Viana tells Folha about the 'problem of the ten martinis'
Reproduction of Marcelo Viana's column in Folha de S. Paulo.
Consider a line segment of length 1. Divide it into three equal parts and remove the middle one: you are left with two segments of length 1/3, right? Next, divide each of these segments into three equal parts and remove the middle one: now you are left with four segments of length 1/9. Again, divide each of these segments into three equal parts and remove the middle one: the result is eight segments of length 1/27. And so on. What remains at the end of infinite operations like this is called the "Cantor set," in honor of Georg Cantor (1845-1918), who introduced this concept in 1883.
In 1974, doctoral student Douglas Hofstadter conjectured that the energy levels of an electron trapped in a crystal under the influence of a magnet form a Cantor set when the magnetic flux is irrational. His fellow physicists thought the idea was bizarre ("numerology!"). And he didn't know how to prove it was true.
Years later, in 1981, two mathematicians met for lunch to discuss Hofstadter's conjecture. Marc Kac (1914-1984) and Barry Simon had been studying a model of the Schrödinger equation and, based on their own work, concluded that Hofstadter was probably right. But they also didn't know how to prove the conjecture: Kac estimated it would be so difficult that he offered to pay ten martinis to whoever could. Simon publicized the offer, and thus the "ten martini problem" was born.
The following year, Simon made partial progress on the problem, and Kac paid for three martinis as a reward. But when Kac died in 1984, the problem remained unresolved, and would remain so for another two decades. Not for lack of trying.
Ukrainian mathematician Svetlana Jitomirskaya had been working on the topic for years. But in 2003 she was discouraged: the previous year, Spaniard Joaquim Puig had solved the problem for most values of magnetic flux. And the worst part was that he had used her work to do it! "I wanted to punch myself: I had done all the hard work, and then he came along and found that beautiful solution."
Enter the young Brazilian mathematician Artur Avila , then only 24 years old, proposing that they work together on the remaining cases of the problem. "I explained to him that it would be difficult and time-consuming, and that nobody cared," says Svetlana. They went ahead anyway, and it worked: soon the two solved one of the most relevant and challenging problems in mathematical physics. The joint work was published in 2005 in the journal Annals of Mathematics , the most prestigious in the field of mathematics. And the solution to the ten martini problem would be one of Artur's most cited achievements when he won the Fields Medal in 2014 .
Did they celebrate with martinis? "Of course. There were lots of celebratory drinks, including martinis," Svetlana clarifies with a smile.