Pure and Applied Mathematics
One of the world's leading number theorists, working on Hilbert's 12th problem.
"Somewhere out there is a theory that would explain my empirical observations, and this theory has yet to be discovered. Mathematics thrives on such mysteries."
Henri Darmon and his girlfriend, Galia, were driving back to Princeton, New Jersey, on a warm Sunday in June after a holiday at Cape Cod. The 20-something couple had been together for two years. They had met in 1991 when Darmon arrived at Princeton University to work as a mathematics instructor and researcher; Galia was studying mathematics as well, finishing her doctorate (PhD).
As mathematicians often do, Henri and Galia whiled away the miles by musing about some of the big puzzles nobody could solve. “So what about Fermat’s last theorem?” asked Galia. “Do you think anyone will ever prove it?” She was referring to a seemingly simple problem that Darmon knew a lot about. His own research on elliptic curves might someday be involved in a mathematical proof. Over the centuries, millions of people had tried to solve French mathematician Pierre de Fermat’s last theorem, which he had scribbled in the margin of a book in 1637, but nobody could do it.
A mathematical theorem is a truth derived from fundamental principles. Fermat’s last theorem relates to another classic phenomenon called the Pythagorean theorem, which says: for the sides of a right triangle, the sum of the squares of the two short sides equals the square of the long side. The figure opposite demonstrates this idea, which was actually discovered by the Babylonians (the ancient civilization that existed in what is now called Iraq) around 2000 b.c. They used it to figure out the relative value of farm fields for tax purposes, but it is attributed to the famous Greek mathematician Pythagoras, who studied it in great detail.
Usually the theorem is stated algebraically: a2 + b2 = c2, where c is the hypotenuse (long side) and a and b are the short sides of the triangle. Lots of numbers satisfy this rule starting with 3, 4 and 5 (replace the b with 3, the a with 4 and the c with 5. Then you get 32 = 9, 42 = 16 and 52 = 25, and since 9 + 16 = 25, it all works out.) Other Pythagorean sets of numbers are 5, 12, and 13 or 7, 24, and 25. Actually there are infinitely many.
This kind of mathematics is called number theory and is Henri Darmon's specialty.
Fermat wondered if the same rule would work for a3 + b3 = c3 or a4 + b4 = c4 and he found out that it didn't. But then he wrote, "It is impossible to separate any power except a square into 2 powers with the same exponent. I have discovered a truly marvelous proof of this, which however the margin is not large enough to contain." To put it another way, he was saying the equation an + bn = cn would never work if n was anything other than 2. And he could prove this. In his life Fermat wrote down many such theorems, but unfortunately he died before he could prove this one on paper. In June 1993, 328 years after his death, this was the only one of Fermat's theorems that had not yet been solved. That is why it's called Fermat's last theorem.
Over the years vast sums of money, at times equivalent to more than US$2 million, had been put forth as the prize for the first person to prove Fermat’s last theorem, so when Henri and Galia talked about it, they were serious. Whoever figured it out would literally become rich and famous. Fermat’s “marvellous” proof, if it was correct, must have been very complicated because none of the old-fashioned methods seemed to work. Darmon knew that the solution would probably use something he was working on, elliptic curves, and this is what he told Galia. “I have a feeling that Fermat will be settled soon,” he said. “Maybe within 10 years.”
Back in 1955 two Japanese mathematicians, Yutaka Taniyama and Goro Shimura, claimed that elliptic curves were related to another mathematical object called modular forms in a profound and surprising way. No one had ever thought this connection between modular forms — functions involving imaginary numbers that result in amazing symmetrical patterns — could have anything to do with elliptic curves, which are standard formulas like y2 = ax3 + bx2 + cx + d.
Left: representation of a modular form. Right: representation of an elliptic curve (in blue). Elliptic curves are not ellipses, though they were originally used hundreds of years ago to help calculate the perimeter of an ellipse. The rational points on an elliptic curve (that is, points that are whole numbers, or ratios of whole numbers) have interesting properties. Any two points can be used to find a third. Number theorists love elliptic curves because they answer many questions about equations and their solutions.
Taniyama and Shimura could not prove their claim. Mathematicians call something like this a conjecture until a proof is found. Interest in the Shimura-Taniyama conjecture had recently increased because in the mid-1980s two other mathematicians (Frey and Ribet) showed that if it were ever proven, a solution to Fermat’s last theorem would automatically result. But because the problem seemed so hard, people didn’t even know where to begin.
“I don’t think anyone will get Shimura-Taniyama in my lifetime, Galia,” said Henri, as he pulled into a parking spot near their apartment in Princeton. He thought it was much harder than Fermat’s last theorem. “I bet Fermat will be solved first,” he said.
The next morning was a Monday and Darmon was in his office at the university. It was a big room in a tower building, with windows looking out over the rural town of Princeton. Just as Darmon was getting down to work, Peter Sarnak burst into the room. Sarnak was a charismatic and assertive mathematics professor at Princeton. He said, “You know Wiles has got it. He’s going to prove Fermat.”
“What?” said Darmon, shocked. He knew Andrew Wiles, another Princeton mathematics professor and one of Peter’s best friends. Wiles was attending a mathematics conference at the Newton Institute in Cambridge, England. Darmon was stunned. Could it be true? “How is he going to do it?” he asked.
Sarnak said, “He’s going to prove Shimura-Taniyama.”
Darmon was now doubly amazed. Not only would Fermat’s last theorem be solved, but so would the Shimura-Taniyama conjecture. “Do you have the manuscript?” he asked Sarnak, figuring he’d need to see the proof to believe it.
“No, I’m sworn to secrecy. I shouldn’t even be telling you, Henri.” Sarnak spoke with a charming South African accent. He said, “Andrew is giving three lectures to build up to it, and he won’t be showing the Fermat proof until his final lecture on Wednesday, so please keep this to yourself. When Andrew gets back, just ask him for a copy.”
Wiles’ proof turned out to be 200 pages long and was so complicated that only a few mathematicians on Earth could fully appreciate it. “In the end, it was not so much that he did it,” says Darmon, “rather it was how he did it.” By bringing together two apparently unrelated areas of mathematics (modular forms and elliptic curves), Wiles had done something much greater than proving one simple theorem. He had performed a kind of grand unification of mathematical thinking. It was a symbolic victory.
“Wiles changed what I thought was doable and psychologically moved me to go in directions where I wasn’t sure I would get an answer,” says Darmon. In the same way, Wiles has inspired millions of other mathematicians to discover many new mathematical relationships in recent years.
Henri Darmon was born in Paris but at the age of 3 his family moved to Quebec City in Canada. His father had a position at Laval University teaching Business Administration. When Darmon was 11, the family moved to Montreal where his father began teaching at McGill University. As a teenager, Darmon liked to travel and while he was on the road he enjoyed obsessing over hard math problems. He would sometimes work all summer on one problem while he travelled. He still does it. "I sometimes get my best ideas on busses and trains," says Darmon.
The summer he was 17, Darmon took a trip to Morocco. He had recently discovered Calculus, the mathematics for describing continuous change. That summer he set himself the personal challenge to find the anti-derivative of sinx/x as he travelled. It looked simple enough, but he spent all summer on this and never found the answer. The reason: there IS no simple solution.