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​The Art of Artur Avila, Brazil’s Math Genius

"Actually, mathematical thinking is crucial and goes beyond what's taught in class. It's essential for the society to develop."
July 5, 2016, 12:00am

"Hmm," mumbled the Carioca mathematician Artur Avila as he stood in the front of a bar in the Leblon neighborhood of Rio de Janeiro. The 36-year-old pointed his index finger at a board with the day's specials. Slightly bent, his head protruding forward, he squinted his eyes and then scratched the back of his neck. He seemed to be dealing with a rather complicated problem. "Risotto…" he sputtered. "Fish…" "Rice with shrimp…" Finally, he said, "Let's go somewhere else." .

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While we walked away, Avila explained to me that he didn't want to eat anything heavy. "My stomach just doesn't respond well," he said with his hand over his tummy. Another less explicit reason we were having trouble finding a place to eat was that he was worried about his physique. Like many of his fellow Cariocas, he values a "healthy body"—which in Brazil, means getting a beach tan and lifting weight. During the few last months, however, Avila hasn't had any time for that. Busy with lectures and conferences, the mathematician just hasn't been able to take care of himself as much as he'd like to. "Going to gym demands a certain organization around schedules, eating on time…" he said. "I lack that discipline, especially with so much traveling."

I guess that's the price of fame. Since August of last year, Avila was named the winner of the Fields Medal, which many pompously refer to as the "Math's Nobel Prize." (The Fields medal is awarded every four years to two, three, or four mathematicians under the age of 40. Nobel, however, is given yearly with no age restriction.) Being the first Latin American to win such an award, the man is under the crushing weight of being an example, an inspiration, and a role model for the young people living in places with no history in the field of mathematics. In Brazil, he's become an evangelist of sorts.

"I've been trying to reach non-mathematicians as means to have a positive effect on Brazil," he told me at bar he frequents, after we'd walked about four blocks looking for a place and he'd rejected the menu of three other restaurants. "Many people who only know math through school think of it as a dead science, as if there's no research behind it," he continued, while the waiter served his dish. Avila ordered a steak, which was spread out on a plate with no side orders, and a beer. "Actually, mathematical thinking is crucial and goes beyond what's taught in class. It's essential for the society to develop."

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Sporting a short-sleeved shirt and stubble, the mathematician fit well around Leblon's young bohemians, who were just starting to fill the streets as the sun hung low in the sky. The mathematician finally started to look like he was feeling at ease as he gulped down his beer. It reminded me of that Ernest Hemingway quote, in which the famed writer said that an "intelligent man is sometimes forced to be drunk to spend time with his fools…" or his stupid, his non-math-related-acquaintances.

Talking to non-mathematicians is a bit of an effort for Avila. When you spend all your time dealing with stuff like Schrödinger's operators, vector analysis, conditional connectors, game theory, transfinite numbers—I got all of those from Wikipedia—it's understandably hard to relate to any average bro who picks his nose and spends hours watching vines on the toilet. For Avila, even answering the question, "How's work?" from a friend outside of the math field is like being given an unsolvable equation.

"It's impossible to tell you what I'm working on right now," he said, with a grin. "If I were a scientist from some other field, a biologist, for instance, I could have a schedule, a clear methodology. But mathematics is just way more unpredictable. We just don't know where it's going. You expect it to lead somewhere, but if it comes to a halt, then that's that."

To Avila, people, have no clue of how complex and wide the research in his field of study is. The result of this disconnect may lie in the belief that, to win a Fields medal, the mathematician has to enter a competition.

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Although Avila emerged victorious in math competitions throughout his youth, Avila didn't have to undergo any test to get his medal. He was awarded for his contribution to the dynamical systems field, which is about as complex as it sounds. He tried to break it down for me with the patience of a saint.

"In dynamical systems, we study themes that evolve during time with a rule that describes the transition of one moment to the next," he said. You could say that it's the rule that allows us to connect past states of systems to future states of systems. Think about the water flowing in a pipe, the number of fish swimming in a lake, the swinging of a clock's pendulum—these are all examples of what could be included in these models.

But Avila's work takes it to the next level of complexity: Chaos. "In the long run, there's chaotic behavior, that we cannot predict." He is seeking to understand what will be the future state of a system with no patterns—like, the shuffling of cards or the smoke from a cigarette—through mathematics. The results of this research can be used to explain complex phenomena from areas vital to our lives; such as turbulence in physics, the financial market in economics, and neurobiology in biology.

By working with chaotic systems, the mathematician has brought order to the field: He's answered some questions that were decades old. And he's created a body of work that is a mix of science and craftsmanship. His style of math isn't just about equations; it's about new discoveries and beauty. "Mathematicians find pleasure, one that is almost artisanal, in getting our hands dirty with abstract dirt," he said. "It's like painting pictures or writing. There are aspects of visualization and creation in math that are as enjoyable as constructing an object. Of course, math has a [language] of its own, which takes a certain effort to [understand]. But the same goes for art, really."

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Those who see Avila working could definitely mistake him for an artist. His daily routine is that of a thinker, someone in search of inspiration, impressions, and insights. He can spend days in deep thought. And he's always looking for creative ideas and solutions for longstanding problems. "In math, the blocking part comes with the central idea, you just can't foresee when it'll happen. You can focus on a theme, which can be effective, but is limited, nonetheless. There are mathematicians who think that most of the job in regards to the formation of ideas is done in our subconscious, not our actual work," he said.

The writing, the calculations, and reviewing of the articles, explained Avila, is the costliest and most predictable part. The beauty of math lies in the unpredictable—in capturing the idea, that moment when the abstract turns sublime. "People have trouble seeing the artistic side of math because they see it as something deeply rooted in that old mechanical form of repetition and doing simple operations," he said. "But that's untrue. That mechanical aspect is barely present."

While sipping on his beer, he summed up what seemed to be his motivation. "I want to create this whole other universe and I'm going to explore it, that's math," he said. "The mathematician is driven by the enjoyment of discovery. You create and you are free." Even with its rules, math, said Avila, is liberating. "The rules just give you the conditions. Even the artist also isn't 100 percent free—everyone sticks to the law of gravity."

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Avila's mathematical prowess manifested at an early age. A good student, he stood out in the subject, and by the time he was 14, entering school competitions, he realized he loved solving problems. Avila stayed up all night solving equations. As a result, he'd wake up late—a habit he happily continues to this day—and skipped classes. His mother never liked the fact that the boy missed class, but never made a fuss about it because Avila had a unique talent. When he stood out at the International Math Olympics at 16 years old, the Brazilian Institute for Pure and Applied Math (IMPA) summoned him. While finishing high-school, the young Carioca was also undertaking his Master's Degree.

Around that time, Avila started to grow rebellious. He frequently ditched classes and earned Fs in his religious courses on purpose. His former school, São Bento, was very conservative—it separated boys and girls and f0rced students to take on ecclesial studies. For those religious classes, Avila left all of his tests blank.

"I had a teacher who thought that the consequences of theology were the fruit of reasoning," he said. "I refused to agree with that. I acted out because that principle was wrong," he told me, with big pauses between his words, as if he was thinking a lot before speaking. The principal gave Avila, the prodigy, a choice: Either he acquiesce, or he was out. Avila opted out.

For the mathematician, it was hard adjusting socially in high-school. He played soccer with his friends and chatted with his peers, but he never really felt a part of any scene. That feeling only got worse due to the fact he was the only child of divorced parents and he lived in an apartment building with no other kids. He had little to no interaction with anyone his own age. So he spent all of his time studying on his own terms, by reading and solving problems. It worked. By the time he was 21, Avila held a doctoral degree.

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Maybe as a reflection of the times he didn't feel tuned in at school, nowadays the mathematician values collectivity in his work. When he was receiving the Fields Award, the presentation complimented the way he's collaborated with several groups over the course of his career.

At 24, Avila started studying the complex field of "interval exchange" when he called up one of the world's leading authorities on the subject to help him out. At the time, Avila was living in Paris, France while he conducted research at the Centre National de la Recherche Scientifique (CNRS). One teacher there insisted he get into the interval exchange field. "He bugged the hell out of me," he said. "Then I started working on it. During that time, a very famous professor who seemed like a nice guy was in France. I picked a fairly well-known problem, which I really had no clue about, and called him for help."

Avila and the professor collaborated for months. In math, he explained, there are no hierarchies. Everyone is pretty much the same. The older professor instructed Avila, who, in exchange, used his outside knowledge, his technical skills, and fresh ideas to help. After several evolutions in the way to approach the problem, the duo managed to solve a question that was more than 30 years old. "It may sound weird, but it works," said Avila, with a faint smile.

So what's next for the mathematician? That's another one that is impossible to answer. As a good man of numbers, he tends to work on various problems at the same time. "Research topics take up several years. But you don't get stuck on only that. You work on other projects, each with its own pace," he said.

Upon noticing his voracity for solving problems, I mentioned Darren Aronofsky's Pi. In the film's climax, a number theorist is so tormented by his work that he ends up drilling a hole into his own skull. I asked Avila if he ever feared he'd turn out like that guy—an obsessed nutcase with a hole in his head.

After one of his trademark pauses—about 15 seconds—he answered me, with the objectiveness you'd expect from a mathematician.

"No."

Translated by Thiago "Índio" Silva