By Andrea Hargraves 
For Artur Avila, one of Brazil's greatest science advocates and the country's most famous mathematician, no one should be proud of not knowing mathematics. “It's like an excuse. It's like someone saying they don't know how to read a restaurant menu because it's 'precise.'”
“I think the way to remove this impenetrable aura is not to kill curiosity, but to allow people to explore and play with mathematics, especially at an early age.”
Ten years after winning the Fields Medal, an honor that is only awarded every four years and to those who have achieved extraordinary accomplishments before the age of 40, Artur Avila has fulfilled the role of being the “face” of mathematics in Brazil and the country’s representative at events abroad.
Avila (45 years old) is currently a full professor at the University of Zurich (Switzerland) and remains affiliated with the Empa Institute (Institute of Pure and Applied Mathematics), in Rio de Janeiro, in the position of extraordinary researcher.
He doesn’t have students in the country yet, but he says he’s open to conversations and co-supervision. He brings his students from all over the world. “It’s essential that each student develops their own way of doing mathematics,” he says. “My role is not to try to create copies of myself. I appreciate independent students who are looking for their own way.” The next step is to go to China and discover the mathematical potential of this Asian country.
In an interview with BinderThe only Latin American to have received the Fields Medal, he talks about the Brazilian panorama and the mathematical problems he faced, always in the field of dynamic systems, those whose state changes over time, and governed by mathematical equations, which described many phenomena, both of the physical world and mathematical entities whose exploration is interesting enough.
“We mathematicians are used to failing over and over again. But we learn to accept these failures as part of the process, knowing that something will work out eventually.”
What changed in your life after winning the Fields Medal?
First, on the mathematical side, my work continues in the same general direction. There is always a natural change in research activities, as new questions arise and mathematics itself evolves. But the Fields Medal has not caused any major changes in the way I do research. What has changed is the reach: today I have access to more people, more students, and different situations that allow me to work with very talented people who want to learn from me. Before, I had a little bit of that, but now it is much more.
How is your relationship with Brazil?
Although I moved to Zurich in 2018, I still have close contact with Brazil, especially through Empa. I participate in events that represent mathematics in Brazil. This helps to show a wider audience, especially potential young talents, that it is possible to do high-level mathematics in Brazil.
Galileo said that the book of nature was written in the language of mathematics. Carl Sagan said that we live in a society that is so influenced by science and technology that it understands almost nothing about science and technology. Does the solution to this problem involve drawing people’s attention to mathematics? What do you think is the way to remove the aura of impenetrability?
People are naturally curious, but at some point that curiosity about mathematics seems to be stifled. I think it’s not healthy for people to say they don’t know anything about mathematics and to be proud of it, as if it were an excuse. It’s like saying they don’t know how to read a restaurant menu because it’s “accurate.” This shouldn’t be a source of pride; everyone should have some level of mathematical knowledge, which is essential for citizenship.
I think the way to remove this impenetrable aura is not to kill curiosity, but to let people explore and play with mathematics, especially at a young age. For example, Math Olympiads start out as fun, but can lead to deeper interest. But in reality, mathematics requires effort to be appreciated. You don’t understand it by looking, you really need to do it – the exercises, for example.
Another interesting point is the general perception of mathematics. People hear about recent discoveries in physics, chemistry and biology, and there is also the Nobel Prize that highlights recent discoveries in these fields once a year. While the contact that almost everyone had with mathematics ends at most in the mathematics of the 17th century.
How do you feel about your current role in showing that a mathematician is not a little white-haired man? Fellow Fields Medalist Marina Viazovska He already said Folha On the importance of these symbols.
The Fields Medal brought greater visibility, and with it the possibility of representing mathematicians to a larger audience. I think that's an important thing, almost everyone can imagine a modern physicist, like Einstein, and it shows that physics is a living science and also a human science. In the case of Maryna Vyazovska, being Ukrainian, she has an even greater symbolic role, because of the war.
On another note, I was recently in China, and I took the opportunity to try to stimulate the development of a network of cooperation with Brazil. China has invested heavily in research and has enormous human capital, so it is interesting to be able to attract students from there and also to be able to send students to work in Chinese institutions.
Can you tell us what you are currently working on?
In recent years, I have continued to focus on dynamical systems, exploring three main directions. One involves classical questions in mechanics and classical differential equations. I also work on complex dynamics. [com números complexos]In an attempt to understand how the geometry of fractal objects, such as the Mandelbrot set, affects higher-dimensional dynamics. The fractals shown in this analysis, in addition to being aesthetically impressive, combine complexity with a partially comprehensible order, allowing for amazing applications in other areas of mathematics.
There is a region in the Mandelbrot complex known as the “Valley of the Elephants,” a particularly interesting part where elephants can apparently be seen parading. This structure can be partially understood, knowing that it is responsible for the formation of other, more complex structures. In a larger dimension, when we look at the complex version of the Hénon attractor, these structures interact with other fractals, producing a variety of dynamic effects.
In addition, I study quantum dynamics in condensed matter models. One might think of the subject of study as the evolution of a system in which there is a large network of atoms, but which is riddled with impurities. One focus of this work is to understand the change in the properties of these systems when the nature of the impurities changes, especially the problems of localization and delocalization of the wave function. These are questions related to physical concepts, such as the properties of metals and dielectrics.
Mathematics is not a language, but ideas and facts, but can't ideas and loose facts also be anything else?
There is a world of mathematical objects that mathematicians conceive of as things in the real world. These objects exist independently of how we describe them, but to communicate these ideas we need a language. So, although mathematics is not a language in itself, it does involve describing ideas that exist in themselves, regardless of how we express them.
Of course, in applications such as physics, things that exist in the real world are taken into account, and mathematics is involved in describing these things. However, mathematical ideas and facts constitute a world that exists in itself and also needs to be described in order to be understood.
It is interesting to take the opportunity to explain what mathematics really consists of. It is not about solving increasingly difficult laws, but about the in-depth exploration of new ideas. Creativity is essential, of course combined with technology, just as it is in the arts, for example.
But there is this bias, it is common to hear people argue against mathematics for creativity. However, it would be good at least to spread the word that mathematics is constantly evolving and that there is much to discover.
What is your routine like? What is the process for resolving these issues?
The routine of a mathematician is strange. At first it can be a bit frustrating because we are used to failing over and over again. But over time, we learn to accept these failures as part of the process, knowing that something will eventually work out. Often, the brain is working in the background, even when we are not consciously thinking about the problem. I like to take walks to clarify my ideas, and when something comes up I go back to pen and paper to test it.
The goal is to understand the problem so well that you don't need to do a lot of math; technical checks can be short, but of course require a great deal of precision. I'm more of a believer in trying to really understand why what we're trying to prove should be true, in a clear way so that technical checks are practically detailed.
How is your relationship with your students?
It is essential that each student develops his or her own way of doing mathematics. My role is not to try to create copies of myself. I value independent students who seek their own way. This approach has been successful, as many of my students already have well-recognized scientific output.
How do you rate the mathematics panorama in Brazil?
This is impressive, especially considering our relatively recent development. Over the past 70 years, Brazil has seen great progress in mathematics, despite the political and economic crises the country has faced. This is proof that we have high-quality mathematical leaders who have been able to create successful paths.
Mathematics has a special advantage: it does not require expensive laboratories or complex inputs. What we need are qualified people and a favorable environment for research. On the other hand, the lack of a state science policy that goes beyond governments remains a major challenge. Science must be a long-term project, and it must be stable, so that we can reap solid results.
How do you assess the impact of initiatives like the Mathematical Olympiad? Do we have a new crop of good mathematicians?
The Olympiads have multiple functions. They not only discover new mathematicians, but also encourage young people to engage with the academic world, even if they are pursuing other careers, such as engineering or physics. Over time, this program has expanded and become more comprehensive.
I remember that this relationship between Emba and the Olympics was important for a long time, and in a certain way it was decisive in my career.
Mr. will be speaking to the youth at ImpaTech. [programa de graduação do Impa] Next week. What message would you like to leave them?
I want to show them what mathematics actually looks like in practice, what it means to think about mathematics, and how that relates to what they will learn. It is completely normal for young people not to have this perspective at this stage of their training. Of course, I also want to send the message that you will need to invest a lot of time and dedication, but the results will come in the end, and it is worth it.
I will also share some lessons I have learned along the way, and some things I have only realized in practice, that may make their journey a little easier. And remember that mathematics is more generally important in life and in the practice of citizenship.
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Artur Avila Cordero de Melo, 45
Professor of Mathematics at the Mathematical Institute of the University of Zurich (UZH) and Extraordinary Researcher at Empa.
PhD in Mathematics from Empa (2001) and Post-Doctorate from the French College (2003). He worked as a researcher at the National Centre for Scientific Research (CNRS, 2003-2018) in France, where he was also Director of Research.
In 2014, he became the first Latin American to receive the Fields Medal, considered the “Nobel of mathematics”, for his work in dynamical systems and spectral theory. He has also received the Salem Prize (2006) and the European Mathematical Society Prize (2008).