Consider a line segment of length 1. Divide it into three parts of equal length and remove the middle part: two pieces of length 1/3 remain, right? Then divide each of these parts into three parts of equal length and remove the middle part: now you are left with four parts of 1/9 length. Again, divide each of these parts into three equal parts and remove the middle part: the result is eight parts 1/27 in length. And so on. What’s left at the end of infinite processes like this is called the “Cantor set,” after Georg Cantor (1845-1918), who introduced the concept in 1883.
In 1974, doctoral student Douglas Hofstadter guessed that the energy levels of an electron trapped in a crystal under the influence of a magnet form a Cantor group when the magnetic flux is irrational. His fellow physicists found the idea strange (“Numerology!”). He did not know how to prove this to be true.
Years later, in 1981, two mathematicians met for lunch and discussed Hofstadter’s conjecture. Mark Kack (1914-1984) and Barry Simon were studying a model of the Schrödinger equation, and based on their own work, they concluded that Hofstadter was probably right. But they also didn’t know how to prove this conjecture: Kak estimated it would be so difficult that he offered to pay anyone who could do it ten martini glasses. Simon announced the offer, and “The Ten-Martini Problem” was born.
The following year, Simon made a partial breakthrough on the problem, and Kak paid three martinis for the result. But when Kack died in 1984, the problem remained open, and would remain so for another two decades. Not for lack of effort.
Ukrainian mathematician Svetlana Zhitomirskaya has been working on this topic for years. But by 2003, I was frustrated: the previous year, Spaniard Joaquim Puig had solved the problem for most values of magnetic flux. What’s worse is that he used her work for this purpose! “I wanted to punch myself: I had done all the hard work, and then he came and found this beautiful solution.”
Here comes the young Brazilian mathematician Artur Avila, then only 24 years old, and proposes to work together on the remaining cases of the problem. “I explained to him that it would be difficult and time-consuming, and that no one cared,” Svetlana says. They went ahead anyway and succeeded: they soon solved one of the most important and difficult problems in mathematical physics. The joint work was published in 2005 in the Annals of Mathematics, the most prestigious journal in the field of mathematics. Solving the 10-martini problem would be one of Artur’s most cited achievements when he won the Fields Medal in 2014.
Did you celebrate with a martini? “Of course. There were lots of celebratory drinks, including martinis,” Svetlana explains with a smile.
However, for the aesthetic sense of mathematicians, the solution was still unsatisfactory, because it consisted of several distinct arguments, each valid for different values of magnetic flux. It would be much better if there was one argument that unites all cases…
The story continues.
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