# Welington Celso de Melo

### Quick Info

Born
17 November 1946
Guapé, Minas Gerais, Brazil
Died
21 December 2016
Rio de Janeiro, Brazil

Summary
Welington de Melo was an outstanding Brazilian mathematician working on dynamical systems. He is perhaps best known as the Ph.D. supervisor of the 2014 Fields Medallist Artur Avila.

### Biography

Welington de Melo was the son of Moacyr Hostalácio de Melo and Maria José Silva de Melo. Although Welington was born in Guapé, a town in the Minas Gerais state in south east Brazil, he never lived there. His father Moacyr was a Collector of the Federal Revenue and later a Tax Auditor of the Federal Revenue. This involved several moves during the years that Welington was growing up and, as a consequence, he attended schools in different towns. First he was at school in Três Corações, Minas Gerais, a larger town than Guapé about 160 km south east of it. He moved from there to the city of Anápolis in the State of Goiás. Another move saw him attend the Colégio Diocezano in Maceió, the largest city of the coastal state of Alagoas. This college, founded in 1902 and run by the Marist Brothers, had around 600 pupils when Welington attended. His next school was the Colégio Estadual in Belo Horizonte, a large city in the Minas Gerais state.

Before completing his secondary school education, Welington began working for the Banco da Lavoura while continuing his final two years of school education studying at night school. The Banco da Lavoura was founded in Belo Horizonte in 1925 and, although it still had its headquarters in that city when Welington was employed there, it had by that time expanded into having branches in most big Brazilian towns. At this stage Welington was intending to have a career as a businessman and he bought a newspaper stand.

In 1965, at the age of nineteen, he decided to give up the idea of a career in business, resigned his position at the Banco da Lavoura, sold his the newspaper stand, and entered the School of Engineering at the Federal University of Minas Gerais. He certainly did not envision a career as a mathematician at this time but was intent on studying electrical engineering. Of course he studied mathematics as part of the engineering course and assisted the Chair of Infinitesimal and Integral Calculus where he organised seminars with professors to study topics of Advanced Calculus. As he approached the last years of the Electrical Engineering course he decided that he wanted an academic career and thought that he would apply to do postgraduate studies in physics. The Seventh Brazilian Mathematics Colloquium was due to be held in Poços de Caldas, Minas Gerais in July of 1969 and, because of his excellent performance in his undergraduate studies, de Melo was awarded a scholarship to attend the Colloquium. One of the four Scientific Initiation Courses at the Colloquium, Análise Geométrica, was being given by Elon Lages Lima, the director of the Instituto de Matemática Pura e Aplicada (IMPA). De Melo attended the course and impressed Lima who invited him to do a Master's Degree in Mathematics at IMPA. He made a quick decision to undertake postgraduate work in mathematics rather than physics.

De Melo graduated with a degree in electrical engineering from the Federal University of Minas Gerais in 1969 and, in the same year, he married Gilza. She had been born in Pedra Azul, in the Jequitinhonha Valley region of the Minas Gerais state, and had studied mathematics at the Federal University of Minas Gerais. She would later work as a systems analyst. In January 1970 de Melo and his wife moved to Rio de Janeiro and, one Saturday morning, he knocked on the door of Jacob Palis in the IMPA. Palis had begun running Dynamical Systems Seminars at IMPA after he returned from California with a Ph.D. in 1969 and they were held every morning except Sunday. De Melo [12]:-
... informed Palis that he had attended Elon Lima's course at the recent Brazilian Mathematical Colloquium in Poços de Caldas. He added that he would very much like to study Dynamical Systems to obtain a Ph.D. in this area.
Palis approached the lecturers at the Federal University of Minas Gerais, asking their opinion of de Melo's abilities [12]:-
... he was informed that de Melo was an excellent student and very determined to learn new relevant topics in mathematics. With all the positive reactions Palis had received, he asked Welington to present one of the important Seminar topics. It was a success! With that, Jacob formalised his request to the Director of IMPA to accept de Melo as a Ph.D. student, under his guidance. A couple of weeks after that, Welington came to Jacob's office to say that he had discovered a serious gap in one of his mathematical papers. Happily, it turned out it was not a serious gap.
De Melo was awarded a CAPES scholarship to fund his Ph.D. studies. CAPES is a Foundation within the Ministry of Education in Brazil whose central purpose is to coordinate efforts to improve the quality of Brazil's faculty and staff in higher education through grant programs. CAPES is particularly concerned with the training of doctoral candidates. De Melo was an outstanding student and was an invited speaker at the first International Symposium on Dynamical Systems, held at the University of Bahia, Salvador, Brazil from 26 July to 14 August 1971 and organised by Elon Lima, Maurício Peixoto and Jacob Palis. De Melo gave the talk Structural Stability on Two-Manifolds. Before the Symposium took place, however, several of the foreign mathematicians who were to speak at it, including Stephen Smale, spent time as visiting researchers at the IMPA in Rio de Janeiro. De Melo said (see [5]):-
What I learned about mathematics without taking any courses, simply talking to people in the corridors, was great.
He was awarded his doctorate for his thesis Estabilidade Estrutural em Variedades de Dimensao Dois in 1972. He published results from his thesis in the 21-page paper Structural Stability of Diffeomorphisms on Two-Manifolds which appeared in the leading mathematical journal Inventiones Mathematicae in 1973. In this paper de Melo writes:-
This paper is based in my doctoral thesis at IMPA under the guidance of J Palis. We are grateful to E Lima, P Mendes, S Newhouse, G L dos Reis, C Robinson, R Williams, and especially to J Palis for many helpful conversations.
An award from the National Research Council of Brazil allowed de Melo to spend from September 1972 to March 1974 undertaking post-doctoral work with Stephen Smale at the University of California, Berkeley. The authors of [6] write:-
Welington was very much influenced by Steve Smale who in the early 1970s was quite interested in economics. Inspired by Steve's work, Welington studied the problem of simultaneously optimising several functions. Forever the pure mathematician, he wrote that this 'is a natural question to consider, even in some basic models in economics'.
Leaving Berkeley, he went to the University of Warwick in England, being there from March 1974 to July 1974. This was to participate in the symposium 'Applications of Topology and Dynamical Systems' which ran 1973-74 and was organised by Chris Zeeman. De Melo was one of several IMPA mathematicians taking part in the symposium, others included Jacob Palis and Ricardo Mañé. De Melo gave the seminar Optimisation of several functions, describing work he had done with Smale at Berkeley. In January 1974, de Melo had been appointed as an Assistant Professor at IMPA and after the University of Warwick symposium he returned to Rio de Janeiro. He was rapidly promoted, becoming an Associate Professor in August 1975 and a Full Professor in September 1980.

A colleague at IMPA encouraged him to take up sailing, and de Melo liked the idea. The colleague had a small sailing boat and a house at Angra dos Reis, on the coast about 150 km east of the city of Rio de Janeiro. De Melo managed to arrange a base there and, in 1979, he bought his own boat [5]:-
De Melo is a man of systematic habits. When he is in Angra, he wakes up early, works in the morning and goes sailing in the afternoon. Although he doesn't take paper and pencil to do mathematics on board, the activity helps to sharpen his reasoning. "It is the time when I organise my ideas," he explains.
In the 1980s Steve Smale bought the 43-foot sailing boat Stardust and decided to take a cruise to the Marquesas Islands in Polynesia. Smale and de Melo were good friends and, knowing de Melo had experience sailing, Smale asked if he would join him. Charles Chapman Pugh (born 1940), a colleague of Smale's at Berkeley who worked on dynamical systems, also joined the sailing trip which involved a three-month cruise on the Stardust from Berkeley to the Marquesas Islands in Polynesia then returning to Berkeley via Hawaii. This took place in 1987 while de Melo had a visiting position at Berkeley.

It was also in 1987 that Melo's paper A finiteness problem for one dimensional maps appeared in the Proceedings of the American Mathematical Society. Gong Fu Liao writes in a review:-
The author discusses the connection between the density of structurally stable maps in the space of unimodal maps of the interval, the finiteness of attractors and the nonexistence of wandering intervals. It is proved that in the space of unimodal maps having an eventually periodic flat critical point, there is a residual subset whose maps have infinitely many sinks and there are also maps having a wandering interval.
De Melo himself writes in [24] about this paper saying that it:-
... is my first contribution to my main field of research: one-dimensional dynamics. In it I discuss the problem of existence of wandering intervals and finiteness of attractors and relate these problems to the density of structurally stable interval endomorphisms.
The most important research contribution made de Melo is, according to his own assessment, the paper Julia-Fatou-Sullivan Theory for real one-dimensional dynamics published in Acta Matematica in 1992 which he wrote jointly with Sebastian van Strien and Strien's Ph.D. student Marco Marten. Referring to this paper as [*], de Melo writes [24]:-
[*] is my most important mathematical contribution: a complete description is given of the topological behaviour of 1-dimensional real dynamical systems. It turns out that the Julia-Fatou-Sullivan theory for conformal mappings of the Riemann sphere is also valid for smooth endomorphisms of the interval (or the circle).

Julia gave the following description of the topological structure of 1-dimensional complex dynamics: every rational map of the Riemann sphere has periodic domains attracting an open set of points, outside this open set there is a closed invariant set, called the Julia set, and maybe some open sets wandering around and being attracted to this Julia set.

The main problems left over were
1) Can a map have infinitely many periodic domains?

2) Do these wandering domains exist?
Fatou showed that there can be only finitely many periodic domains and Sullivan showed that these wandering domains do not exist.

The first result about the topological structure of 1-dimensional real dynamics was obtained by Denjoy and Poincaré. Poincaré gave a description of the topological structure of the dynamics of circle diffeomorphisms. As in the theory of complex dynamics the existence of wandering domains played an important role. Denjoy showed that they cannot exist for $C^{1+b\nu}$ diffeomorphisms of the circle. The topological work of Poincaré could not be generalised to endomorphisms of the interval (or circle). The topological description of the dynamics of interval dynamics was done by Milnor and Thurston. The description was exactly like the one given by Julia and again the only thing left over was to answer the questions above.

A sequence of authors achieved partial results on the non-existence of wandering intervals (Denjoy, Guckenheimer, van Strien and de Melo, Lyubich and Blokh). The general non-existence of wandering intervals was proved in [*]. Furthermore [*] also gives the finiteness of periodic attractors. As a corollary of the theory developed in [*] we got the classical Denjoy-Schwarz theorem under much milder smoothness conditions.
Since sailing played such a large part in de Melo's life, let us say a little more about it. In 1991, he purchased the Doisdu, a sailing boat about 10 metres long and with capacity for eight people. Since 2006 the boat was moored beside a pier attached to his house in a condominium in Angra. We have already explained that sailing and mathematics were strongly connected in his life and it is interesting how many leading mathematicians sailed with him at various times [5]:-
Whenever he takes the sailboat out, de Melo likes to set himself a little mathematical constraint. "Each time I try to make a path that is homotopically non-trivial," he explained. Faced with the stunned silence of the interlocutor, he tried to expand: it is equivalent to circumventing at least one island. He was proud to have already sailed those waters in the company of five Fields medallists: René Thom (1958), Stephen Smale (1966), Sergei Novikov (1970), Jean-Christophe Yoccoz (1994) and Stanislav Smirnov (2010). Artur Avila - now sixth on the list - has a photo on his IMPA page in his swimming trunks aboard the Doisdu, but he is not exactly a sailor. "He is very much from the city," lamented his advisor de Melo.
The reference to de Melo's Ph.D. student Artur Avila, who won the Fields Medal in 2014, requires some further details. In 1995, when he was sixteen years old, Artur Avila received a scholarship to start studying for a Master's Degree at the IMPA in Rio de Janeiro while still attending high school in Rio de Janeiro. For the M.S. degree Avila was advised by Elon Lages Lima but soon came into contact with de Melo. He said (in 2017) [1]:-
Remembering Welington is remembering someone who was fundamental for our institution [IMPA] and for many people. He showed, in addition to exceptional mathematical talent, an enormous commitment to IMPA. He wanted IMPA to succeed. His contribution was not just mathematics. ... He was an extremely rigorous mathematician, committed to the highest quality of the work carried out at IMPA. He wanted the work to be at the highest possible level, but, because of this rigour, his reputation scared students a little. I found it a bit risky to enrol, I think in 1997, in a course taught by Welington at IMPA. With great caution, I did not enrol in the course, but went to attend classes. He turned out to be, unlike his reputation, very nice to me. At the end of the course, he said that I had done well. ... He asked me to go on his boat in Angra. He provided the food, the beer. But the students had the task of getting the barnacles out of the hull. He even provided the appropriate instruments.
Avila asked de Melo if he would be his Ph.D. advisor and he began his studies in 1998. A conference on dynamical systems was held at IMPA from Monday 11 May to Friday 15 May 1998. De Melo invited Mikhail Lyubich, a Ukrainian mathematician who was co-director of the Institute for Mathematical Sciences at Stony Brook University, New York, to the conference. At this conference it was arranged that early in 1999 de Melo and Avila would travel to Stony Brook and discuss a Ph.D. topic for Avila with Lyubich. Before this, however, de Melo went to the International Congress of Mathematicians in Berlin where he was an invited speaker. He gave the lecture Rigidity and Renormalization in One Dimensional Dynamical Systems. His lecture has the following Abstract:-
If two smooth unimodal maps or real analytic critical circle maps have the same bounded combinatorial type then there exists a $C^{1+\alpha}$ diffeomorphism conjugating the two maps along the corresponding critical orbits for some α > 0. The proof is based on a detailed understanding of the orbit structure of an infinite dimensional dynamical system: the renormalisation operator.
Avila and de Melo travelled on United Airlines to JFK Airport in February 1999, then they went on to Long Island [17]:-
Just before welcoming the Brazilians to Stony Brook, Lyubich had written a series of articles in which he proved his most important results. "Very few people really understood what it was about", he commented, "and Welington was a notable exception. It was his proposal that Artur explore this line of research." ... The three spent a month tossing ideas back and forth, in a style of doing mathematics that only requires a blackboard, chalk, and space to walk back and forth. The daily conversations took place in the Institute's rooms, at Lyubich's house, in restaurants or during walks through the woods around the university. ... "Occasionally, amazed, I realised that Lyubich and I were a little behind Artur," recalled de Melo, "he was so young. .. I would forget about it but then be scared." ... After a month of intense discussions, the trio had a clear strategy for resolving the problem that consumed them, but the proof was still out of reach. There was an obstacle that refused to budge. Lyubich and de Melo decided to leave it in the boy's hands. "That was in March", remembers Artur. "I kept the problem in my head and a few months later, in September or October, I had a weird idea."
Avila was awarded his Ph.D. by IMPA in 2001 for his thesis Bifurcations of unimodal maps: the topologic and metric picture. He went on to win a Fields Medal in 2014 and perhaps, despite all his other achievements, de Melo will be best known as the Ph.D. supervisor of a Fields Medal winner.

Welington de Melo published seven books, each co-authored with one of the three mathematicians Jacob Palis, Sebastian van Strien and Edson de Faria. You can see more information about these books at THIS LINK.

Many honours were given to de Melo including full membership of the Brazilian Academy of Sciences in 1991, the award of the National Order of Scientific Merit in 1996, the Gran Cruz do Mérito Científico of the Ministry of Science and Technology in 2002, The World Academy of Sciences Award in Mathematics in 2003, and membership of The World Academy of Sciences in 2005.

During his career, de Melo made many research visits. For example in 1977 he visited France spending time at the University of Lille and at the Institut des Hautes Études Scientifique. In 1982 he was in both Chile and France, then in the following year he went to the International Centre for Theoretical Physics in Trieste, Italy. He continued regular visits abroad, mostly to either the United States of France but a few to Holland or Belgium. He was an invited speaker at many conferences around the world, for example 'Dynamical Systems and Turbulence' at the University of Warwick in 1980, the London Mathematical Society Symposium in Durham, England in 1988, a 'Workshop on Dynamical Systems' in Portugal in 1992, an 'International Conference on Dynamical Systems and Chaos' in Tokyo, Japan, in 1994, a conference on dynamics in Hong Kong in 1996 and the conference 'Contemporary Problems in Dynamical Systems' in Nizhny Novgorod, Russia in the same year. In fact around 40 conference lectures up to 2009 are listed in [24], where details of the conference, its location and the title of the lecture given by de Melo are given.

Writing about de Melo's life outside mathematics, Edson de Faria and Sebastian van Strien explain in [6]:-
Welington met his wife Gilza while he was still quite young. They were inseparable, truly adored each other and even felt comfortable squabbling in front of others. No doubt, Gilza was instrumental in Welington's success. Usually Welington and Gilza lived relatively modestly. However, occasionally Welington would love to eat out in famous restaurants and talked about these memorable meals for years after. Welington loved to be near the sea and in particular to sail and the sense of adventure and freedom.
In early December 2016, shortly after celebrating his 70th birthday, de Melo suffered a heart attack and was taken to hospital. He died around two weeks later, while still in hospital, of complications arising from his heart condition. Only a short time before, 14-18 November at IMPA, there had been a conference 'New Trends in One-dimensional Dynamics' to celebrate his 70th birthday. The editors of [10] write:-
The occasion was particularly gratifying for us because of the active participation of a number of experts in this field, many of Welington de Melo's co-authors and all his former doctoral students. Collecting the articles for this volume was initially intended as an opportunity to celebrate the success of the meeting and Welington de Melo's joy at the excellent scientific level of the lectures and the friendly atmosphere that week. Unfortunately, Welington de Melo passed away a month after the meeting. He is sorely missed by the Brazilian mathematical community, and the publication of this volume became more a tribute to Welington de Melo. His role in the development of mathematics is indisputable, especially in the area of low-level dynamics, and his legacy includes, in addition to numerous articles with fundamental contributions, books that are mandatory references for beginners in this area. Welington had only seven formal Ph.D. students, and we were both honoured of being two of them. It is worth mentioning that Artur Avila, the 2014 Fields Medal winner, was also Welington's Ph.D. student.
We should say a little about a project undertaken by de Melo's wife Gilza. She knew that the Jequitinhonha Valley region where she was born and brought up had high rates of poverty and unemployment. She wanted to improve this by starting an educational project which, not surprisingly, was based on mathematics. After getting financial support from the IMPA, she began with a pilot project in June 2011 in five cities in the Jequitinhonha Valley. For four hours on a Saturday morning, pupils wishing to take part were coached for the Brazilian Public School Mathematics Olympiad. Details of this project are given in [8].

### References (show)

1. Artur Avila homenageia Welington de Melo, Institute of Pure and Applied Mathematics (1 August 2017).
https://impa.br/en_US/noticias/artur-avila-homenageia-welington-de-melo-no-31o-coloquio/
2. P Bonfert-Taylor, Review: Mathematical tools for one-dimensional dynamics (2008), by Edson de Faria and Welington de Melo, Mathematical Reviews MR2455301 (2010a:37090).
3. K M Brucks, Review: One-dimensional dynamics, by Welington de Melo and Sebastian van Strien, SIAM Review 37 (3) (1995), 461-462.
4. B Esteves, Morre Welington de Melo, O Matemático Velejador, piauí 123 (December 2016).
5. B Esteves, Velejador Regrado: No mar, passeios não triviais, piauí (August 2014).
6. E de Faria and S van Strien, Welington de Melo (1946-2016), in M J Pacifico and P Guarino (eds.), New Trends in One-Dimensional Dynamics: in Honour of Welington de Melo on the Occasion of His 70th Birthday, IMPA, Rio de Janeiro, Brazil, 14-18 November 2016 (Springer, 2021)
7. A Georgescu, Review: Geometric Theory of Dynamical Systems: An Introduction, by Jacob Palis, and Welington de Melo, Bulletin mathématique de la Société des Sciences Mathématiques de la RépubliqueSocialiste de Roumanie 32 (80) (1) (1988), 92.
8. Matemática traz novas perspectivas no Vale do Jequitinhonha, Olimpíada Brasileira de Matemática das Escolas Públicas.
http://www.obmep.org.br/noticias.DO?id=558
9. Morre o matemático Welington de Melo, orientador do medalha Fields, O Globo (22 December 2016).
10. M J Pacifico and P Guarino (eds.), New Trends in One-Dimensional Dynamics: in Honour of Welington de Melo on the Occasion of His 70th Birthday, IMPA, Rio de Janeiro, Brazil, 14-18 November 2016 (Springer, 2021).
11. Palestra de Artur Avila celebra memória de Welington de Melo, IMPA (27 July 2017).
https://impa.br/en_US/noticias/palestra-de-artur-avila-celebra-memoria-de-welington-de-melo/
12. J Palis and F Lenarduzzi, Welington de Melo and Jacob Palis: Their First Meeting, Some of Their Work on Structural Stability and a Lifetime of Friendship, in M J Pacifico and P Guarino (eds.), New Trends in One-Dimensional Dynamics: in Honour of Welington de Melo on the Occasion of His 70th Birthday, IMPA, Rio de Janeiro, Brazil, 14-18 November 2016 (Springer
13. J Palmer, Review: Mathematical aspects of quantum field theory, by Edson de Faria and Welington de Melo, SIAM Review 54 (1) (2012), 199-203.
14. P Piccione (ed.), Welington de Melo - Selected Works (Springer, 2022).
15. F Przytycki, Review: One-dimensional dynamics, by Welington de Melo and Sebastian van Strien, Mathematical Reviews MR1239171 (95a:58035).
16. J Robbin, Review: Geometric Theory of Dynamical Systems: An Introduction, by Jacob Palis, and Welington de Melo, The American Mathematical Monthly 91 (7) (1984), 448-449.
17. J M Salles, Artur tem um problema: Como se forma um grande matemático, Instituto de Matemática e Estatística Universidada de São Paulo (13 September 2011).
18. S Sedini, Welington de Melo, Instituto de Estudos Avançados de Universidade de São Paulo (17 September 2014).
http://www.iea.usp.br/pessoas/pasta-pessoaw/welington-de-melo
19. W D van Suijlekom, Review: Mathematical aspects of quantum field theory (2010), by Edson de Faria and Welington de Melo, Mathematical Reviews MR2683277 (2012d:81227).
20. R B Walker, Review: Geometric Theory of Dynamical Systems: An Introduction, by Jacob Palis, and Welington de Melo, Mathematical Reviews MR0669541 (84a:58004).
21. Welington Celso de Melo, The World Academy of Sciences.
https://twas.org/directory/de-melo-welington-celso
22. Welington Celso de Melo, CNPq (8 October 2015).
http://buscatextual.cnpq.br/buscatextual/visualizacv.do;jsessionid=1E3594BEEEB60FE9773C91334D2254BD.buscatextual_4
23. Welington Celso de Melo, escavador (August 2022)
https://wikijii.com/wiki/Welington_de_Melo
24. Welington de Melo CV, yumpu.
25. Welington de Melo, Academia Brasileira de Ciências.
https://www.abc.org.br/en/membro/welington-celso-de-melo/
26. S Zakeri, Review: One dimensional dynamics: The mathematical tools, by Edson de Faria and Welington de Melo, Mathematical Reviews MR1851555 (2002f:37077).

### Additional Resources (show)

Other pages about Welington de Melo:

Other websites about Welington de Melo:

### Honours (show)

Honours awarded to Welington de Melo

### Cross-references (show)

Written by J J O'Connor and E F Robertson
Last Update February 2023