16/07/2025

Folha: 'The mathematician, the letter, and the horseshoe'

Photo: Maurício Peixoto

Reproduction of Marcelo Viana's column in Folha de São Paulo.

My friend Jair Koiller, a visiting professor at the State University of Rio de Janeiro, visibly proud, handed me the latest issue of Notices, the monthly journal of the American Mathematical Society. His pride is more than justified, as the magazine features on its cover an article by him recounting a recent, and very surprising, discovery about a pivotal episode in the history of mathematics in Brazil.

To understand this, we need to go back in time to the end of the 1950s, when young Brazilian mathematicians Elon Lages Lima and Maurício Peixoto, from the newly created Institute of Pure and Applied Mathematics (IMPA), became acquainted with the equally young American Steve Smale, a professor at the University of Chicago.

By this time, Smale was already one of the most brilliant mathematical leaders of his generation, a specialist in topology. Inspired by the pioneering advances that Mauricio had been making in dynamical systems, he also began working in that area. Then the two Brazilians invited him to visit IMPA, which he did at the end of 1959.

Perhaps this was the most fruitful period of his extraordinary mathematical career. During the six months he spent with his family in Copacabana, Smale proved the Poincaré conjecture, one of the most important and famous problems in mathematics, in dimension 5 or higher. For this feat, he would be awarded the prestigious Fields Medal in 1966.

The problem, formulated by Henri Poincaré at the beginning of the 20th century and which had already been tackled by several excellent mathematicians without success, concerns the mathematical characterization of what a sphere is. Poincaré had in mind spheres of dimension 3, but Smale's originality lay in considering the question in higher dimensions (where it should be more difficult!) and solving it. Poincaré's original problem was only solved in 2003 by the Russian Grigory Perelman, and the case of dimension 4 remains open.

As if that weren't enough, during this period Smale made another mathematical discovery that was as, or even more, impactful than Poincaré's conjecture: he found a mathematical object, the "horseshoe," which today forms the basis of the entire theory of dynamical systems.

It all began with a letter dated February 20, 1960, in which his colleague Norman Levinson drew attention to work in which he had shown that differential equations can have an infinite number of periodic solutions and that, in fact, this happens frequently. This result by Levinson went completely against Smale's intuition and, even worse, directly contradicted his first work on dynamical systems, which he had just published.

In a remarkable effort to resolve his perplexity at this situation, Smale was led to the discovery of the horseshoe, which changed the course of mathematics in Brazil and the world. The account of what followed has already entered the history books, deservedly so, not least because its repercussions went beyond academia and reached even the world of geopolitics.

Read the full article on the Folha de São Paulo website.