26/07/2019

An engineer and his intangible works.

Ilustração: Caio Borges_2019

Reproduction from Piauí magazine.

By Fernando Tadeu Moraes – Finalist in the Mathematics category of the 2019 IMPA-SBM Journalism Award

Few mathematicians have the distinction of having made a decisive contribution to a field of research, one capable of changing its course and dividing it into a before and an after. Mauricio Peixoto, who died on April 28th at the age of 98 in Rio de Janeiro, did that and more. His theoretical work undoubtedly "changed the history of the field of dynamical systems," stated Marcelo Viana, director of the Institute of Pure and Applied Mathematics (IMPA), which Peixoto helped found in the 1950s, today one of the rare centers of excellence – in any field of knowledge – based in Brazil.

The results of Mauricio Peixoto's research, in addition to boosting and revitalizing the study of dynamical systems worldwide, would also change the institutional history of mathematics in Brazil: his area of expertise would become the field in which some of the best Brazilian mathematicians, such as Viana himself and Artur Avila, winner of the Fields Medal in 2014, would make their most important discoveries. If Brazil has relevance in the current geopolitics of mathematics, it is because of decades of accumulated studies in this very specific area, a tradition largely founded by Peixoto.

The field of "dynamical systems" seeks to understand the evolution of processes over time – such as everyday climate variations, the spread of epidemics, or the balance of planets in the Solar System – and the subtle relationships of order and apparent disorder existing in the dance of interdependent elements of each of these systems, whether they are stars, cells, or molecules. In studying them, mathematicians seek to discover patterns and regularities even where at first glance only confusion is apparent – or, conversely, they seek to show how seemingly simple and stable systems are subject to suddenly losing any regularity, becoming chaotic, or falling apart.

The origin of this research dates back to the end of the 19th century, when some physicists and mathematicians, including the Frenchman Henri Poincaré, focused on a question that was both intriguing and frightening: Is the Solar System stable? We know that there are quite complex gravitational interactions between the planets and satellites that compose it, in addition to those established between the Sun and each of the bodies that move around it. If this set of bodies in which the Earth is included, with all these complex force relationships, suffers some kind of disturbance, will it tend to return to its normal "functioning"? Or could the regular dance of the planets and satellites around the Sun be disrupted? In general, any system will be stable, that is, it will return to its organizational pattern even after a small disturbance. Or not – that is, if it is shaken, it will become disorganized, like a system of planets in which one of them is hit by an asteroid and some go flying through space, escaping their orbits.

The idea of structural stability first appeared in a mathematical paper in the late 1930s, published by two Soviets, Aleksandr Andronov and Lev Pontryagin. However, the term was only coined in 1949 by the American of Russian origin, Solomon Lefschetz.

Peixoto encountered the concept in 1955 while conducting research at the newly created IMPA library. His interest was immediate. Shortly afterward, he wrote a letter to Lefschetz, outlining his ideas on the subject, and entrusted his friend, the mathematician Leopoldo Nachbin, with delivering it to the American during a congress in Mexico. Lefschetz received the message, was enthusiastic about what he read, and, in his reply to the Brazilian, invited him to spend some time at Princeton University in the United States, where he taught. It was in Princeton, in the late 1950s, that Peixoto completed the manuscript of his first article on structural stability. This and two other papers , published shortly afterward, form the core of the Brazilian's fundamental contribution to mathematics.

A phenomenon is considered structurally stable when small changes in the law that describes it do not produce significant variations in its evolution. Marcelo Viana uses a culinary analogy to clarify the concept: "Most cooking recipes don't go wrong if we're a little imprecise about the ingredients; they are stable." The same applies to a pendulum. Minute changes in its size, weight, or shape will not change the fact that it will oscillate for a period until it stops, due to friction with the air.

Peixoto's first breakthrough was correcting the existing definition of structural stability at the time – his proposal became canonical and is still taught in books and courses today. “He found the correct definition,” explained researcher Benar Fux Svaiter, from IMPA. “In mathematics, having the correct definition is often the key to opening certain doors. This facilitated the formal treatment of various issues and put things in tangible terms, allowing mathematicians to work within a new conceptual framework.” Svaiter is currently preparing a book that will bring together selected texts by the mathematician.

In subsequent articles, Peixoto proposed and demonstrated the theorem that entered mathematical literature bearing his name. “Basically, he showed that phenomena that can be described on surfaces like the sphere, that is, by means of two variables, are practically all structurally stable. In more modern language, we say that systems with two degrees of freedom are not chaotic,” they will not become definitively disorganized when subjected to small perturbations, explained Viana. “And why is this important? Because in the analysis of any physical phenomenon, one seeks to find laws that are structurally stable. Is this characteristic common? Is it uncommon? Peixoto's theorem tells us that, for systems with two degrees of freedom, this will essentially always happen.”

According to the director of IMPA, in addition to proving a great result, Peixoto also boosted the field of dynamical systems by inspiring the American Stephen Smale, one of the greatest mathematicians of the 20th century, to delve into it. In *The Mathematics of Time *, Smale describes his meeting with the Brazilian, which took place at Princeton University: “Around 1958, I met Mauricio Peixoto. We were introduced by Elon Lima, who was finishing his Ph.D. with Ed Spanier [ at the University of Chicago ] […], and he showed me his results on structural stability in two dimensions. I was immediately enthusiastic, not only about what he was doing, but also about the possibility of, using my knowledge of topology, generalizing his work,” that is, making it valid for other dimensions, larger than that of surfaces like the sphere.

"From there, things took off," Viana summarized.

The meeting would also bear fruit for IMPA and Brazilian mathematics. At the invitation of the Brazilians, Smale spent six months of 1960 at the Rio de Janeiro institute. During this period, he proved results that would earn him, in 1966, the most prestigious award in the discipline, the Fields Medal, and which had the side effect of putting IMPA and Rio de Janeiro on the world map of mathematics.

The Brazilian researcher who had inspired Smale, however, would soon miss the boat on the development of the field. "After these two critical, I would even say historic, interventions, Mauricio's role in dynamical systems became primarily symbolic," Viana said.

Thus came true the prediction of his friend and mentor Lefschetz, recounted by Peixoto himself in his acceptance speech for the Academy of Sciences of the Developing World award in Beijing in 1987. When complaining to the American that no one was interested in the topic of structural stability, the Brazilian heard the following: “No, Mauricio, that’s not a problem, it’s your good fortune. Try to work as quickly and hard as you can on this subject, because the day will come when you won’t understand a single word of what they’re saying about structural stability. That happened to me in topology.”

Peixoto's pioneering spirit made him an unavoidable example for researchers in the country, Viana asserted. “Imagine how difficult it was to be a mathematician in Brazil in the 1950s. Then along comes a guy who, with limited training, without any guidance, and studying in a newly created library, chooses and solves an important problem, thus gaining international recognition. That can only serve as an inspiration.”

Mauricio Matos Peixoto was born on April 15, 1921, in Fortaleza. At the age of 9, he arrived in Rio de Janeiro with his entire family after his father, the governor of the state of Ceará, was deposed by the Revolution of 1930. In the then federal capital, he was a student at Colégio Pedro II and, later, at the School of Engineering of the University of Brazil.

Peixoto's interest in mathematics, curiously, arose after he failed the subject at age 11. "I came from Ceará to Pedro II extremely weak in mathematics and, at the end of the year, I failed," he would recount years later. A family friend, a student at the engineering school and also from Ceará, helped the boy prepare for the make-up exam. "We started from scratch, and I was dazzled by the classes; even then I decided I was going to study something involving mathematics," he stated in an interview included in the book IMPA 50 Years .

Later, in engineering school, he became friends with Leopoldo Nachbin, who would become one of the country's greatest mathematicians. It was also during this time that he met Marília Chaves, his first wife, whom he married in 1946.

Surprisingly, the man who changed the history of dynamical systems not only lacked a degree in mathematics but never completed a doctorate in the field. Peixoto began the course at the University of Chicago in 1949, but abandoned his studies after taking and passing a competitive exam for a professorship at the School of Engineering. He began teaching at the institution in 1952. “I never regretted abandoning my doctorate, because a professorship at the University of Brazil was very difficult to obtain, as it was a lifetime appointment; it wasn't an opportunity to be wasted.”

In the same year he became a full professor, he founded IMPA with Nachbin and Lélio Gama. In the book about the institute's fiftieth anniversary, one of its first researchers, Elon Lima, pointed out what, in his opinion, differentiates the Rio de Janeiro center from similar institutions. “At IMPA, its researchers have always had the vision that it is not necessary to learn a tremendous amount of mathematics to do high-level, meaningful research. One can set aside many formalisms, general, abstract and complex theories, and concentrate on relevant, basic problems, and still succeed – the greatest example of this is Professor Mauricio Peixoto.”

According to Lima, this attitude towards mathematics contrasted with that of Nachbin, the other great figure at the institution in its early days. “Professor Nachbin had a somewhat different vision because he had a more French-style background, meaning he had to learn a lot of things […]. [ He saw ] mathematics as a formal system, while Mauricio had a vision closer to that of an engineer. [ Mauricio ] did pure mathematics, but with a much more classical perspective. And with that, he had considerable success. At IMPA, Mauricio's vision ended up prevailing.”

Added to this mathematical pragmatism is an uncompromising commitment to excellence. "He could very well have done mediocre math," Viana imagined. "What controls were there at the time? What competition? None of that existed. But he chose the path of quality. This example permeated the institution from the very beginning."

This characteristic was already present in the first doctoral thesis that Peixoto supervised at IMPA – the first also defended at the institute – in the early 1960s. The work carried out by Ivan Kupka gave rise to the Kupka-Smale Theorem, another important result in the area of dynamical systems. “It was a beautiful way for an institution to begin training students,” declared Viana. “Mauricio set the bar very high.”

Although he spent most of his academic life associated with IMPA, Peixoto also had some periods of professional activity away from the institution, such as his time at Brown University in the United States (1964-68) and at the University of São Paulo (1973-78), in addition to a stint as president of CNPq, the National Council for Scientific and Technological Development (1979-80). He also held the presidency of the Brazilian Mathematical Society (1975-77) and the Brazilian Academy of Sciences (1981-91). In 1974, he became the second Brazilian invited to be a speaker at the International Congress of Mathematicians, the most important event in the discipline, held every four years since 1897. The first Brazilian had previously been Nachbin.

It was also during the 1970s, in São Paulo, that Peixoto became close to the mathematician Alciléa Augusto, who had studied at IMPA in 1962 and 1963 under his guidance. She describes her husband as a thoughtful, serene man with simple habits, who preferred listening to speaking and enjoyed the company of younger people. "It was common for him to arrive home enthusiastic, talking about some meeting he had had at IMPA." Until a few months before his death, Peixoto continued to frequent the institution, where he maintained an office.

His intellectual longevity was astonishing. He produced mathematics and wrote articles until he was 93 years old. “He worked hard, often staying up all night,” his wife recalled. His dedication was matched by his scientific ambition. Viana recounts that, in 2010, he took on the task of preparing a volume of Peixoto's selected works, a tribute to his 90th birthday. “When I presented the idea to him, I found it strange, because he didn't seem enthusiastic,” he recalled. “I approached him a second time, at IMPA, and he did everything to avoid the subject. When I insisted, a third time, the surprise came. He said that, of course, he was happy and grateful for the tribute, but asked if it wouldn't be possible to wait another year or two, as he was writing an article that would be the culmination of his work.” Peixoto's last work was published in 2014. The book that Viana couldn't finish in 2010, now under the care of Benar Fux Svaiter, should be published in the coming months by the prestigious Springer publishing house.

Besides formulas and numbers, Mauricio Peixoto appreciated literature and classical music. “He loved Bach's cantatas and Beethoven's symphonies,” Alciléa recounted. He also admired poetry, especially that of Carlos Drummond de Andrade and Rainer Maria Rilke. His other pleasure was spending weekends at his farm in the rural area of Petrópolis. “There he transformed. He would walk all day around the property, open roads, and even built a house when he was over 80 years old. He loved trees, and he planted thousands of eucalyptus trees on the farm, as it was the species that best supported that land.”

In the region where the property is located, known as Estrada dos Contrões, Peixoto financed the construction of a small educational institution with the money he won in 1969 from the Moinho Santista Prize. The Marília Chaves Peixoto Municipal School, inaugurated in 1971 and named in honor of his first wife – who died prematurely in 1961 – operated for forty years. “During all that time, Mauricio ensured there was a teacher there. He had a personal obsession with improving basic education. At that time, there were many illiterate people in the region; the nearest school was about 10 kilometers away,” stated Alciléa. Although the space is no longer used as a school, it remains available to the community. Today, the Brejal Microbasin Association operates there, taking care of the preservation of local springs.

Peixoto's health deteriorated suddenly at the beginning of this year. "Starting in February, he began to get more tired and less interested in things," said Alciléa. He was hospitalized – for the first time in his life – in mid-March. On April 17, he returned to the hospital, where he died eleven days later. He left behind his wife, four children, and eight grandchildren.

From a young age, he had said he wanted to study music, but he never managed to fulfill that wish. "Now, at the very end, we started talking again about the possibility of him taking piano lessons," his wife said. There wasn't time. On the day of Mauricio Peixoto's death, one of his grandsons wrote, in a text dedicated to his grandfather: "Even though you lived 98 years, I think it wasn't enough."