowing.” It derives from the imagination, and declares that “such property belongs to such object, but I can’t prove it.” Alexander Grothendieck, perhaps the greatest visionary of the second half of the past century, and one of the only people for whom the label “genius” does not seem hyperbolic, defined it as the ability to “feel the rich substance behind a statement.” One looks to a place where there are only fragments and suddenly perceives a body, each piece connected to another by subjacent structures.

Intuition can lead to epiphanies – and mathematics is full of them. “Everyone has their own story,” says Marcelo Viana, a researcher at Impa and one of Artur’s coauthors. In a brief, classic essay, Poincaré describes at least four of his exemplary insights, one of them responsible for the first discovery that brought him renown and glory. It happened when he was chatting with a friend about an unrelated topic and about to catch the bus. The instant his foot touched the step, he knew that he knew. Epiphanies cannot be forced. They are always concise, they fall like a bolt of lightning and they produce an unerring sense of certainty. Artur had long ago set aside the problem that came to his mind that sleepless night. Poincaré did not stop for a second to reflect on what had just happened to him –he kept on chatting.

Another great French mathematician, Laurent Schwartz, wrote that the process of mathematical discovery is like coffee percolating. At first, the hot water cannot penetrate the thick layer of grounds. Bit by bit, the principal vein bifurcates, and little filaments make their way through along other paths. Nothing happens – until, all at once, the liquid overcomes the obstacle and springs from the other side. The idea emerges.

Artur spends long periods idle, days or weeks, “and then an idea comes and the exhaustive work begins.” One of his strategies is working on several problems “of different flavors” at the same time. When one stalls, he attacks another. Now and then he’ll say that he was lucky, because he thinks that the idea came to him out of the blue, or he went down an improbable path that led to a solution. But of course the ideas were percolating. His wife has a clear opinion on the matter: “The harder Artur works, the luckier he is.”

Thousands of ideas will occur to the mathematician over the course of his productive life. Everyone says that the main criterion for immediately recognizing the superiority of those that win out is that they are beautiful. Mathematicians are closer to artists than to engineers. “Imagine two entirely different things, created independently,” proposes Artur, “and imagine that for some mysterious reason, you discover that they are part of a single thing.” Unity is one of the hallmarks of a mathematician’s sense of beauty.

The ideas discussed in Stony Brook have grown enormously over recent years. “In Artur’s hands,” Mikhail Lyubich says, “they are becoming a universal tool, a sort of glue that sticks together a number of apparently unrelated problems.” Great mathematicians are sometimes compared to pioneers and colonizers. The former ones throw themselves into unthought lands and establish an outpost there; the latter connect these islands of thought to the continent of the discipline. Artur is a colonizer.

“He has the ability to discover unexpected relationships between things, and that’s what mathematicians like,” says Viana. For mathematicians, nothing is more wondrous than the intuition of a totality. Aesthetic sense is the ability to see wholeness, which is the sieve separating the wheat from the chaff. The only objects that pass are those that, in their beauty, declare: *I exist*. “We spend our lives thinking about exquisite objects,” says Yoccoz, smiling beatifically. “The aesthetic pleasure is comparable to that of music.” Great mathematicians are aesthetes. By the enthusiasm with which they speak of what goes through their heads, it’s as if there were music all around them and we non-mathematicians were all deaf.

** “M**athematics is infinite rigor,” says Artur. The definition helps to clarify his aversion to discussing topics he hasn’t thought through. It’s not uncommon to hear him respond, “I don’t know” or see him reflect before replying to a trivial question. When he speaks, his words tend to land on the bulls-eye, without skidding. He demands utter precision in the use of words. As he was shocked that the state of Mato Grosso do Sul had included homeopathic remedies in the list of drugs for fighting swine flu, it seemed reasonable to assume that he was skeptical of homeopathy. “No,” he said with a smile. “The principle of homeopathy is absolute dilution, so at the end of the process there’s not a molecule left of the active ingredient. I’m not skeptical of homeopathy. Skepticism implies doubt, and I have no doubts in this case.”

Insistence on rigor seems to be one of the innate traits that, if insufficient, are at least necessary for one to evince a vocation for mathematics. The symptoms come early, and, in Artur’s case, led to his leaving (before they expelled him) one of the best high schools in Rio de Janeiro, the city where he was born.

His father Raimundo was from the state of Amazonas and grew up tending a riverside plot of manioc. 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, 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, who was still small when they separated. He was raised by his mother.

At age 6, Artur enrolled in São Bento, a Catholic school in downtown Rio that usually comes in first in most rankings of the best schools in Brazil. At age 5 he was already reading math books, and, as the curriculum struck him as tedious, he went looking for more advanced didactic material. At age 13, he was good at history and science. Social issues interested him, and for some time Artur thought that journalism might be a good option for a career. From his father he inherited a love of Rio football club Vasco da Gama, and asked his mother to take him to the team’s practice sessions. He could afford the luxury; school was hardly challenging him.

Artur would have graduated from São Bento without breaking a sweat if it weren’t for religion class. For the first time, at age 14, he ran into an obstacle. The difficulty wasn’t the content, but the nature of the discussion. He soon concluded that he was being taught bad philosophy. “They treated God like a matter of logic. I couldn’t accept that, independently of whether I believed in God or not,” he recalls. “If the priest said, ‘These are the doctrines of the Church,’ okay. But they were suggesting that logical reasoning necessarily led to the existence of God. That was a philosophical argument, and without the counterargument it seemed fallacious. I wanted the refutation, and the refutation of the refutation.” Since none of those was on the way, the Benedictines suggested that he leave the school. “It was a relief,” he says.

While still at São Bento, Artur was lucky enough to come across a teacher who told him about the local Math Olympiad. In his first competition, at age 13, he did “relatively well.” If he harbored any doubts about his problem-solving talents, they were set aside. Later that year, he won his first Olympic medal in the national version of the competition. His bronze medal in 1992 would become gold in ’93, ’94, and ’95.

From São Bento he went to Santo Agostinho, another high school with a sterling reputation. Exact sciences seemed increasingly easy, and he started skipping class in order to sleep in, a habit he has always zealously cultivated. The syllabus irritated him: “People didn’t study for the sake of knowledge, but to pass the test. In the schedule, after physics there came 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.” He was 16.

Later in 1995, Artur was on the Brazilian team for the International Math Olympiad, the stiffest international competition for high schoolers. Each country sends a maximum of six representatives, chosen from among the most talented in the nation. Countries like China, Russia, and the United States treat the competition as a matter of state. The teams are formed through an extremely rigorous selection process and trained by specialized mathematicians over a weeks-long immersive period. Brazil, at the time, just rounded up its talents and put them on a plane, at most giving them a list of exercises a few days beforehand. The tests were held in Toronto, with 73 participating countries and 412 contestants. Artur got five of the six problems right, and, along with 29 other youths (none of them Brazilian), went home with a gold medal. When he arrived, Impa immediately offered him a scholarship.

Artur started frequenting Impa and entered a master’s program while still in Santo Agostinho, which he would finish along with high school. (He skipped regular undergraduate work.) He hadn’t showed up to high school regularly in a year. As students can only miss a certain number of classes before flunking, there arose the question of how many absences he accrued. Artur thought – longer than the question demanded. “If I write that you skipped 50% of your classes, would you find it odd?” He looked to the side, took off his glasses and rubbed his eyes, a gesture that invariably follows a question that he finds trivial or tedious. “There’s no need to be rigorously precise,” I suggested. “It’s hard for you to ask me to not be rigorously precise – 50% is a precise figure. Say that I skipped 30-50% of my classes.”

** T**he main Impa building has long terraces that sprawl along the fringes of the Tijuca Forest. The bathrooms are sparkling. Along the corridors, one overhears Spanish, English, and French, as well as Portuguese. With a bit of patience, you can pick up Russian, Farsi, Chinese, and German. Impa was born in 1952 through a CNPq initiative, the national agency for the promotion of research that had been created the year before. Its first years were spent in a classroom on loan from the Instituto de Física, and after a series of moves it wound up in a building of its own, in the Horto neighborhood behind Rio’s Botanical Garden. The institution focuses on training master’s and doctoral students across a variety of mathematical fields, in addition to promoting the refinement of the teaching of mathematics, with classes and teacher-oriented publications.

It is, in every respect, the best teaching institution in Brazil. No other research center enjoys such international prestige. Impa publishes or is cited frequently in the best mathematical journals across the world. Some of the 324 PhDs it has produced are on the cutting edge of the field. None of them paid a cent to study there. Impa is the result of a rare conjunction of factors: successful public policy and administrators endowed with ambition, pragmatism, and technical competence, plus mathematics’ singular status as both low-budget and apolitical.

Created by mathematicians Lélio Gama, Maurício Peixoto, and Leopoldo Nachbin, Impa has managed to establish contact with the most talented mathematicians of the time. Peixoto and Nachbin were renowned researchers, and, thanks to their intellectual prestige, they, much like the tropical delights of Rio de Janeiro, managed to draw a constant flow of great mathematicians to lecture or research in the city. Researchers such as the French mathematicians Laurent Schwartz and René Thom, or American Stephen Smale, gave talks or taught at Impa. (Smale made two of his greatest discoveries during his stay in Rio). All three are Fields medalists and belong on any list of the most important mathematicians of the second half of the past century.

Impa’s excellence is tied to the name of Jacob Palis, considered the greatest mathematician in Brazil. Palis carried the founders’ project onward. During his tenure, the number of students and foreign mathematicians increased significantly. Welington de Melo came to Impa in 1970. “The sheer amount of math that I learned in these corridors was spectacular. Jacob created that environment.” This was the period of the military dictatorship; while most universities were stricken with intellectual restrictions, Impa grew stronger – abstract ideas never bothered those in power, be they generals or commissars. Mathematics can flourish under any political system, and since it only requires pencil and paper – or even less (Archimedes drew in the sand) – it can be practiced under extreme conditions. French mathematician Jean Leray revolutionized the field of topology from a prison during World War II.

Mathematics reacts more to intellectual resources than material ones – and the former were available in Rio. Impa is to mathematics as Jamaica is to athletics: it doesn’t compete in every field, which allows it to achieve exceptional performance in just one. The institute staked out excellence in aperiodic dynamical systems. Stephen Smale is one of the totemic fathers of the field. He was Jacob Palis’ thesis adviser, who advised Welington de Melo, who advised Artur Avila.

** T**he line of cars waiting to enter the parking lot at Pontifícia Universidade Católica (PUC), in Rio’s neighborhood of Gávea, can stall out at times. On that Thursday in September, I had scheduled a chat with mathematician Nicolau Corção Saldanha at 11:30 AM. It was 11:15 and my car wasn’t budging. It would be appropriate to call and apologize for any possible delay. “What’s your car?” Nicolau (everyone calls him by his first name) asks. “I’ll be there in a minute. We’ll talk in the car. I won’t need a blackboard anyway.” In a few seconds he appeared, a figure in shorts and tennis shoes. For vegetarian motives, he doesn’t wear leather. He got into the car and suggested that we drive around Leblon.

Nicolau Saldanha is 45 years old and was the first Brazilian to win a gold medal in an International Math Olympiad. He competed in 1981, in Washington, and solved all the problems. That exceptional talent would lead him to PUC, where he completed his master’s, and then to Princeton, one of the great international centers for mathematics. There he studied with one of the greatest mathematicians of the last 50 years, William Thurston, his doctoral adviser. During his time at Princeton he was considered one of the most brilliant students around.

Nicolau is very pale and has the fatigued air of someone who sleeps little. He speaks softly, somewhat ethereally, and his features recall those of a Romantic composer given to preludes. When Artur came to Impa, Nicolau was one of the first professors he met. “The class was massive,” Nicolau recalls. “I’m known for giving challenging tests. The grades were generally very poor, but Artur’s performance was extraordinary. I only noticed him because of how well he did, but I’m not even sure that he came to class. I don’t remember him there. He never asked a single question.”

In the office at Impa that he occupies during the months he spends in Brazil, in shorts, a T-Shirt, and sandals – his uniform while in Rio – Artur recalls: “The class was called ‘Introduction to Real Analysis,’ and it was very important for me.” Up until then, what he knew about math boiled down to winning competitions. Nicolau’s course was far more than that. At each class he was presented with complex ideas which could only be dealt with by reevaluating one’s way of thinking. Nicolau allowed the students to use books and notes on their tests. Artur only brought a pen. “I only used what I had in my head,” he says, “because the answers weren’t in the books. That’s what was cool about Nicolau’s tests: you had to have an *idea*.” The passing grade was 3. “Artur scored 12, the highest possible,” Nicolau recalls, “the second-best scored 7; the third, 5.” Artur did indeed go to class. If he never caught his teacher’s eye, it’s because he didn’t open his mouth. “I kept quiet because I wanted to be very sure of what I would say,” he explains. “It takes a while to be halfway sure, which is the least you need to do research.”

Since the previous year, when he won gold in Toronto, Artur had become the most valuable member on the Brazilian Olympic team. Everything was set for them to travel to India, the host of the 1996 Olympiad; then, a few days before their departure, he let them know that he wasn’t going. Brazil would probably miss out on another gold – and up until then the country had only won five, including his. The organizing committee, made up of influential mathematicians, pressured him to go, but Artur wouldn’t yield. The team had to go without him.

Between Canada in 1995 and India in 1996, something very important had happened: he had started at Impa. “I had no idea of what doing mathematics was. I looked and I said: *this* is it.” He no longer had any taste for the competition. “There, everything has a solution, and the best part of math is uncertainty: you can spend years wrestling with something that may never be solved.” The haste of the thing also bothered him. Mathematicians don’t need to make urgent decisions, and none of them will be forced to prove a conjecture by the end of the month. “Mathematics is done with time; there’s no pressure. And I like to reflect,” says Artur.

Quitting the Olympiad was the first decision from the mathematician that Artur Avila would come to be. “If I’m right,” says Elon Lages Lima, his master’s adviser, “that shows clearly that at age 17 he already knew how you go about building a career. He wasn’t interested in winning a prize he already had. He had more important things to do.”

** A**rtur works in a sad little room in one of the ugliest buildings in Paris, a workspace for 300 mathematicians. There are just two desks, a blackboard, a metal cabinet, and a telephone he doesn’t know how to work. Until 2008, he had been employed as

*chargé de recherche*, research fellow, in association with the CNRS Laboratory for Probabilities and Aleatoric Models. The sad little room was new, and came along with a promotion. In October 2009, he was starting out the academic year in France as one of the youngest mathematicians to take on the post of

*directeur de recherches*, Director of Research.

Besides a bump in his salary, this just meant switching floors. “Researchers who are promoted are sent to another city, where they have to work and teach. In my case, they know that I want to stay here, and so the bureaucratic solution was to have me change floors.” Changing floors means being allocated to a different laboratory – in this case, the Institut de Mathématiques de Jussieu. “If some researcher complains about their transfer and uses me as an example, they’ll always be able to say that the paper trail shows that I moved, too.” Since mathematicians are independent, working how and when they please – “it would be scandalous if someone tried to tell me which problems to attack” – nothing much changes in practice.

This privilege is one of the concessions that CNRS has made in order to avoid the risk of losing him. The premature promotion was driven by an email that landed in Artur’s boss’s inbox: it was from Yale University, demonstrating an interest in hiring him. He probably wouldn’t accept, because he likes France. “I make twice the minimum wage, but it’s enough,” he says. “I don’t need much. I find it nice to live in a place with good public schools, healthcare, public transit. A society where researchers make two minimum wages and nobody makes thirty interests me.” He also appreciates the civilized custom of eating serenely. “Americans eat while they walk,” he says. He is horrified when he gives talks in the United States and they don’t invite him to dinner afterwards.

Artur’s only job is producing mathematics. “I’m quite detached from concrete things. I’d be reluctant to say that what I do is *useful*.” However, questions about the usefulness of mathematical research have begun cropping up from French bureaucrats. *What have you done to improve the world? What have you done for the economy?* “There’s some pressure from the French government. Subtle, but it’s there.”

For most people, math’s usefulness seems self-evident: bridges, economic projections, computer algorithms. A good deal of mathematicians find such applications uninteresting. “It is what is commonplace and dull that counts for practical life,” wrote Hardy in . “The mathematics which can be used ‘for ordinary purposes by ordinary men’ is negligible, and that which can be used by economists or sociologists hardly rises to ‘scholarship standard,’” he wrote. “The ‘real’ mathematics of the ‘real’ mathematicians, the mathematics of Fermat and Euler and Gauss and Abel and Riemann, is almost wholly ‘useless.’”

Hardy’s stance is exaggerated – to be sure, great mathematicians have dedicated themselves to what he might call useful mathematics – but they do not go against a certain existing consensus amongst his colleagues in the field. For many, the topic doesn’t even merit discussion. Kepler needed the properties of the ellipse in order to produce the laws of the movements of the heavenly bodies; but the Greeks had studied it merely because the form struck them as beautiful. The brutal difficulty of a problem is enough reason for someone to dedicate their life to solving it. It falls to mathematicians to take reason to its ultimate consequences, seeing just how far it can go. The uses for their work come along later – when they come at all.

In 1998, at age 18, Artur Avila started his PhD. “Very brilliant kids tend to be irritating, with a tendency to show off all the time,” recalls Welington de Melo. “That wasn’t the case with him. He rarely asked questions, but when he did, you had to take his question home with you and think about it over the weekend.” But it wasn’t the good questions or high grades that first drew Melo’s attention. There was something else, much rarer and much more important – Artur wasn’t interested in secondary problems. He seemed to only go after fundamental ones. Back then, Melo was working on a paper he felt was important. The main part of the job was done, but there were some secondary aspects that he had yet to deal with. Melo thought that Artur, at his age, would take this as a challenge. He proposed that Artur deal with the secondary problems and publish the article with him. “Imagine, he was just a teenager…” Artur refused. Melo smiles: “It wasn’t a central problem.”

Mathematicians speak of beauty, but also of *good taste*, both of which define one’s ability to detect what is truly important. From very early on, Artur displayed a keen intuition for great problems. Elon Lages Lima sees this as his greatest strength: “Artur has a much clearer vision of the role of a mathematician than most of the brilliant students who studied here. We exist to solve the problems that have never been solved. That’s not a deliberate choice. It’s like an animal’s hunting instinct. They do it because it’s inside them.”

Everything in Artur Avila’s life is oriented towards efficiency. The apartment in Rio, in a small building with no elevator, one block from the beach, is Spartan. The bookcases have no books, and the walls have no paintings. One table, a few chairs. An eternally unmade bed and a flat-screen on the wall. Over the years, he progressively abandoned nearly everything in favor of devoting his energies to his wife, cooking (“Nobody can spend years in Paris without becoming civilized,” says his coauthor Marcelo Viana), reading about politics online, and mathematics. It’s been years since he’s watched a Vasco game. He doesn’t go to the movies because he distrusts film critics. He prefers old films, since “if they’ve made it this far, that means they must be good.”

When he heard that I was reading the hulking 528-page autobiography of Laurent Schwartz, he suggested, in all sincerity: “Why don’t you just start in the middle, like I do?” Once Artur used the words: “The book I read.” *The* book? Just one? “The last one was in 2000, on a plane. It was that one by Oscar Wilde…” *The Picture of Dorian Gray*? “Yeah. I started in the middle, read a bit, it got kind of mysterious, and then I went back to the beginning.” He never finished. He doesn’t read technical texts. His favorite way of studying, he says, is by chatting.

Artur doesn’t check baggage at the airport. Whatever doesn’t fit in his carryon he leaves in the terminal’s trashcan, so he’s not left waiting at the luggage carousel. He has no keepsakes, wardrobes, or excess of any sort. He doesn’t like teaching and has practically no students. “At this point, it’s excruciating to have to explain all the details.” He doesn’t waste time. He thinks mathematics and travels to do mathematics. From January to August in 2009, he spent time at the University of Maryland, Stony Brook (NY), Caltech (CA), Irvine (CA), Northwestern and Chicago (IL), Stony Brook (again); he also gave seminars in Marseilles and Germany. Before the year was out, he’d go to Chile and return to France.

His ability to churn out work is astonishing. Whether alone or in collaboration, he has published around 40 articles in international journals, 1,112 pages in all. Eleven of them appeared in the three most prestigious journals in the world, the *Annals of Mathematics, Acta Mathematica*, and *Inventiones Mathematicae*. “No other Brazilian mathematician has or has ever had a list of publications the size of Artur’s, in their entire career. It’s truly impressive, even on an international level, for people his age,” Melo says. Artur has one clear objective: to neither waste his energies nor squander the vigor of his youth. He operates like a ballistic missile. “When you’re young, you may have more energy available in order to brutally attack your object,” he says.

Nicolau Saldanha, whom everyone refers to as one of the most brilliant mathematicians in Brazil, is impressed by how his former student works and publishes. “He takes tremendous interest in writing, unlike many other colleagues, who prefer thinking mathematics over writing mathematics.” Looking out the car window, he goes on: “Deep down, that describes me too. I have trouble in writing for publication. I’m satisfied with just having solved a problem, that’s enough. It’s for me; I don’t need to know if it’ll make the world better or not,” he says, as if, for him, mathematics were more of a pleasure than a profession.

Artur is a consummate professional. He doesn’t enjoy writing, but knows that the task is part of the job. “Revising isn’t exciting. I have ideas that I’ve presented dating back to 2003 that I haven’t had time to put on paper. When you solve a problem, the rest stops being interesting. When they ask, I say that the proof’s on my laptop.” The volume of published articles would indicate the contrary. Artur not only publishes, but also takes care to write well: “I need the reader to be interested in my topic. That’s what I do. That means constructing the narrative carefully.” He has never reread his doctoral thesis because “the style is dreadful.”

His professional discipline springs from a profound love for what he studies. One of the few recent films he’s seen is *Good Will Hunting*, by Gus Van Sant, in which Matt Damon plays a math prodigy. Artur hated it. “The guy doesn’t like his topic. It was like a competition – who can solve this problem first, who’ll solve that one. He could be studying anything, nothing was specific. No mathematician is like that. We study things because they interest us,” he says. He, for example, doesn’t particularly like algebra, in which everything is perfectly certain. He prefers more flexible objects that have small doses of uncertainty, errors that he will have to control so they don’t propagate. “I like to joke that I don’t really like the equals sign. I prefer approximations.”

Often, the most valuable thing a mathematician can teach isn’t his or her proofs, but the way he or she thinks. In the second year of his PhD program, Artur met up with Lyubich in Rio. It was a decisive conversation. “I started asking questions, and he, who was a really important mathematician, started thinking out loud,” he says. Artur was mesmerized. Lyubich wasn’t afraid of making mistakes. Freedom led his train of thought down purely speculative paths. You could venture out there, through the most fantastical dimensions of mathematics, guided by intuition alone. “It was a surprise,” Artur recalls. “I saw how an exceptional mathematician thought. I went and asked Welington if working with Lyubich would be possible.” Soon enough, he was at Stony Brook.

When, months later, Artur presented his strange idea for getting around the obstacle keeping them from their proof, Melo and Lyubich were taken aback: “It was something outside our usual toolbox,” Melo explains. “It’s like a piano: you can only play those keys, there aren’t any other ones, but sometimes someone shows up with something that’s not on the scale.”

Part of the project developed at Stony Brook would be included in Artur’s doctoral thesis, which he defended in 2001 at age 21 before an illustrious committee comprised of Marcelo Viana, Jacob Palis, and two foreign mathematicians, Yoccoz (France) and John Milnor (US), both Fields winners. Later that year he went to France, as a researcher at the Collège de France, and grew closer to Yoccoz. “It was very important. Since I don’t read the technical literature, I only knew about the things being discussed at Impa. I was very ignorant, and it was only then that I realized how big mathematics is.” He spent five years in France.

In 2006 he received a fellowship from the Clay Mathematics Institute, a private foundation in the United States that supports and promotes mathematics. The fellowship is offered to extremely promising young mathematicians. They have total freedom: the winner may live wherever he or she likes and is guaranteed not only a good salary, but also rent money and funds for work travel. Artur took unpaid leave from CNRS – his employer after the Collège de France – and went back to Brazil, where he stayed, within Impa, for the three years of his fellowship.

Staying in Brazil is not an ideological choice. “I’m not a nationalist. I’m no cheerleader for Brazil, but I want the math around here to move forward. I like to stay close to my Brazilian collaborators” – Gugu and Marcelo Viana chief among them – “and it’s also good because there’s a ton of stuff happening out there. It’s important to take time to stop and think. You can do a lot in isolation.” The Clay fellowship expired in July 2009, but CNRS allowed him to spend another semester in Brazil. When at home, he is paid by Impa alone. When he’s in France, his only salary comes from CNRS.

The money around here is a good sight better. Artur has tried to seduce foreign professors to come to Brazil with what he calls “salary in kilos of filet mignon.” “There, filet mignon is around 40 euros, here it was at 17 reais the other day. In France, with all the discounts, a researcher like me earns around 2,500 euros [a month]. In Impa, a guy who’s just starting earns 2,887 euros [per month].”

** J**acob Palis has an extraordinary influence on Brazilian mathematics. From ’99 to ’02 he was the president of the International Mathematical Union, the organization responsible, among other things, for awarding the Fields Medal at its quadrennial congresses. There is no Swedish Academy for mathematics. There’s the IMU.

Palis has an office on the top floor of Impa, an institution he speaks about with uninhibited passion. He is 69 years old, jovial, tall, fit, and jowly, with a mischievous smile. He seems to find the world quite fun. On the wall behind his desk, he appears in photos alongside the great and the good. Eminent mathematicians; first-class scientists; then-Chinese president Jiang Zemin.

He is furiously energetic. The current president of the Brazilian Academy of Sciences, he is constantly interrupted by two cell phones and two landlines that ring off the hook. It’s almost always about pending political issues – more research funds, putting pressure on the minister to loosen the purse strings.

As far as Jacob Palis is concerned, Artur is absolutely the direct product of Impa, one of the highest points in the history of the institution. There have been others. “My student Ricardo Mañe, who only became famous after he died very young, which only shows that we’re on the margins. If he’d been at some university like Princeton or Harvard, he’d probably have won the Fields. Marcelo Viana was also on the short list for the medal. There’s Gugu, who’s extraordinary.” Right now, however, it seems that all bets are on Artur Avila.

Artur says that the decision to go to France was his, but Palis suggests that there was an institutional strategy behind it. “It was hardly an innocent move when we supported Artur’s going to France. Spending half the year there and half the year here helps to raise his profile.”

Next August, between the 19^{th} and the 27^{th}, the city of Hyderabad, India, will host the IMU’s International Congress of Mathematicians. There will be 20 sessions, each dedicated to a specific field in mathematics, plus twenty open talks in which a mathematician addresses the entire community of his or her peers.

“It’s an immense honor,” says Marcelo Viana, the only Brazilian to have had this privilege. Artur will be one of the speakers at the event. “At his age, it’s absolutely uncommon. The speakers are invited because their discoveries are so important that their work is considered an advance for mathematics as a whole.” Viana says that three field specialists put Artur’s name forth. “Three that I know of,” he says. “That means that at least three fields of mathematics consider that Artur’s contributions were fundamental for them.”

** I**n his office at Impa, always epically messy, Artur averts his gaze from the massive window and the springtime sunlight streaming through it. He drank a bit too much the night before. Impa had organized a barbecue to celebrate two important international prizes recently won by their researchers. The first, by Gugu; the second by Artur, awarded by the French Academy of Sciences to mathematicians under age 35 who have made significant contributions in their field. He only found out that the prize existed – not to mention the 30,500 euros that go along with it – when they told him that he’d won.

In India, Artur will speak about a variety of aspects of his work, including Schrödinger. By September, he had finished most of parts I and II of his global theory. An insuperable problem was keeping him from coming to part III, the final section. “One day I hope to solve it,” he’d said back then. Now, nursing his headache and shading his eyes from the sun, he smiles: “An idea popped up this week. I woke up in the middle of the night and there was a sort of magic.”

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