# Maurício Matos Peixoto

### Quick Info

Born
15 April 1921
Fortaleza, Ceará, Brazil
Died
28 April 2019
Rio de Janeiro, Brazil

Summary
Maurício Peixoto was a Brazilian mathematician who did outstanding work on the structural stability of dynamical systems. In a series of four papers he developed what today is known as Peixoto's Theorem.

### Biography

Maurício Peixoto was the son of José Carlos de Matos Peixoto (1884-1976) and Violeta Rodrigues de Matos. José Carlos de Matos Peixoto was professor of Roman Law at the Federal University of Rio de Janeiro and from 1928 to 1930 was governor of the State of Ceará. He married Noeme Câmara in 1914 and then his second marriage was to Violeta Rodrigues de Matos Peixoto, daughter of journalist and author Tibúrcio Rodrigues and Elvira Garcia. In 1930 an armed Revolution led to the President being deposed and José Carlos de Matos Peixoto was deposed as the governor of the State of Ceará. Humiliated by the new regime, he left Ceará and went with his family of eight children to Rio de Janeiro, where he devoted himself to law and teaching.

Maurício was nine years old when the family moved to Rio de Janeiro in 1930 but in 1931 he was sent back to Fortaleza to complete his primary education. Back in Fortaleza he lived with an aunt until he finished his primary schooling at the end of 1932. Returning to live with his parents and siblings in Rio de Janeiro he began studying at the Colégio Pedro II in 1933. This college, the third oldest in Brazil founded in 1837, took pupils from the full range of ages. Most of the students belonged to the country's economic and political elite. His performance at the Colégio Pedro II was poor, however, the result of the primary schooling he had in Fortaleza being of a much lower level than that in Rio de Janeiro, and he failed his examinations. His father decided that his son needed better teaching and employed Nelson Chaves, who was a student at the Escola Nacional de Engenharia. This tutor did an excellent job as Peixoto explained [12]:-
He was an intelligent young man, with very clear ideas about mathematics. It was a very important event, because it changed my life, in a certain sense. ... and I was blown away by his teaching; at that time I decided that I would study something that involved mathematics.
With Chaves' help, Peixoto easily passed the examinations at the Colégio Pedro II and spent six years at the school. In 1938 he entered the Universidade do Distrito Federal to study mathematics. This university had been founded in Rio de Janeiro only three years before, in 1935, on the initiative of the Secretary of Education, Anísio Teixeira. This university, with both administrative and pedagogical autonomy, employed professors with a wide range of views, a fact that generated fierce clashes with the Federal Government. The Universidade do Distrito Federal was forced to close in 1939 when Peixoto had only completed one year of his studies. He explained [12]:-
As a career as a mathematician did not exist, I went to study the subject in the only place where it was taught, that is, at the School of Engineering.
In 1939 Peixoto enrolled at the National School of Engineering at the University of Brazil in Rio de Janeiro. Marília de Magalhaes Chaves and Leopoldo Nachbin were both his fellow students, all three beginning their studies in the same year. Peixoto said in the interview [12]:-
During the five years of the engineering course, Leopoldo and I were inseparable companions. ... Marília's influence was very strong during these "golden years" ... Both Marília Chaves and Leopoldo Nachbin were important influences in my becoming a mathematician. This in the sense of trying to live for and by mathematics.
In 1943 Peixoto graduated with an engineering degree but he never intended to have a career as an engineer. Mathematics was always the subject for him and throughout his engineering course both he and Nachbin studied advanced mathematics together, often in the municipal library or at the university. Peixoto also attended lectures at the National Faculty of Philosophy at the University of Brazil by Luigi Sobrero and Gabriele Mammana. Luigi Sobrero (1909-1979) had been born in Turin, Italy, and was awarded an engineering degree from the University of Rome in 1931 and a mathematics degree two years later. Between 1939 and 1943 he was a visiting professor of theoretical physics at Rio de Janeiro. Gabriele Mammana (1893-1942) was also an Italian mathematician who had graduated from Pisa and then became Mauro Picone's assistant at the University of Catania. He taught in Livorno, Cagliari, Catania and Naples before teaching at the Federal University of Rio de Janeiro between 1939 and 1942. He was an analyst working on differential equations. Peixoto also gave private mathematics lessons throughout his time as an undergraduate.

On 6 September 1946 Peixoto married the mathematician Marília de Magalhaes Chaves in Rio de Janeiro. Marília had been born in Sant'Ana do Livramento, Rio Grande do Sul, Brazil on 24 February 1921, to parents Tullio de Saboia Chaves (1884-1971), Professor at the Surgical Medical School at Rio de Janeiro, and Zillah da Costa Magalhaes. Maurício and Marília had two children, Marta Chaves Peixoto (born 31 March 1949) and Ricardo Chaves Peixoto (born 24 March 1953).

After graduating, Peixoto was an assistant to the professor of Rational Mechanics at the National School of Engineering. He continued to undertake research on mathematics studying differential equations involved in dynamics. He planned to compete for professorships and developed a thesis for this purpose. He began publishing mathematics papers in 1946 and published the 68-page book Sistemas nao holônomos in 1947. In the same year, he began working at the Mathematics Centre of the Fundação Getúlio Vargas. This higher education institution had been founded in 1944 with the aim of stimulating development in Brazil. At the Mathematics Centre there were lectures and much discussion between an enthusiastic group of young mathematicians. While at the Fundação Getúlio Vargas he met several leading mathematicians, for example Oscar Zariski, who came to give lectures. The journal Summa Brasiliensis Mathematicae which began publishing in 1945 came out of discussions at the Fundação Getúlio Vargas. Peixoto published the paper On the existence of derivative of generalized convex functions in that journal in 1948. His 66-page work Convexidade das curvas was also published in 1948. Edwin Beckenbach writes in a review:-
... there is presented a brief discussion of the basic theory of convex functions as developed by Jensen and others, followed by a more extensive treatment of generalised convex functions. The latter development contains contributions which the author himself has made.
This work was produced in competition for a professorship in the School of Agronomy. He did not win the professorship but the thesis was accepted and it entitled him to a doctorate given the regulations in force in Brazil at this time.

Marshall Harvey Stone and Adrian Albert, both from Chicago, came to Rio de Janeiro on a State Department programme and Peixoto, together with his friend Leopoldo Nachbin, decided to try to obtain a Buenos Aires Convention scholarship to study at Chicago. Both were eventually awarded a scholarship. Peixoto said [12]:-
It was a State Department grant, which cost us almost daily trips to Ministry of Foreign Affairs for about a year; the aid came from the Buenos Aires Convention, through which the American government gave scholarships to Latin Americans.
On 21 September 1949, Peixoto was elected to the Brazilian Academy of Sciences. After the award of the scholarship, Peixoto and his wife travelled to Chicago in 1949. They both attended the International Congress of Mathematicians held in Cambridge Massachusetts from 30 August to 6 September 1950. Peixoto gave the presentation Note on uniform continuity in the Analysis Section on the morning of Wednesday 6 September. On 18 September 1950 they flew from New York to Rio de Janeiro but on 5 October 1950 Peixoto returned without his wife on a flight from Rio de Janeiro to New York on his way back to Chicago. In Chicago he undertook research aiming for an American Ph.D. but, before completing his thesis, decided to return to Brazil and apply for a professorship at the University of Brazil. He flew back from New York to Rio de Janeiro on 7 June 1951. He said [12]:-
I never regretted having dropped out of the doctorate, because a professorship at the University of Brazil appeared very infrequently, since it was for life; it was not an opportunity to be wasted.
He presented his thesis Equações gerais da dinâmica (1951), in the competition for the Chair of Rational Mechanics at the University of Brazil. He defended this thesis in 1952 and was appointed to the chair. Also in 1952 the Institute of Pure and Applied Mathematics (IMPA) was founded by Peixoto and his colleagues Leopoldo Nachbin and Lélio Gama.

In 1955, in the library of the IMPA, Peixoto came across a 1952 article by H F DeBaggis entitled Dynamical Systems with Stable Structure [19]:-
The encounter, apparently accidental, followed by the active browsing through it, had launched Peixoto's insightful mathematical intuition. At home, he worked on the article until late hours. Afterwards he also discussed it with his wife Marília C Peixoto, also an engineer, mathematician and his assistant in the Chair of Mechanics. They decided that, in due time, he should prepare a presentation on Structural Stability in the Seminar he directed, which had the participation of Marília, selected students and teaching assistants of the Chair of Mechanics.
Solomon Lefschetz wrote about DeBaggis (see [19]):-
Father Henry DeBaggis, an Assistant Professor of Mathematics at Notre Dame and a PhD. under Karl Menger at Notre Dame also, joined our Project in 1949 and was a member for two years 1949-51. While his thesis was on Hyperbolic Geometry, he had little taste for that subject and wished to change over to Differential Equations. A reading of an appendix in Andronov and Chaikin 1949 recently appeared, awoke his interest in structural stability.
Peixoto began developing DeBaggis's ideas and wrote to Lefschetz in 1956 explaining the project he had set himself. Lefschetz was immediately interested and wrote back to Peixoto suggesting he apply for a Research Associate visiting position at Princeton. With support from the Brazilian National Council for Scientific and Technological Development, he visited Princeton in 1957-58. Peixoto wrote [13]:-
Lefschetz offered me a table in his spacious office and we talked at length about mathematics, mathematicians and everything else under the sun. We became true friends as much as his eminence could allow, and also the fact that, at seventy-three, he was more than twice my age. Lefschetz was very fond of Pontryagin and told me that at the age of 14, Pontryagin had lost both his eyes in an accident with a stove. Lefschetz himself had been in an accident in 1910, aged twenty-six, and had both arms amputated just below the elbows. He then decided to become a mathematician. In fact, before the summer of 1958, Lefschetz had met Pontryagin only on a few occasions during a conference in Moscow in the early 1930s. They talked at length about mathematics and Lefschetz was tremendously impressed. He referred to the young Pontryagin of those days in hyperbolic terms: "Everything he touched turned to gold."
Peixoto, Lefschetz and Pontryagin were all at the International Congress of Mathematicians held in Edinburgh, Scotland, from 14 August to 21 August 1958. Peixoto delivered the short communication On structural stability. In a published version of his talk he writes:-
It is only fair to acknowledge here my indebtedness to Professor S Lefschetz, who followed closely the development of this work, and to Professor H Whitney for his kind advice. Both read a first draft of the manuscript and suggested many valuable improvements ...
Peixoto wrote about meeting Pontryagin at the Congress [13]:-
During the International Congress of Mathematicians in Edinburgh in 1958, Lefschetz took the initiative to arrange a meeting between me and Pontryagin and offered to serve as an English-Russian interpreter. The idea was that I would communicate to Pontryagin the content of 'On structural stability' that had not yet been published. From my point of view, the meeting was not very productive. Pontryagin was not interested in the matter ...
Articles that Peixoto published in 1959, 1962 and 1963 contain Peixoto's Theorem. In 1969 he was awarded the Moinho Santista Award by the Bunge Foundation for Peixoto's Theorem: the citation states:-
... the theorem of Peixoto on the structural stability in two-dimensional varieties inspired the mathematician S Smale to create the general theory of dynamic systems.
In [13] Peixoto writes about meeting Smale:-
Another mathematician who believed in structural stability, and who would become the central figure in the qualitative theory of ordinary differential equations, was Steve Smale. I was introduced to him by Elon Lima at Princeton in the summer of 1958 shortly after I returned from Edinburgh. Before that, some time ago, and through Lima himself, I was already aware that Smale had great promise. Later, in evocative notes, Smale wrote about our meeting at Princeton and describes his reaction upon learning of the results that were supposed to appear in my Annals article: "Through Lefschetz, Peixoto became interested in structural stability and showed me his own results on structural stability in the disk (in an article due to appear in the "Annals of Mathematics, 1959"). I was immediately excited, not only by what he was doing, but by the possibility that, using my experience in Topology, I could extend his work to the n th dimension."
In 1986 Peixoto received the TWAS prize from the Third World Science Academy. The following citation, which was read on the podium by Shiing-shen Chern at the opening ceremony, was engraved on the medal that was presented to Peixoto by the Pakistani theoretical physicist and a Nobel Prize laureate Abdus Salam, President of the Third World Science Academy:-
For his pioneering and fundamental study of the structural stability of dynamical systems, in particular, for proving that flows on surfaces are generally structurally stable.
In his acceptance speech, Peixoto said [13]:-
I would like to point out that this work on structural stability was basically carried out in several articles mentioned below, one of which was in collaboration with my first wife Marília [Structural Stability in the plane with enlarged boundary conditions (1959)], who did not live to see the end of this adventure. However, her influence in those golden days, decisive and already distant, was great.
Let us return to Peixoto's career. He continued to hold the professorship of Rational Mechanics at the University of Brazil until 1964 when he accepted an invitation by Lefschetz to a professorship at Brown University in Providence, USA. It may have been the coup by the armed forces in Brazil which happened in April 1964 and the following military dictatorship, that was a factor in his decision to leave the country and accept the professorship at Brown University. He continued to hold this position until 1970 when he returned to the Institute of Pure and Applied Mathematics in Rio de Janeiro. In 1971 he organised the first International Symposium on Dynamical Systems, bringing together world leading experts, held at the University of Bahia, Salvador, Brazil from 26 July to 14 August. Peixoto edited the conference proceedings and wrote the Preface, see THIS LINK.

In 1973 Peixoto was appointed as a professor at the Institute of Mathematics and Statistics at the University of São Paulo. He continued in this post until 1978 after which he returned to the Institute of Pure and Applied Mathematics in Rio de Janeiro. In 1974 he gave an invited 30-minute lecture at the International Congress of Mathematicians held in Vancouver, Canada, from 21 August to 29 August. His lecture with title On Bifurcations of Dynamical Systems begins:-
We survey here a certain line of developments of bifurcation theory of dynamical systems on a compact differentiable manifold $M^{n}$. The word bifurcation was introduced by Poincaré in connection with differential equations of celestial mechanics depending on a real parameter $\lambda$. He focused on a certain feature of the equation, say an isolated closed orbit, present for $\lambda = \lambda_{0}$. If when $\lambda$ varies past $\lambda_{0}$ this closed orbit disappears or undergoes a sudden topological change, $\lambda_{0}$ is said to be a bifurcation value; otherwise it is an ordinary value of the parameter. Traditionally bifurcation theory has been developed from this local point of view. After the emergence of the generic theory of dynamical systems via the coming of age of the concept of structural stability it became natural to take a more general and global approach to bifurcation theory ...
He served as President of the Brazilian Mathematical Society (1975-1977), President of the National Research Council (1979-1980), and President of the Brazilian Academy of Sciences (1981-1991).

Peixoto's first wife Marília died in 1961 and he married twice more, to the mathematician Maria Lúcia Alvarenga and to the mathematician Alciléa Augusto.

In 2010 Peixoto was 89 years old. In that year Marcelo Viana, the director of the Institute of Pure and Applied Mathematics, wanted to select Peixoto's most important articles and publish them in a book. Peixoto suggested that Viana wait for a few years since he felt the work he was doing at the time should be included. In fact Peixoto published four papers in 2011 and his final paper was published in 2014. He died in Rio de Janeiro at the age of 98.

### References (show)

1. Aos 98 Anos, Morre Maurício Peixoto, Fundador do Impa, Brazilian Academy of Sciences (29 April 2019).
2. CV - Maurício Peixoto, Universidade do Minho, Braga, Portugal.
https://w3.math.uminho.pt/~web/dynappl2008/public_html/abstracts/Mauricio_Peixoto_CV.pdf
3. C Fioravanti, Maurício Peixoto foi um dos criadores do Impa, Pesquisa FAPESP (June 2019).
https://revistapesquisa.fapesp.br/dedicacao-plena-a-matematica/
4. D F Freire and A J M de Queiroz, As Contribuições Demaurício Peixoto Para A Matemática Brasileira, Boletim Cearense de Educação e História da Matemática 7 (20) (2020), 57-66.
5. Maurício Matos Peixoto, Brazilian Academy of Sciences.
https://www.abc.org.br/membro/mauricio-matos-peixoto/
6. Maurício Peixoto, founder of IMPA, dies at 98, Instituto de Matematica Pura e Aplicada.
7. L F Mello, Estruturalmente Estável (April 2019).
https://www.researchgate.net/publication/332767792_Estruturalmente_Estavel?
8. Morre o matemático Maurício Peixoto, um dos fundadores do IMPA, GALILEU (28 April 2019).
9. Morre no Rio o matemático Maurício Matos Peixoto, G1 Rio (28 April 2019).
https://g1.globo.com/rj/rio-de-janeiro/noticia/2019/04/28/morre-no-rio-o-matematico-mauricio-matos-peixoto.ghtml
10. Morre o matemático Maurício Peixoto, um dos fundadores do Impa, BEM PARANÁ (28 April 2019).
11. M M Peixoto, A A Pinto and D A Rand (eds.), Dynamics, Games and Science I: DYNA 2008, in Honor of Maurício Peixoto and David Rand, University of Minho, Braga, Portugal, September 8-12
12. M M Peixoto, Maurício Matos Peixoto, in J Palis, C Camacho and E L Lima, (eds.), IMPA 50 anos (IMPA, Rio de Janeiro, 2003), 240-250.
13. M M Peixoto, Acceptance speech, TWAS Award 1987, in Proceedings of the Second Conference Organised by the TWAS (World Scientific, Singapore), 1989), 600-614.
14. Peixoto, Mauricio Matos, TWAS Directory.
https://twas.org/directory/peixoto-mauricio-matos
15. C M da Silva, Politécnicos ou matemáticos?, História, Ciências, Saúde - Manguinhos, Rio de Janeiro 13 (4) (2006), 891-908.
16. C M da Silva, Two-point boundary value problems: some recent contributions of M M Peixoto, Computational and Applied Mathematics 20 (1-2), 2001), 245-256.
17. J Sotomayor, On Maurício Matos Peixoto and his Mathematical Work (July 2019).
https://www.researchgate.net/publication/335128763_On_Mauricio_Matos_Peixoto_and_his_Mathematical_Work
18. J Sotomayor, Introduction: A few words about Mauricio Peixoto on his 80th birthday, Computational & Applied Mathematics 20 (1-2) (2001), 3-9.
https://www.ime.usp.br/~sotp/peixoto.pdf
19. J Sotomayor, On Maurício M Peixoto and the arrival of Structural Stability to Rio de Janeiro, 1955, Annals of the Brazilian Academy of Sciences 92 (1) (2020), 1-6.
20. J Sotomayor, On Maurício Matos Peixoto and his Mathematical Work on Differential Equations, 1955 - 1965, Revised Version, Instituto de Matemática e Estatística Universidade de São Paulo (29 July 2020), 1-25.
https://www.researchgate.net/publication/343303903_On_Mauricio_Matos_Peixoto_and_his_Mathematical_Work_on_Differential_Equations_1955_-1965_REVISED_VERSION/citations
21. J Sotomayor, On a meeting of Maurício M Peixoto with José Tola Pasquel in Lima, November 1961 (1961).
https://www.researchgate.net/publication/335798384_ON_A_MEETING_OF_MAURICIO_M_PEIXOTO_WITH_JOSE_TOLA_PASQUEL_IN_LIMA_NOVEMBER_1961
22. J Sotomayor, A Tribute to Mauricio M Peixoto on the occasion of his 90th birthday (June 2018).
https://www.researchgate.net/publication/325934473_A_Tribute_to_Mauricio_M_Peixoto_on_the_occasion_of_his_90th_birthday
23. J Sotomayor, R Garcia and L F Mello, Maurício Matos Peixoto, Revista Matemática Universitária 1 (2020), 1-22.
24. J Sotomayor, Baggis Peixoto (August 2019).
https://www.researchgate.net/publication/335465373_BAGGIS_PEIXOTO
25. J Sotomayor, On an encounter of two men of mathematics in Lima (October 2020).
https://www.researchgate.net/publication/346455504_On_an_encounter_of_two_men_of_mathematics_in_Lima
26. J Sotomayor, A mathematics lesson (August 2019).
https://www.researchgate.net/publication/335465271_A_MATHEMATICS_LESSON