# Marília de Magalhaes Chaves Peixoto

### Quick Info

Born
24 February 1921
Santana do Livramento, Brazil
Died
5 January 1961
Rio de Janeiro, Brazil

Summary
Marília Chaves Peixoto was the first Brazilian woman to be awarded a Ph.D. in mathematics. In 1951 she was the first Brazilian woman to be elected a member of the Brazilian Academy of Sciences. She made substantial contributions to what is now known as Peixoto's Theorem on the structural stability of dynamical systems.

### Biography

Marília Chaves Peixoto was the daughter of Tullio de Saboia Chaves (1884-1971), Professor at the Surgical Medical School at Rio de Janeiro, and Zillah da Costa Magalhaes (1899-2001). Tullio de Saboia Chaves had studied medicine at the University of Geneva in Switzerland returning to Brazil in 1918. Tullio Chaves and Zillah Magalhaes were married on 31 May 1920, in Rio de Janeiro. They had three children, Marilia de Magalhaes Chaves, the subject of this biography, Lúcia de Magalhaes Chaves (18 May 1924 - 25 February 1980), and Livio de Magalhaes Chaves (1926-2016).

Manchete was a Brazilian weekly news magazine published from 1952 and in its first year, in the fifth issue on 24 May 1952, it published an interview with Marília Chaves Peixoto. She spoke of her upbringing in this interview, saying (see for example [16]):-
I was lucky, my parents encouraged me. So much so that I went to a high school in Santana do Livramento, where girls couldn't study. By the kindness of the priests, I attended secondary school as a private student and was able to take the tests with the boys. Then the family helped me to study what interested me.
Later she went to Rio de Janeiro and spent at least one year at the Colégio Andrews since the Brazilian newspaper in 1939 lists her as graduating as an outstanding student from the College. The Colégio Andrews had been founded by Isabella Robinson Andrews in 1918 and she was still running the school when Marília Chaves studied there. It offered a secular education to both boys and girls, which was unusual in Brazil at this time. When Marília Chaves studied there the college had around 1500 pupils in Primary, Secondary and "Superior" courses, which prepared students, both boys and girls, to enter the schools of Medicine, Law and Engineering. Marília was by this time intent on studying mathematics and was taking the "Superior" course at the Colégio Andrews preparing to enter the National School of Engineering at the University of Brazil in Rio de Janeiro. In fact Marília qualified for entry in 1939 to the National School of Engineering and achieved third place among the 73 students approved for entry. This was unusual in Brazil at this time [16]:-
Women were not absent in the building of the scientific field of mathematics in Brazil, but only a small number of them had access to specific training. As Brazilians only had access to mathematics through the Polytechnic Schools, and this was traditionally a male stronghold, it was only with the emergence of faculties of philosophy, in the 1930s, that women really began to occupy lecture theatres. The presence of women in engineering schools was not very common at the beginning of the century, but in 1939 Marília de Magalhaes Chaves was enrolled at the National School of Engineering ...
At the National School of Engineering, Maurício Matos Peixoto and Leopoldo Nachbin were both her fellow students, all three beginning their studies in the same year. Maurício Peixoto said in the interview [14]:-
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 Marília Chaves graduated with a civil engineering degree but she never intended to have a career as an engineer. Mathematics was always the subject for her and throughout her engineering course Marília Chaves, Maurício Peixoto and Leopoldo Nachbin all studied advanced mathematics. They attended lectures at the National Faculty of Philosophy at the University of Brazil given 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.

On 6 September 1946 Marília Chaves married the mathematician Maurício Matos Peixoto in Rio de Janeiro. Maurício and Marília had two children, Marta Chaves Peixoto (born 31 March 1949) and Ricardo Chaves Peixoto (born 24 March 1953).

In 1949, Marília Peixoto took part in the competition to become a professor of Infinitesimal Calculus at the National School of Engineering. She submitted the thesis On the inequalities $y''' ≥ G(x, y, y', y'')$ for the competition for the post for which she was the only candidate. Her thesis, published as the paper [4], was reviewed by Edwin Beckenbach in [2]. The description in [2] is technical, but a much less technical description in given in [1]:-
Her competition thesis for Licentiate Teaching constitutes an original research contribution on third-order differential inequalities. In this work, she managed to extend to the case of third-order inequalities theorems that had been demonstrated in the case of second-order differential inequalities by her husband, Maurício Matos Peixoto.
This thesis was accepted for a doctorate in mathematics making Marília Peixoto the first Brazilian woman to be awarded a Ph.D. in mathematics.

Marília Peixoto was appointed to the professorship at the National School of Engineering where she taught differential and integral calculus. She taught courses on other topics as well, for example she taught a Modern Algebra course in 1952 at the Centro Brasileiro de Pesquisas Físicas. She also held seminars on topological groups at this Brazilian Centre for Physical Research and in 1951 delivered the course 'Introduction to General Topology'.

Marília Peixoto and her husband travelled to Chicago in 1949. They both attended the International Congress of Mathematicians held in Cambridge Massachusetts from 30 August to 6 September 1950. Perhaps due to an error by the Congress organisers, Marília's name only appears as her husband's partner and not as a participant in her own right. On 18 September 1950 they both flew from New York to Rio de Janeiro but on 5 October 1950 Maurício Peixoto returned without his wife on a flight from Rio de Janeiro to New York on his way back to Chicago.

On 1 July 1951 Marília Peixoto was elected to the Brazilian Academy of Sciences. The president of the Academy made a speech in which he said [1]:-
Marília Chaves Peixoto has, among other merits, that of being the first woman to cross the austere thresholds of the Brazilian Academy of Sciences.
Marília Peixoto was elected as an associate member of the Brazilian Academy of Sciences becoming the first Brazilian woman to be elected to the Academy. In fact Marie Curie had been elected as a foreign associate member of the Brazilian Academy of Sciences in 1926.

In 1955 she published the 130-page book Cálculo vetorial . This book is aimed at students of engineering and certainly does not prove all the results stated but is rich in applications of vector calculus. In the book, she notes that there are several different ways to approach vector calculus, the geometric point of view, the point of view of abstract algebra, or that of invariants of transformations. There are many exercises, for example: "An observer is on a train that moves horizontally with speed $v$. Rain falls vertically with speed $u$. At what angle to the horizontal does the observer see the rain fall?" Three years after Marília Peixoto died, the book was republished with a Preface by Maurício Peixoto. He writes in the Preface:-
Clear, methodical, objective and well-prepared, it is a good reflection of her didactic qualities and her exemplary dedication to teaching.
In 1955, in the library of the Institute of Pure and Applied Mathematics, Maurício Peixoto came across a 1952 article by H F DeBaggis entitled Dynamical Systems with Stable Structure [18]:-
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.
We see that Marília Peixoto and her husband worked together of the structural stability of dynamical systems and she would make a substantial contribution to the result that today is known as Peixoto's Theorem, which gives the characterisation of structurally stable systems in two-dimensional manifolds. Solomon Lefschetz was very interested in their work and Maurício and Marília Peixoto, with support from the Brazilian National Council for Scientific and Technological Development, visited Princeton in 1957-58. Renate Watanabe and her husband, the Brazilian physicist Shigueo Watanabe, were also visiting Princeton and Renate writes about becoming friends with Marília Peixoto while there (see, for example, [16]):-
I chose to attend a mathematics course at Princeton which was at a suitable time, that is, at a time when Shigueo could be with the children (Ernesto was a new born). I went to talk to Professor Donald Spencer to ask what were the prerequisites needed to follow his course. His answer was: none. I went to the first class. Marília and Maurício were there, but I didn't know them yet. In less than 10 minutes I realised that I didn't know what he was talking about. Later, ... Ricardo [Marília Peixoto's son] and Shigueo Jr met in the same kindergarten class and that's how Maurício, Marília, Shigueo and I got to know each other. Instead of belittling me for my ignorance of mathematics, Marília started teaching me. I remember that she taught me, among other things, the first notions about sets. It was thanks to this initial impulse that I started to attend some courses and it was thanks to Marília that I did well in the initial course, here in São Paulo, which gave rise to GEEM [the Mathematics Teaching Study Group].
In 1959 Marília Peixoto and her husband published the joint paper Structural stability in the plane with enlarged boundary conditions. The article was sent for publication on 16 September 1958 and was published on 30 June 1959. See reference [12] for the paper and [10] for a review of the paper. This was one of four papers on the structural stability of dynamical systems, the other three being by Maurício Peixoto, which produced what today is known as Peixoto's Theorem. Maurício Peixoto said in 1986 [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 was great in those golden days, decisive and already distant days, in the autumn of 1957 at Princeton.
After this address by Maurício Peixoto, Laurent Schwartz said (see [16]):-
Since you speak about your first wife Marília, let me add that I knew her at the same time as I got to know you when I came to Rio de Janeiro in 1952; and already at this time it was very well known that you were both very good mathematicians.
Marília Peixoto died on 5 January 1961. Later that year Maria Laura Mozinho gave a lecture at Brazilian Academy of Sciences in which she said (see [17]):-
Seeing that calm girl, with very large and expressive eyes, it was necessary for someone to whisper to us that she was an outstanding Professor of Calculus and Mechanics at the National School of Engineering, energetic and, at the same time, generous, in addition to having an acute intelligence that turned to mathematical research.
A number of honours have been given to Marília Peixoto after her death. A street in her home town of Sant'Ana do Livramento is named Rua Marília Chaves Peixoto. It is a predominately residential street in the Prado district of the town. Marília's husband Maurício Peixoto founded the school 'Municipal School Marília Chaves Peixoto' in the Controes region of Petrópolis, in 1971. In 1969 he had been awarded the Moinho Santista Award by the Bunge Foundation for Peixoto's Theorem and he used the money from the award to fund the school. Alciléa Augusto, Maurício's third wife, explained (see for example [16]):-
Back in 1945, Maurício, Marília's fiancé at the time, got to know the Controes region, in the confines of the municipality of Petrópolis, a rural region, where Marília's father, Dr Túlio Chaves, had a farm. ... Maurício realised that many of the local residents were illiterate, and that the nearest school was about 10 km away. ... On land donated by Dr Túlio, the Municipal School Marília Chaves Peixoto was inaugurated in 1971. ... For almost 40 years, in this single building, with kitchen and cafeteria, more than 300 students attended primary school. ... Maurício bought a bicycle for one teacher, paid for petrol for another. ... and provided lunch when the city failed to do so. ... In Controes, there are no more illiterate people!
The Brazilian politician and university professor Dorinha Seabra Rezende introduced in the Chamber of Deputies a Resolution to give the name Marília Chaves Peixoto to Plenary 13 of Annex II of the Chamber of Deputies in 2020. Here is part of the Resolution [15]:-
.. in addition to the unquestionable value of the research she developed, Marília Peixoto bravely broke through a virulent and ingrained discourse that dislodged women from dedicating themselves to certain areas of knowledge and certain work activities, a discourse normally based on the idea of existence of an unequivocal relationship between biological attributes of gender, in which the female is limited to the exercise of domestic functions and a very small number of professions, usually with lower social prestige and remuneration. By becoming a leading professional in an area considered, until then, as exclusively male, mathematician Marília Peixoto opens a precedent of dignity for all Brazilian women, who still remain in the fight for equality in the field of formal study and work.

### References (show)

1. Academia Brasileira de Ciências, Jornal do Comércio, Rio de Janeiro (1 July 1951), 7- 8.
2. E F Beckenbach, Review: On the inequalities y''' ≥ G(x, y, y', y''), by M Chaves Peizoto, Mathematical Reviews MR0031969 (11,235a).
3. M F Cavalari, Marília Chaves Peixoto (1921-1961), Mulheres Pioneiras na Matemática no Brasil, Revista do Professor de Matemática.
https://rpm.org.br/cdrpm/80/2.html
4. M Chaves Peizoto, On the inequalities y''' ≥ G(x, y, y', y''), Annals of the Brazilian Academy of Sciences 21 (1949), 205-218.
5. D F Freire and A J M de Queiroz, As Contribuiçoes Demaurício Peixoto Para A Matemática Brasileira, Boletim Cearense de Educação e História da Matemática 7 (20) (2020), 57-66.
6. Marília Chaves Peixoto, pioneira da ciência no Brasil, Vermelho (4 January 2016).
https://vermelho.org.br/2016/01/04/marilia-chaves-peixoto-pioneira-da-ciencia-no-brasil/
7. Marília Chaves Peixoto, Brazilian Academy of Sciences.
https://www.abc.org.br/membro/marilia-chaves-peixoto/
8. Marília Chaves Peixoto - Cientista Brasileira - 100 Anos, Classe News (16 May 2021).
https://classenews.com/noticia/970195/marilia-chaves-peixoto-cientista-brasileira-100-anos
9. K Kishi, Divulga Cientista - Marília Chaves Peixoto, primeira doutora em matemática no Brasil, www.blogs.unicamp.br.
https://www.blogs.unicamp.br/cienciaemrevista/2015/08/17/divulga-cientista-marilia-chaves-peixoto-primeira-doutora-em-matematica-no-brasil/
10. L Marcus, Review: Structural stability in the plane with enlarged boundary conditions, by M C Peixoto and M M Peixoto, Mathematical Reviews MR0107067 (21 #5794).
11. M S A Nunes, A desigualdade de gênero na matemática: aspectos históricos e atuais (Joao Pessoa, 2021).
https://repositorio.ufpb.br/jspui/bitstream/123456789/20616/1/MSAN06082021.pdf
12. M C Peixoto and M M Peixoto, Structural stability in the plane with enlarged boundary conditions, Annals of the Brazilian Academy of Sciences 31 (1959), 135-160.
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. 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.
15. D S Rezende, Draft Resolution to give the name Marília Chaves Peixoto to Plenary 13 of Annex II of the Chamber of Deputies (2020).
16. M Silva da Silva, Marília Chaves Peixoto - Uma Matemática Brasileira à Sombra, XIII Seminário Nacional de História da Matemática (14-17 April 2019) 150-171.
17. C M da Solva, Politécnicos ou matemáticos?, História, Ciências, Saúde - Manguinhos 13 (4) (2006), 1891-1908.
18. 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.