Marcelo Viana: A difficult equation

The director of IMPA talks about Brazil's entry into the international elite of mathematics research, and math education problems in the country's schools

Imagem: Léo Ramos ChavesThe next few months are going to be a lot of work for Marcelo Viana, director of the Institute for Pure and Applied Mathematics (IMPA), in Rio de Janeiro. In addition to leading the renowned academic institution, the resident of Rio, raised in Portugal, is also general organizer of the 28th International Congress of Mathematicians (ICM 2018), the largest and most important meeting in the field, which takes place every four years. For the first time, the event is being held in Brazil, in Rio de Janeiro, from August 1 to 9. When this interview is published, he is likely to be one of the few people in the world who knows the names of the winners of the Fields Medal, the most prestigious award in mathematics. As is common practice, the winners of this honor and other prizes awarded by the International Mathematical Union (IMU) are announced during the event. Brazilian mathematician Artur Avila received the Fields medal during the 2014 congress, held in Seoul, South Korea. “I have 20 hotel rooms in Rio reserved in my name to house the winners and other VIPs,” says Viana, with the slight Portuguese accent of one who spent his childhood and youth in Porto, where he received a degree in mathematics before returning to live on the other side of the Atlantic.

Math done at Brazilian universities and research centers like IMPA arrives at ICM 2018 with its prestigious reputation already established. The IMU has just elevated Brazil into the group of 11 countries that make up the elite of world mathematics research, such as the United States and France, the two biggest superpowers in the field. The application for admission was put forward in 2017 by IMPA and the Brazilian Mathematical Society (SBM).

Although academic research in the field may be going well, the same cannot be said of basic mathematics education in Brazil, given the poor performance of the country’s students in international math tests. In this interview, Viana, a specialist in the area of ​​dynamical systems, speaks about these two sides of Brazilian mathematics, which can simultaneously produce both a winner of the Fields Medal and children who don’t know how to do basic arithmetic.

Age 56
Dynamical systems
Mathematics degree from the University of Porto (1984); Doctorate in mathematics from IMPA (1990)
Scientific production
54 articles in scientific periodicals, 11 books authored or edited, 9 book chapters, advisor for 36 doctoral and 19 master’s theses

What kind of work was done with the IMU to get Brazil accepted into the elite group in mathematics research? Was it a job of convincing them or were there technical prerequisites that the country had already achieved?
There aren’t any explicit requirements; but the assessment is based on the country’s mathematical performance. We created a dossier with more than 30 pages in which we presented the reasons that justified our entry into Group 5. In this dossier, for example, there is a diagram in which we show the number of international articles in mathematics that have had a Brazilian author during the last 30 years. We went from 253 articles in 1986 to 2,349 in 2016. We are talking about all of Brazil and not just IMPA. But this isn’t even my favorite number in the dossier. About 30 years ago, the country produced 0.4 out of every 100 scientific articles on mathematics published around the world. It was a very low number, since both the population of Brazil and our GDP represent about 2.8% of the world total. Our goal in math research also needed to be at that level. Now we’re up to 2.4 articles out of every 100 produced in the world. In the dossier, there are other parameters like these. There are data on research, events, postgraduate education, basic education, about popularizing mathematics, and on the Math Olympics. The evaluation isn’t made using a formula. Mathematicians don’t much like numerical evaluations. It’s an overall assessment, subjective, but it has to be convincing. There are another ten countries in Group 5. We’re going to fit right in. We’re there beside the United States and France, two great powers, and Germany and the United Kingdom. These are the four largest delegations attending the ICM 2018. Brazil is the fifth. Group 5 also includes China, Russia, Canada, Japan, Israel and Italy.

But, in part, isn’t the large Brazilian delegation at the congress due to the event being held here?
There is one factor that helps us mathematicians measure this. Every four years there is this international congress. Everyone can attend, but only those invited can lecture. You cannot apply to be a speaker. About 200 speakers lecture at the congress, 20 at plenary sessions and the others at specific sessions in various areas of mathematics. Our first speaker at a congress was Leopoldo Nachbin [1922–1993], in 1962. The second was Maurício Peixoto in 1978, sixteen years later. Then we had sporadic participation in the congress until at some point we started having a speaker at every event. I was Brazil’s first plenary speaker, in 1998. At this year’s congress, there will be thirteen Brazilian mathematicians giving lectures. So it is true that there is a “home court” effect; whoever hosts the congress has more speakers than normal. But I’ll give you a counterpoint. Four years ago, the congress took place in South Korea, a country respected in mathematics. Do you know how many Koreans gave lectures at the event? Four.

Are you saying that Brazilian mathematics is better than South Korea’s?
I’m not saying that, but you understood me. The IMU is structured into groups. When Brazil joined the union in 1954, it was in Group 1, which is the lowest. The countries in this group pay one unit of IMU membership dues [€1,395 in 2018] and are entitled to one vote at the union’s general meeting. Group 2 gets two votes and pays four dues units and so on. In 2005, on the initiative of Jacob Palis [Brazilian mathematician, past IMPA director, and former IMU general secretary and president], we applied for group 4, where we were until the beginning of this year, alongside countries such as South Korea, Poland, Sweden, India, and Switzerland. It was already a very honorable position for a country that, 60 years ago, had almost no mathematics. Now we’re in group 5, which is the highest, with five votes and we will be paying 12 dues units. Our membership dues are paid by the Brazilian Mathematical Society, which is the organization that represents Brazil in the IMU.

Imagem: Léo Ramos Chaves Students and teachers attend a lecture in the auditorium at IMPAImagem: Léo Ramos Chaves

The starting point of the dossier is 1986, when you left Portugal and moved to Brazil. What was the state of Brazilian mathematics at that time?
IMPA already had a great reputation. Names such as Jacob Palis, Welington de Melo, and Paulo Sad were already known abroad. But IMPA was much smaller than it is today, in every aspect, even in its scope of activities. We were beginning to work in basic education, with the projects for improving teachers created by Elon Lages Lima, who was also director of the institute. The Math Olympics were much smaller. Later, in the 2000s, we became organizers of the OBMEP [Brazilian Public Schools Math Olympics], in addition to running the OBM [Brazilian Mathematical Olympics, open to all schools]. We started supporting regional Olympics and sending students to competitions abroad. Popularizing mathematics is one of our priorities today. In the last 30 years, there has been a quantitative and qualitative change at IMPA. The number of scientific researchers has more than doubled. The number of students has almost tripled. There has been growth, and at the same time IMPA’s image has been increased abroad. This had a lot to do with Jacob’s activities, but it wasn’t just that. Then in 2014 Artur Avila received the Fields Medal, and we won the right to host the ICM 2018. These two recent events boosted IMPA to an extraordinary level in world mathematics.

How would you evaluate Brazilian mathematics research done outside of IMPA?
Historically, Brazilian mathematics has radiated out from two centers: IMPA and USP. The impact of USP, and also of UNICAMP, is as evident in São Paulo as it is in other parts of the country. In the 1990s, there was a process of capillarization, with the emergence of postgraduate courses at educational centers in the North, Northeast, and Central West regions of the country. Today all of the northeastern states have postgraduate degrees in mathematics recognized by CAPES [Coordination for the Improvement of Higher Education Personnel]. This is relatively new, since it happened during the 2000s. So now we can say that mathematics is reasonably well distributed throughout the country and has the potential to grow. We have researchers with international projects and collaborations, but the institutions haven’t taken a prominent position on the global scene yet. For example, it’s important for our institutions to be more consistent in having and maintaining bilingual sites. At IMPA we’re putting a lot of effort into this area.

In 2016, you received the Louis D. Grand Prix Scientifique from the French Academy of Sciences, awarded for the first time in mathematics, and to a Brazilian researcher. Previously, in 2002, you were considered as one of the possible winners of the Fields Medal. What was it like living with the possibility of winning that medal?
I was born in 1962. So I could have won the medal until 2002. At the time I was in a very active period. The monetary value of the prize itself is low, about €10,000. But, since it’s a prize given to mathematicians with a maximum age of 40, it’s a career hallmark for the rest of their lives. I know a mathematician who won the medal and told me that his salary increased substantially after he left his home country and went to work in the United States. In emotional terms, winning the Fields Medal probably produces a greater impact than the Nobel, which is normally won near the end of one’s career. But, answering your question, there is no official list of candidates for the medal. In fact, no one presents himself as a candidate. I know I was nominated for it in 2002, but I never stopped living my life because of it, either before, or after not winning. In 2002, there were only two winners of the Fields Medal [the French mathematician Laurent Lafforgue and Russian Vladimir Voevodsky, 1966–2017]. The number of winners can be between two and four, with a strong tendency towards it being four. But this is a decision that the committee of medalists makes and the IMU has to follow it. It’s been years since there’s been a consensus on the winners.

“Mathematics research done in Brazil produces 2.4 out of every 100 articles published in international journals”

What’s the makeup of the committee that decides the winners?
The IMU is managed by an executive committee, elected at the general assembly to a four-year term. The executive committee appoints the major academic committees and doesn’t participate in the choices after that. This is a device for avoiding conflicts of interest. In the case of the Fields Medal, a committee of ten or twelve people is formed which is usually presided over by the president of the IMU. In 2002, Jacob Palis was the president and abstained from participating on the committee to avoid a conflict of interest. So the secret is to put mathematicians on this medal-selection commission which no one can find fault with. The committee decides and its decision is absolute. The system creates some tensions. For many years, the 40-year age limit for the medal winner wasn’t a written rule.

Why was this rule created?
Canadian mathematician John Fields [1863–1932] created the prize with the money left over after the international congress of mathematicians in Toronto, in 1924. His goal was to encourage young mathematicians. But he didn’t say more than that. At the 1936 congress the first committee for awarding the medal was created, which interpreted the term “young” as someone up to 40 years old. The committees that followed maintained this tradition, but it was not written. Even after the IMU wrote this rule in explicitly, tension still existed. In the 1990s, the rule came close to being broken. In 1993, the British mathematician Andrew Wiles discovered the proof for the greatest unsolved theorem, the Fermat problem, which had been unsolved for more than 300 years. The issue was that he would be 41 at the following IMU congress in 1994 when the next medal could be given. If there was ever a time to break the age-40 rule, that was it. But what happened? A few months before the congress, they discovered a hole in his proof of the theorem. Wiles gave a presentation at the congress, but the proof was not yet complete. It took him another year to finally solve the problem, but by then he was 42. At the next congress in 1998, he would be 45 years old, and that was just too late. I think that since then, the age-40 limit has become a rule written in stone—and healthier for it. After the age of 40, you no longer have to keep thinking about the prize because you really can’t win it.

Do we have Brazilian candidates for the Fields Medal this year?
I believe so. Ethically, I wouldn’t want to mention a specific name, it creates enormous pressure and expectation. I would say there are ten significant candidates worldwide. After the death last year of Iranian mathematician Maryam Mirzakhani, who, in 2014, was the first woman to win the Fields medal, it’s possible that the committee will give the prize to another female mathematician.

“I would say that there are ten significant candidates worldwide, including Brazil, who could win this year’s Fields Medal”

IMPA is a non-profit NGO that depends mainly on federal government funds. Have cuts to the science and technology budget during recent years affected the institution?
There have been cuts; a budget of R$39 million was proposed by the MCTIC [Ministry of Science, Technology, Innovation and Communications] for IMPA this year, less than half of what we normally receive, in addition to the MEC [Ministry of Education] contribution, which was R$26.5 million. I was worried for a time if we would be able to do both the 2018 OBMEP, which costs R$45 million, and the international congress, which will cost about R$15 million. A good part of the congress should be funded by the attendees’ registration fees, but we still don’t know how many people will attend the event. We’re hoping for 5,000 people at the congress, but it could be less. The first congress to reach that number was the one in South Korea. Registration fees are relatively high; US$450 for researchers and US$200 for students. If 3,000 show up I’ll be satisfied. At the end of last year we signed an agreement for a supplementary budget with MEC and MCTIC, which will be giving us more funds. But money is still short and we anticipate some cuts. In 2017, our budget was right around R$100 million. But there was no international congress to be organized.

Why do people tend to find math boring? Don’t mathematicians do a good job of selling it?
Certainly, mathematicians are terrible marketers. Mathematics is not an easy subject to teach for two main reasons. For young children, math isn’t boring. I see this in my own children, although I know they may not represent typical children. The math of children ages 5 to 7 is for counting toys or the number of slices in a pizza. This math matches the child’s interests. Of course, this isn’t always going to be the case. There comes a time in school where the subject becomes more abstract. There is no avoiding this. That’s when we lose our audience—unless the teacher can show that mathematics has something to do with things that are interesting to the child or young person, who wants to have fun. This is the first reason for not liking mathematics. The second is that mathematics deals with a set of sequenced learning. When you lose one part of this knowledge, either it’s recovered quickly or the next part becomes incomprehensible. But this is not an impossible problem to solve. Several countries have achieved this.

Which, for example?
The Nordic and Asian countries, generally. Each of them bets on their strong point. There are even cultural explanations behind this phenomenon. One of the things that impressed me the most at the opening of the 2014 congress was a video the South Koreans made. They wanted to emphasize that today they are doing well and that 50 years ago, when the country was in a bloody war, the schools didn’t stop working. The difficulty of mathematics isn’t mysterious. It is recognized, as well as its solutions. We have to have well-trained teachers, who work under good conditions, who really enjoy teaching, and are well compensated for it. We are lacking in every one of these criteria.

Imagem: Personal archive The dynamical systems group at IMPA during the late 1990s: mathematicians Carlos Gustavo Moreira, Jacob Palis, Welington de Melo, and VianaImagem: Personal archive

OBMEP is considered to be the largest Math Olympics in the world, with 18 million participants. It helps discover math talent, but the performance of Brazil’s high school students in mathematics remains weak in international tests such as the PISA, the Program for International Student Assessment. Why doesn’t the success of our Math Olympics transfer to the performance of these students?
The Math Olympics is very successful at what it can do: discovering talents, creating a dynamic of interest around mathematics. There are studies that show that when a school engages in the Math Olympics, student performance improves. But the Olympics can’t solve Brazil’s problems by itself. The number-one problem of education is teacher training, which, in general, is quite poor. About 5% of licensed math teachers are educated at public universities, 15% through public distance learning courses and 80% at private colleges. A few of the private colleges are good. The 5% that are educated at public universities, in general, don’t end up in the classroom. The Math Olympics recognizes and supports teachers. But we are talking about a few, not the mass of teachers. Besides education, there is the question of teacher appreciation. OBMEP gives the teachers motivation—it could be a medal, or just the satisfaction of seeing their students get excited. But this incentive has to be structural, from the schools themselves. There’s an idea prevalent in Brazilian schools that everyone must be equal, that to incentivize performance is discrimination, or worse, that it’s meritocracy, as if rewarding merit were bad. From 2003 to 2012, Brazil was the country that grew fastest in mathematics on the PISA test [from 356 to 391 points]. But I won’t hide the fact that the 2015 results were a disappointment [dropping to 377 points]. Still, there was an improvement in 2015 compared to the 2003 performance. I want to believe that 2015 was a hiccup and we are going to continue to improve again.

But the math performance of Brazilian students on the PISA is really bad. The country ranked 66th among 70 countries in 2015.
Yes, we’re pretty lousy. But we had improved more than 30 points between 2003 and 2015, which was not a trivial gain. I advocate, for example, the creation of a national exam, such as the OAB [Brazilian Bar Exam], to certify math teachers.

Let’s talk a bit about your personal history. Why did your Portuguese parents immigrate to Brazil?
It was a tradition in my family. My father, my grandfather, and my great-grandfather all came to Brazil to earn a little money. They were from northern Portugal, from Póvoa de Varzim, the hometown of Eça de Queiroz [Portugal’s most famous novelist], near Porto. They were farmers or fishermen, sometimes both. They grew vegetables, potatoes, carrots… My father, Joaquim, arrived in Brazil in 1952, returning to Portugal five years later to get married. But he called off the engagement, met my mother, Isaura, and married her. They came to live in Rio, where they stayed for five years. I was born in 1962, but at that time the situation here wasn’t too advantageous and my mother convinced my father to return. I returned with her at 3 months, and my father returned a short time later. I learned to speak, and grew up in Portugal. There, I’m Brazilian. Here, Portuguese. I’m a foreigner wherever I am. I did my studies there and got my degree at the University of Porto. My two brothers were born in Portugal, and still live there.

Imagem: Personal archive Viana (left) and Lorenzo Díaz at IMPA in 1987, when both were doctoral candidatesImagem: Personal archive

Were your parents educated?
I’m the first generation of my family to go to college. My father was a chauffeur; he even worked in construction here in Rio. On his entrance card into Brazil, he was described as a carpenter. It was a fabrication—carpenter was a kind of upgrade for him. He was a farmer. My mother was a primary school teacher. She is retired now, and my father has passed away.

Were you interested in math at school?
I was a good student in general. The subject I liked most was math. When I was 15, my mother asked me what I wanted to be. I said I wanted to be a mathematician and go to the top of the field. She was impressed by my certainty. I did my undergraduate degree and graduated in 1984. The following year, there was a one-week conference at the University of Coimbra. I arrived on Monday and signed up at the last minute to present a paper. I was then introduced to the star of the event, Jacob Palis, who was to give the final presentation, and he asked me if I was speaking. I said yes, my presentation would be at 8:30 p.m. on Friday. I was thinking that by then everyone would have gone away, and, thank God, no one would see my presentation. I was extremely nervous, but Jacob said he would stay and watch. And he actually did. At the end of my talk, he invited me to come to IMPA. The first thing I said was that I would need a scholarship.

Did he know you were Brazilian?
He did. But at the time, I had a totally Portuguese accent, and I felt Portuguese.

“The number-one problem of education in Brazil is teacher training, which, in general, is quite poor”

Did you already have research results to present at the Coimbra event?
Right after I graduated, I got a stipend from the university to spend two weeks in Paris with a French mathematician, Adrien Douady [1935–2006]. When I got there, the customs officer asked me about my visa [at the time Brazilians needed a visa to enter France]. “Oh, um, I don’t have one,” I said. He said I needed it. I asked him where I could get one. He directed me to the airport administration. Then I began to suspect that things were not going to go well. I didn’t know what a visa was. None of my friends had a visa.

You lived in Portugal, but only had Brazilian documents?
No one told me that I’d need a visa. I was 22 years old. I started to think things were weird when I heard an airport clerk ask if I was dangerous. I gathered my courage and went back to the customs officer to find out what was going on. I was held the entire day in the airport waiting room. I had an interesting anthropological experience. A guy came over and asked if I had a few coins for the pay phone. I gave him some change and he gave me a pack of cigarettes. At the time, I still smoked. We started chatting and I discovered that he was a Moroccan pot dealer who was being held like I was. He was quite calm. The worst moment was when the customs officer came to send me back to Portugal. The drug dealer began saying: “Let the kid go, he’s good people.” I mean, I had a Moroccan drug dealer as my defender! I returned to Lisbon and had to figure out how to get back to Porto. The following Monday I bought a ticket and applied for a visa at the consulate. Exactly a week later, I was back at the same Orly airport, scared to death. This time everything went fine, and I stayed in Paris for a week with Adrien Douady. He proposed some problems for me to think about; I managed to solve one of them and I went to present the results in Coimbra.

Can you explain what that problem was?
It was a thing in my area of dynamical systems. I started to study the subject when I was an undergrad. At the time, there were great advances being made in the study of fractals, and Douady was one of the great renovators of this area of ​​research. In simple language, in the area of ​​dynamical systems there’s a phenomenon, and a known law that describes the evolution of this phenomenon. In the solar system, for example, with the planets spinning around the Sun, it’s Newton’s law of gravity. But usually, knowing the law doesn’t speak directly to what will happen within that system. There is a mathematical formula that needs to be solved. It’s necessary to try to extract information from the law. The area of ​​dynamical systems encompasses a set of techniques and results that help with this process. In general, the equations are not simply solved with one formula or one solution. There is a kind of law we call iterations. This type of law is a formula that says that if you are currently in state X, the next time you will be in the state f of X. And, in the following, in the state f of f of X. And so on in succession. We must always apply this transformation. But the point is to know what happens at the end, where that sequence is going to stop. If this law describes an ecological system, what we want to know is if, in the end, these species will still be alive or extinct. In my first paper I studied a type of transformation.

Weren’t there better opportunities in Europe than coming to study mathematics in Brazil?
There was practically no doctoral program in mathematics in Portugal, and I knew I would have to leave the country. At that time Portugal was still negotiating to enter the European Union and there were few scholarships available. Most were of foreign origin, from the British Council or the Fulbright Foundation. Just look at the irony: most of the few scholarships available were for Portuguese students. I was Brazilian and was eligible for fewer than half of them. I hadn’t yet naturalized as Portuguese because I didn’t want to do military service; today I have dual citizenship. I already knew of IMPA’s reputation and had studied Jacob’s books when he proposed that I come to Rio on a doctoral scholarship. The proposal was interesting. Brazil has always had a tradition of granting scholarships based on the merit of the student, without looking at the color of their passport. For me, that was unimportant, because I was Brazilian. But this is an important feature of our post-graduate system. At IMPA, half of the students are foreigners, almost all of them from Latin America. Half of them stay in Brazil and half return to their home countries, which is great in both cases. Those who return home are our ambassadors.

Today we talk a lot about algorithms, artificial intelligence, big data. How do you balance research in applied and abstract mathematics?
In math, you never know what will result in an application. This is a key rule: you have to let human beings exercise their creativity. When the advances of mathematics are applied, they’re invisible. Without math, tomography wouldn’t exist. There is a lot of math in the way the signal from a soccer game is broadcast, for example. Even I wasn’t aware of it.

How so?
A few years ago, an American mathematician of Belgian origin, Ingrid Daubechies [now at Duke University and the first woman president of the IMU, from 2011 to 2014] gave a talk at IMPA and spoke on this topic. Among other things, she helped create the JPEG 2000 protocol, an image compression standard. She studies wavelets: a mathematical object, a ripple, a vector space of infinite dimension. The concept of wavelets was something conceived in mathematics without any kind of application in mind. Today it is a tool used in high definition broadcasts. It works like this: a grass soccer field is almost all the same, and the formula—the algorithm of these wavelets—automatically recognizes this resemblance and decreases the resolution of that part of the image. There is no need to show too many details in the grass field. But when the camera focuses on Neymar’s face, for example, the resolution gets higher and there are more details. This system greatly reduces the data size of the signal to be transmitted. Who’s ever heard mathematicians advertising this? This is a very intelligent method for automatically adjusting image resolution based on what you actually want to see. But if society doesn’t know about this, it doesn’t value math. Nor can we demand a priori that research has to have applications in mind to be encouraged.

What are the major challenges of mathematics research today?
For me, the big challenge is to create tools for dealing with new study subjects that didn’t exist 30 years ago. That is, to mathematize these new subjects. I can cite two or three scientific challenges that will only be solved when they’re solved mathematically. Today, however, we’re still not even close to solving them. In physics, for example, some recent theories, such as quantum field and string theory, seem to me to be halfway between alchemy and magic. They are theories that researchers use as tools to do calculations, but which don’t have a rational foundation that makes sense. This is also true of all of quantum mechanics, in concepts such as entanglement. Welington de Melo gave courses on quantum field theory for years. One day I asked him what it was all about and he said something like this, quoting another mathematician: “It’s a non-existent theory about a subject that no one understands.” When talking about the future of mathematics, I like to tell a story that’s a combination of big money, intelligence, and the American way of thinking. James Simons, a good American mathematician, headed to the financial market and became a billionaire. He created a private foundation to finance science. A few years ago, he asked Ingrid Daubechies what he could do to help mathematics. She suggested setting up an institute to study large masses of data. I don’t like using the term “big data.” He then set up a new institute, funded by the Simons Foundation. This is a new kind of mathematics, one that will extract information from large masses of data.