**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.

- Although Welington was born in Guapé, a town in the Minas Gerais state in south east Brazil, he never lived there.
- This involved several moves during the years that Welington was growing up and, as a consequence, he attended schools in different towns.
- This college, founded in 1902 and run by the Marist Brothers, had around 600 pupils when Welington attended.
- 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.
- 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.
- 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.
- 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.
- 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.
- De Melo was awarded a CAPES scholarship to fund his Ph.D. studies.
- 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.
- He was awarded his doctorate for his thesis Estabilidade Estrutural em Variedades de Dimensao Dois Ⓣ(Structural Stability in Manifolds of Dimension Two) in 1972.
- 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.
- 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'.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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).
- Since sailing played such a large part in de Melo's life, let us say a little more about it.
- Since 2006 the boat was moored beside a pier attached to his house in a condominium in Angra.
- "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 wanted the work to be at the highest possible level, but, because of this rigour, his reputation scared students a little.
- 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.
- 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.
- The proof is based on a detailed understanding of the orbit structure of an infinite dimensional dynamical system: the renormalisation operator.
- "Very few people really understood what it was about", he commented, "and Welington was a notable exception.
- 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.
- Lyubich and de Melo decided to leave it in the boy's hands.
- 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.
- 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.
- 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.
- 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.

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

View full biography at [MacTutor](https://mathshistory.st-andrews.ac.uk/Biographies/De_Melo/

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive