A German mathematician, Gerd Faltings, is this year’s winner of the Abel Prize, an honor that is regarded as mathematics’ version of the Nobel Prize. Dr. Faltings, 71, is best known for solving a problem that had puzzled mathematicians for decades. He showed that a class of equations possessed a finite number of solutions.
The Norwegian Academy of Science and Letters, which manages the Abel Prize, announced the honor on Thursday morning.
“He’s a towering figure in number theory,” said Helge Holden, chairman of the prize committee.
Number theory is a branch of mathematics that studies the properties and relationships of integers.
“His ideas and results have reshaped the field, settling major longstanding conjectures while also establishing new frameworks that have guided decades of subsequent work,” the prize citation said.
In the 1980s, Dr. Faltings became interested in a problem that had been first described nearly six decades earlier.
The problem involves Diophantine equations, named after Diophantus of Alexandria, a third-century Greek mathematician.
Diophantine equations consist of polynomial expressions, like the equation of a line, ax + by = c, where the coefficients are integers. Sometimes integer solutions exist; sometimes they do not.
Dr. Holden gave an example: If you want to buy something with coins, there may be several ways to come up with the exact amount using quarters, dimes, nickels and pennies. But if you do not have any pennies, you cannot pay exactly 17 cents, for instance.
Another example is the Pythagorean theorem, which describes the relationship between the lengths of the sides of a right triangle: a² + b² = c². It is possible to find a right triangle with sides all of integer lengths — for example, 3, 4 and 5 satisfy the equation. Indeed, there are an infinite number of right triangles with integer lengths.
In 1922, a mathematician, Louis Mordell, put forth a conjecture, something that he believed to be true but for which he could not offer a mathematical proof. He suggested that for certain more complex equations, the number of rational solutions — those that can be written as fractions — is finite.
For more than half a century, other mathematicians also failed to find a proof.
Dr. Faltings was drawn to the problem after conversations with Lucien Szpiro, a French mathematician.
“He had some ideas about this,” Dr. Faltings said in an interview. “I thought these ideas are interesting, although I didn’t expect to prove the conjecture. But I thought something interesting will come out.”
Dr. Faltings cracked the problem, publishing his proof in 1983. The Mordell conjecture, now known as the Faltings theorem, showed this result using a connection between number theory and geometry.
For the proof, he had to first prove two other significant conjectures, and he used a novel approach instead of a more obvious strategy — known as Diophantine approximation — that other mathematicians had tried to employ.
“That was as shocking as a math result can be in our small community and gave him instant fame,” Dr. Holden said.
For mathematicians, knowing that there are a finite number of solutions “changes everything,” Dr. Holden said, even though there is still usually no procedure to find the solutions or even figure out exactly how many there are.
In 1986, Dr. Faltings was one of the recipients of the Fields Medals, which, at the time, were the most prominent awards in mathematics. Every four years, the medals are awarded to a small number of mathematicians age 40 or younger for groundbreaking work early in their careers.
In 1989, another mathematician, Paul Vojta, came up with another proof of Faltings theorem by using the more traditional Diophantine approximation approach. Dr. Faltings reviewed Dr. Vojta’s proof and was able to extend it to a more general theorem, providing deeper insights about the structure of rational numbers.
There is no Nobel Prize in mathematics, but in 2002, the Norwegian Academy of Science and Letters established the Abel Prize, which is similar in setup to the Nobels.
Unlike Nobel laureates, who are often surprised with middle-of-the-night phone calls just before the honors are publicly announced, Dr. Faltings learned of his honor last week.
Although retired, Dr. Faltings still regularly goes to his office at the Max Planck Institute for Mathematics in Bonn, Germany. He was summoned to a colleague’s office under the pretext of a meeting. After he sat down, a perplexed look crossed his face as a woman on the videoconference call introduced herself as Marit Westergaard, the secretary general of the Norwegian academy.
She told him he was the recipient of this year’s Abel Prize.
Previous laureates include Andrew J. Wiles, who proved Fermat’s last theorem, and John F. Nash Jr., whose life was portrayed in the movie “A Beautiful Mind.”
The honor is accompanied by 7.5 million Norwegian kroner, or about $780,000. The award ceremony will be held in Oslo in May.
During the video conference call, Dr. Faltings thanked Dr. Westergaard and said the Abel was an unexpected surprise.
“I’m getting old, and I thought I’m beyond the time for such prizes, but apparently not,” Dr. Faltings said. “I guess I have to rent a tuxedo.”
Kenneth Chang, a science reporter at The Times, covers NASA and the solar system, and research closer to Earth.
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