Maurício Peixoto
Encyclopedia
Maurício Matos Peixoto, born on April 15 of 1921, in Fortaleza
, Ceará
, is a Brazil
ian engineer who pursued a bright career as a mathematician
. He pioneered the studies on structural stability
, and is the author of Peixoto's theorem
.
Once, while talking with his mentor, Solomon Lefschetz
, Maurício Peixoto commented that no one cared about structural stability
of dynamical systems 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."
Lefschetz's support was very important to Peixoto at the time. In 1957, Peixoto went to research the subject with Lefschetz at the Princeton University
, where he spent uncountable hours talking to the Russian professor about Mathematics and other subjects. Despite of the great age difference (Peixoto was 36 years old and Lefschetz 73), they became good friends.
With Lefschetz incentive, Peixoto wrote his first paper on structural stability, that would be later published on the Annals of Mathematics
, of which Lefschetz was editor. In 1958, they went to the International Mathematics Congress, in Edinburgh
, Scotland
, where Lefschetz introduced Peixoto to the Russian mathematician Lev Pontrjagin, whose work on dynamical systems was used by Peixoto as a basis for his studies. Pontrjagin, 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
, the American mathematician spent six months at (Instituto de Matemática Pura e Aplicada - Institute of Pure and Applied Mathematics (IMPA)- at Rio de Janeiro
). Through Smale, Peixoto would meet the French
mathematician René Thom
, who would help Peixoto to formulate his theorem
, that was finalized during Thom's visit to IMPA.
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 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."
Fortaleza
Fortaleza is the state capital of Ceará, located in Northeastern Brazil. With a population close to 2.5 million , Fortaleza is the 5th largest city in Brazil. It has an area of and one of the highest demographic densities in the country...
, Ceará
Ceará
Ceará is one of the 27 states of Brazil, located in the northeastern part of the country, on the Atlantic coast. It is currently the 8th largest Brazilian State by population and the 17th by area. It is also one of the main touristic destinations in Brazil. The state capital is the city of...
, is a Brazil
Brazil
Brazil , officially the Federative Republic of Brazil , is the largest country in South America. It is the world's fifth largest country, both by geographical area and by population with over 192 million people...
ian engineer who pursued a bright career as a mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
. He pioneered the studies on structural stability
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 C1-small perturbations....
, and is the author of Peixoto's theorem
Peixoto's theorem
In the theory of dynamical systems, Peixoto theorem, proved by Maurício Peixoto, states that among all smooth flows on surfaces, i.e. compact two-dimensional manifolds, structurally stable systems may be characterized by the following properties:...
.
Once, while talking with his mentor, Solomon Lefschetz
Solomon Lefschetz
Solomon Lefschetz was an American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear ordinary differential equations.-Life:...
, Maurício Peixoto commented that no one cared about structural stability
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 C1-small perturbations....
of dynamical systems 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."
Lefschetz's support was very important to Peixoto at the time. In 1957, Peixoto went to research the subject with Lefschetz at the Princeton University
Princeton University
Princeton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....
, where he spent uncountable hours talking to the Russian professor about Mathematics and other subjects. Despite of the great age difference (Peixoto was 36 years old and Lefschetz 73), they became good friends.
With Lefschetz incentive, Peixoto wrote his first paper on structural stability, that would be later published on the Annals of Mathematics
Annals of Mathematics
The Annals of Mathematics is a bimonthly mathematical journal published by Princeton University and the Institute for Advanced Study. It ranks amongst the most prestigious mathematics journals in the world by criteria such as impact factor.-History:The journal began as The Analyst in 1874 and was...
, of which Lefschetz was editor. In 1958, they went to the International Mathematics Congress, in Edinburgh
Edinburgh
Edinburgh is the capital city of Scotland, the second largest city in Scotland, and the eighth most populous in the United Kingdom. The City of Edinburgh Council governs one of Scotland's 32 local government council areas. The council area includes urban Edinburgh and a rural area...
, Scotland
Scotland
Scotland is a country that is part of the United Kingdom. Occupying the northern third of the island of Great Britain, it shares a border with England to the south and is bounded by the North Sea to the east, the Atlantic Ocean to the north and west, and the North Channel and Irish Sea to the...
, where Lefschetz introduced Peixoto to the Russian mathematician Lev Pontrjagin, whose work on dynamical systems was used by Peixoto as a basis for his studies. Pontrjagin, 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
Brazil , officially the Federative Republic of Brazil , is the largest country in South America. It is the world's fifth largest country, both by geographical area and by population with over 192 million people...
, the American mathematician spent six months at (Instituto de Matemática Pura e Aplicada - Institute of Pure and Applied Mathematics (IMPA)- at Rio de Janeiro
Rio de Janeiro
Rio de Janeiro , commonly referred to simply as Rio, is the capital city of the State of Rio de Janeiro, the second largest city of Brazil, and the third largest metropolitan area and agglomeration in South America, boasting approximately 6.3 million people within the city proper, making it the 6th...
). Through Smale, Peixoto would meet the French
France
The French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semi-presidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan France...
mathematician René Thom
René Thom
René Frédéric Thom was a French mathematician. He made his reputation as a topologist, moving on to aspects of what would be called singularity theory; he became world-famous among the wider academic community and the educated general public for one aspect of this latter interest, his work as...
, who would help Peixoto to formulate his theorem
Peixoto's theorem
In the theory of dynamical systems, Peixoto theorem, proved by Maurício Peixoto, states that among all smooth flows on surfaces, i.e. compact two-dimensional manifolds, structurally stable systems may be characterized by the following properties:...
, that was finalized during Thom's visit to IMPA.
For his theorem, Peixoto won the Bunge Foundation Award in the year of 1969. According to Bunge Foundation
Bunge Limited
Bunge Limited is a Bermudan food conglomerate with its headquarters in White Plains, New York. As well as being a leading global soybean exporter it is also involved in food processing, grain trading, and fertilizer...
, "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 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."