09/04/2025

Folha: 'Henri Poincaré, champion of intuition in mathematics'

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

In 2012, I was invited by the Brazilian Mathematical Society to give a lecture marking the centenary of the death of Henri Poincaré (1854–1912). For a couple of months, I immersed myself in the task, reading books, consulting colleagues, and researching on the internet. I started out thinking I knew a lot about his work, but I ended up much more humbled by the magnitude of his scientific legacy.

Great mathematicians can be classified into two main styles: problem solvers, who resolve conjectures and prove difficult theorems, and pathbreakers, who discover new mathematical ideas and connections between them. Brazilian Artur Avila , winner of the Fields Medal in 2014, belongs to the first style. Poincaré is a perfect example of the second. With his extraordinary intuition, he opened new paths in mathematics that are still being explored today.

His mathematical legacy is extraordinary: he contributed to almost every discipline of mathematics and created others—the theory of automorphic functions, algebraic topology, dynamical systems—in addition to paving the way for the theory of functions of several complex variables and for asymptotic analysis.

His contribution to other sciences is equally remarkable. He influenced the development of physics in his time, actively participating in major debates. He revolutionized celestial mechanics, discovering the phenomenon of 'chaos'. He found new equilibria in celestial bodies and proposed a new mechanism for the formation of binary stars. And he was one of the creators of the theory of special relativity , which bears Einstein's name.

His first important work was the discovery of a type of function which he called 'Fuchsian', in homage to his German colleague Lazarus Fuchs (1833–1902). He himself recounts, in words that demonstrate the elegance of his style: "For 15 days I strove to prove that such functions did not exist. I was very ignorant. Every day I sat at my desk and spent an hour or two trying various combinations, and I arrived at no result."

Until one evening, "against my usual habit, I drank black coffee and couldn't sleep. Ideas kept flowing in. I felt them clashing with each other, until two of them settled down, so to speak, to form a stable combination. By morning, I had proven the existence of a class of Fuchsian functions. All that remained was to write up the results, which took only a few hours."

To read the full article, visit the newspaper's website.