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Maurício Matos Peixoto, (April 15, 1921 – April 28, 2019), was a Brazilian engineer and
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
. He pioneered the studies on
structural stability In mathematics, structural stability is a fundamental property of a dynamical system which means that the qualitative behavior of the trajectories is unaffected by small perturbations (to be exact ''C''1-small perturbations). Examples of such q ...
, and was the author of Peixoto's theorem.


Biography

Maurício Peixoto, born in Fortaleza in 1921 to José Carlos de Matos Peixoto and Violeta Rodrigues Peixoto, pursued a career in mathematics since his adolescence. To fulfill his goal, Peixoto enrolled in engineering at the School of Engineering of the current Federal University of Rio de Janeiro (UFRJ). While still working at the School of Engineering around 1955, Peixoto came into contact with the work of De Baggis, a student of
Solomon Lefschetz Solomon Lefschetz (; 3 September 1884 – 5 October 1972) was a Russian-born American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear ordinary differential equatio ...
. Through this work, he learnt about the problem of structural stability of dynamic systems. After some correspondence with Solomon Lefschetz regarding this problem, Peixoto was invited and went to Princeton University in 1957 to work as Lefschetz's student on the topic of structural stability of differential equations. Despite the great age difference (Peixoto was 36 years old and Lefschetz was 73 years old), Peixoto and Lefschetz became good friends. Once, while talking with Lefschetz, Peixoto commented that no one cared about
structural stability In mathematics, structural stability is a fundamental property of a dynamical system which means that the qualitative behavior of the trajectories is unaffected by small perturbations (to be exact ''C''1-small perturbations). Examples of such q ...
of
dynamical systems In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...
and that was the main problem in working with it. But to Peixoto's surprise Lefschetz's answer was no less than "No Mauricio, this is no trouble, this is your luck. Try to work as hard and as fast as you can on this subject because the day will come when you will not understand a single word of what they will be saying about structural stability; this happened to me in topology." With Lefschetz incentive, Peixoto wrote his first paper on structural stability, that would be later published on the
Annals of Mathematics The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as t ...
, of which Lefschetz was editor. In 1958, they went to the International Mathematical Congress, in
Edinburgh Edinburgh is the capital city of Scotland and one of its 32 Council areas of Scotland, council areas. The city is located in southeast Scotland and is bounded to the north by the Firth of Forth and to the south by the Pentland Hills. Edinburgh ...
,
Scotland Scotland is a Countries of the United Kingdom, country that is part of the United Kingdom. It contains nearly one-third of the United Kingdom's land area, consisting of the northern part of the island of Great Britain and more than 790 adjac ...
, where Lefschetz introduced Peixoto to the Russian mathematician Lev Pontryagin, whose work on
dynamical systems In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...
was used by Peixoto as a basis for his studies. Pontryagin, though, showed no interest whatsoever in Peixoto's work. Back to Princeton, Peixoto met Steve Smale, the mathematician that would later become a reference in dynamical systems. Smale was interested in Peixoto's work and realized he could extend his own based on it. Their contact intensified and, when Peixoto came back to
Brazil Brazil, officially the Federative Republic of Brazil, is the largest country in South America. It is the world's List of countries and dependencies by area, fifth-largest country by area and the List of countries and dependencies by population ...
, the American mathematician spent six months at the
Instituto Nacional de Matemática Pura e Aplicada The ''Instituto Nacional de Matemática Pura e Aplicada'' (National Institute for Pure and Applied Mathematics) is considered to be the foremost research and educational institution of Brazil in the area of mathematics. It is located in the cit ...
(Institute of Pure and Applied Mathematics or IMPA) at
Rio de Janeiro Rio de Janeiro, or simply Rio, is the capital of the Rio de Janeiro (state), state of Rio de Janeiro. It is the List of cities in Brazil by population, second-most-populous city in Brazil (after São Paulo) and the Largest cities in the America ...
. Through Smale, Peixoto would meet the French mathematician
René Thom René Frédéric Thom (; 2 September 1923 – 25 October 2002) was a French mathematician, who received the Fields Medal in 1958. He made his reputation as a topologist, moving on to aspects of what would be called singularity theory; he became ...
, who would help Peixoto to formulate his
theorem In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...
, that was finalized during Thom's visit to IMPA.


Awards

For his theorem, Peixoto won the Bunge Foundation Award in the year of 1969. According to Bunge Foundation, "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 1986, Peixoto was awarded the
TWAS Prize This is a list of recipients of the TWAS Prize, awarded annually by The World Academy of Sciences (TWAS) since 1985. Summary Agricultural Sciences Biology Chemistry Earth Sciences Engineering Sciences Mathematics ...
by the Third World Science Academy, "for his fundamental and pioneer studies on structural stability in dynamical systems, in particular for proving that surface flows are generically structurally stable."


Personal life

Peixoto married engineer and mathematician Marília Magalhães Chaves (who was known professionally by her married name) Marília Chaves Peixoto in 1946, with whom he had two children. The couple collaborated in their work, jointly published "Structural Stability in the plane with enlarged boundary conditions" in 1959, before Marília Chaves Peixoto's early death in 1961. It would be a key paper which led to the development of Peixoto's theorem.


References

{{DEFAULTSORT:Peixoto, Mauricio 1921 births 2019 deaths People from Fortaleza 20th-century Brazilian mathematicians Instituto Nacional de Matemática Pura e Aplicada researchers TWAS laureates Federal University of Rio de Janeiro alumni