Gerd Faltings, mathematician who proved the Mordell conjecture, wins the Abel Prize at the age of 71

Gerd Faltings, mathematician who proved the Mordell conjecture, wins the Abel Prize at the age of 71

By Joseph Howlett edited by Clara Moskowitz

This year’s Abel Prize, an annual lifetime achievement award for mathematics awarded by the Norwegian Academy of Sciences and based on the Nobel Prize, has been awarded to Gerd Faltings, a German mathematician best known for proving the influential Mordell conjecture in 1983. This “conjecture” has since been named after him.

The award joins a pile of accolades Faltings, 71, has accumulated during his long career. That list includes the Fields Medal, mathematics’ most coveted award, which Faltings won at age 32. “Near the beginning of my career, I got the Fields Medal. And toward the end, I get the Abel Prize,” Faltings says. “It’s a nice duality.”

Falting’s theorem is about curves. Often these can be described with simple equations with two variables that are multiplied and added together. Map the solutions of such an equation onto a coordinate grid, and they will form a line or an ellipse or a more complicated, convoluted curve.

On supporting science journalism

If you like this article, please consider supporting our award-winning journalism by subscribes. By purchasing a subscription, you help secure the future of impactful stories about the discoveries and ideas that shape our world today.

Since the dawn of mathematics, people have been looking for a rarefied subset of these solutions—”rational” points on the curve, where the coordinates are integers or fractions. These particular points have rich and complicated relationships that belie a hidden order that mathematicians aim to uncover.

But there is an infinity of curves out there, and it seemed impossible to nail down all their rational points – until Falting’s theorem, that is. He proved that if the equation of a curve has a variable raised to a power higher than 3, it must have a finite number of these points. Only lines, squares (like circles) and cubic equations can have an infinite number.

The proof is considered a cornerstone of arithmetic geometry, the field that studies curves and shapes represented by this type of equation.

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

Mathematicians are still working out the ramifications of the theorem, which was originally hypothesized 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.

The theorem that bore his name was just one of Falting’s many mathematical achievements. 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 for studying 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, NJ, near the end of January—just as a winter storm blanketed the Northeast in feet of snow. – We had nothing else to do but sit down and discuss mathematics, said Helge Holden, the committee’s chairman, at the Abel symposium, an event held the following week. “The hotel was short on supplies, so the bread got drier and drier.”

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

The field of mathematics has changed in many ways since Faltings made his major contributions. He does not envy today’s mathematicians who race to tackle the richest open problems, he says. “Now it seems that on anything interesting there is a huge group of people doing things,” he says. “I’m kind of glad I don’t have to compete with them.”

In terms of excitement about this pinnacle enterprise, Faltings doesn’t betray much, even by the stoic standards of German mathematicians. “I’m old, and a lot of things have happened in my life, so I don’t jump around,” he says. “But it’s a very nice thing.”

It’s time to stand up for science

If you liked this article, I would like to ask for your support. Scientific American has served as an advocate for science and industry for 180 years, and right now may be the most critical moment in its two-century history.

I have been one Scientific American subscriber since I was 12 years old, and it helped shape the way I see the world. SciAm always educates and delights me, and inspires a sense of awe for our vast, beautiful universe. I hope it does for you too.

If you subscribe to Scientific Americanyou help ensure our coverage is centered on meaningful research and discovery; that we have the resources to report on the decisions that threaten laboratories across the United States; and that we support both budding and working scientists at a time when the value of science itself is too often not recognised.

In return, you receive important news, captivating podcasts, brilliant infographics, can’t-miss newsletters, must-see videos, challenging games, and the world of science’s best writing and reporting. You can even give someone a subscription.

There has never been a more important time for us to stand up and show why science is important. I hope you will support us in that mission.