In Folha, Viana talks about the history of the Theory of Dynamical Systems.
Reproduction of Marcelo Viana's column in Folha de São Paulo.
In 1953, mathematician Maurício Peixoto, one of the founders of IMPA (Institute of Pure and Applied Mathematics ), began a pioneering study in the relatively new area of dynamical systems . By the end of the decade, he had already proven that most two-dimensional differential equations have a finite number of periodic solutions and are stable: the solutions are not essentially affected when we slightly modify the equation.
Our American colleague Steve Smale soon became interested in these results. In a paper published in early 1960, he generalized the type of equations considered by Peixoto to higher dimensions and conjectured that these equations, which the Frenchman René Thom called "Morse-Smale equations," would still constitute the majority of all differential equations in any dimension.
But shortly afterward, while visiting IMPA at the invitation of Peixoto and Elon Lages Lima, Smale received a letter from his colleague Norman Levinson, dated February 20th, which changed everything: Levinson had discovered, some time ago, that there are many differential equations with an infinite number of periodic solutions. Therefore, Smale's conjecture could not be true!
It was an embarrassing setback, but one that proved very fruitful: Smale reformulated his theory and, in a masterful work published in 1967, laid the foundations for the modern theory of dynamical systems, which has become one of the most active in all of mathematics today.
By this time, Smale already had as his first doctoral student a brilliant young Brazilian, recommended by Lima and Peixoto: Jacob Palis . In his thesis, Palis greatly generalized Peixoto's theorem and gave a rigorous basis to part of Smale's original intuition, proving that every Morse-Smale equation, in any dimension, is stable.
Strangely, at a time when written correspondence was usually carefully preserved, Smale did not take such precautions regarding Levinson's letter: questioned over the years, he never knew its whereabouts and didn't even remember if he had replied. The famous letter was presumed lost…
Imagine the surprise when, more than 60 years later, Jair Koiller found it among Peixoto's papers bequeathed to the Museum of Astronomy's archive after his death in 2019.
Even more surprising because, while going through the 15 boxes of legacy documents, Jair also found a second letter from Levinson whose existence had been completely forgotten!
Read the full article on the Folha de São Paulo website.
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