The Mordell conjecture—now known as Faltings’s theorem—concerns the number of special points on a curve

By Joseph Howlett edited by Clara Moskowitz

A man sitting with hands folded on a bench outside a wooden shed.

At age 71, German mathematician Gerd Faltings was awarded the Abel Prize today. Peter Badge/Typos1/The Abel Prize

Mathematics

Join Our Community of Science Lovers!

Sign Up for Our Free Daily Newsletter

This year’s Abel Prize, an annual lifetime achievement award for mathematics that is bestowed by the Norwegian Academy of Science and Letters and was modeled on the Nobel Prize, has been given to Gerd Faltings, a German mathematician who is most famous for proving the influential Mordell conjecture in 1983. That conjecture has since been named “Faltings’s theorem” after him.

The award joins a heap of accolades Faltings, age 71, has piled up over his long career. That list includes the Fields Medal, math’s most coveted prize, which Faltings won at age 32. “Near the beginning of my career, I got the Fields Medal. And near the end, I’m getting the Abel Prize,” Faltings says. “It’s a nice duality.”

Faltings’s theorem is about curves. Often, these can be described by simple equations with two variables that are multiplied and added together. Chart the solutions of such an equation on a coordinate grid, and they’ll form a line or an ellipse or a more complicated, twisty curve.

Since the beginning of math, people have been looking for a rarified subset of these solutions—“rational” points on the curve, where the coordinates are integers or fractions. These special points have rich and complicated relationships that bely a hidden order that mathematicians aim to uncover.

But there are an infinity of curves out there, and nailing down all their rational points seemed impossible—until Faltings’s Theorem, that is. He proved that if a curve’s equation has a variable raised to a power higher than 3, then it must have a finite number of these points. Only lines, quadratics (such as circles) and cubic equations could have an infinite number.

The proof is considered a cornerstone of arithmetic geometry, the field that studies curves and shapes represented by these types of equations.

“It’s absolutely fundamental,” says Noam Elkies, a mathematician at Harvard University, about Faltings’s proof. “The fact that Mordell’s conjecture is now a theorem and all of the structures he developed have informed a lot of the work in nearby fields that’s happened since.”

Mathematicians are still working out the consequences of the theorem, which was originally conjectured by Louis Mordell in 1922. Just a few weeks ago mathematicians announced that they had found an actual limit on how many rational points curves can have.

Profile shot of older man sitting on a sofa reading a newspaper.

Peter Badge/Typos1/The Abel Prize

The theorem bearing his name was only one of Faltings’s many mathematical accomplishments. These include an expansive generalization of the theorem from curves to multidimensional shapes, which he proved in 1991, and major contributions to an important field known as “p-adic Hodge theory,” which provides methods to study such shapes and the equations that form them.

The five-member committee convened to make the decision at the Institute for Advanced Study in Princeton, N.J., near the end of January—just as a winter storm blanketed the Northeast in feet of snow. “We had nothing else to do than just sitting down and discussing mathematics,” said Helge Holden, the committee’s chair, at the Abel Symposium, an event that was held the following week. “The hotel was running low on supplies, so the bread became drier and drier.”

The choice is never easy, says Holden, whose four-year term as chair is ending this year. But their selection is tough to contest. “Gerd Faltings is a towering figure in arithmetic geometry,” Holden says. “His ideas and results have reshaped the field.”

The field of mathematics has changed in many ways since Faltings made his major contributions. He doesn’t envy today’s mathematicians racing to tackle the richest open problems, he says. “Now it seems, on everything interesting, there's an enormous bunch of people who do things,” he says. “I’m sort of happy that I don’t have to compete with them.”

As far as excitement at this capstone achievement goes, Faltings doesn’t betray much, even by the stoic standards of German mathematicians. “I’m old, and many things have happened in my life, so I don’t jump around,” he says. “But it’s a very nice thing.”

(http://www.autoadmit.com/thread.php?thread_id=5847479&forum_id=2#49755283)