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

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

By Joseph Howlett edited by Clara Moskowitz

The Abel Prize for this year, a prestigious lifetime achievement award in mathematics presented by the Norwegian Academy of Science and Letters and inspired by the Nobel Prize, has been awarded to Gerd Faltings. Faltings, a German mathematician, is renowned for proving the Mordell conjecture in 1983, which is now known as Faltings’s theorem.

This accolade adds to a series of honors Faltings, aged 71, has amassed throughout his career. Among these is the Fields Medal, the most coveted award in mathematics, which he received at the age of 32. “Early in my career, I won the Fields Medal. Now, as my career is winding down, I am receiving the Abel Prize,” Faltings remarks. “It’s a nice symmetry.”

Faltings’s theorem deals with mathematical curves, which can often be represented by simple equations involving two variables. When these equations are plotted on a coordinate grid, they create lines, ellipses, or more intricate, twisting curves.

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Throughout history, mathematicians have sought out a unique subset of solutions called “rational” points on a curve, where the coordinates are integers or fractions. These points have complex interconnections that mathematicians strive to understand.

Despite the infinite nature of curves, identifying all their rational points was once deemed impossible—until Faltings’s Theorem. He demonstrated that if a curve’s equation involves a variable raised to a power exceeding 3, it can only have a finite number of rational points. Only lines, quadratics (such as circles), and cubic equations can have an infinite number.

This proof is regarded as a foundational element in the field of arithmetic geometry, which examines curves and shapes through such equations.

“It’s absolutely fundamental,” notes Noam Elkies, a Harvard University mathematician, regarding Faltings’s proof. “The transition of Mordell’s conjecture into a theorem and the frameworks he established have significantly influenced related fields.”

Mathematicians continue to explore the theorem’s implications, originally proposed by Louis Mordell in 1922. Recently, mathematicians declared they had determined a precise limit on the number of rational points a curve can possess.

Faltings’s theorem is just one among his many achievements. His other significant contributions include a broad generalization of the theorem to multidimensional shapes in 1991 and vital work on “p-adic Hodge theory,” which provides tools to examine such shapes and the equations that define them.

The selection committee, consisting of five members, met at the Institute for Advanced Study in Princeton, N.J., in late January amid a snowstorm that blanketed the Northeast. “There was nothing else to do but sit and discuss mathematics,” said Helge Holden, the committee’s chair, during the Abel Symposium the following week. “The hotel was running low on supplies, making the bread progressively drier.”

Selecting the winner is always challenging, according to Holden, whose four-year term as chair concludes this year. However, their choice is hard to dispute. “Gerd Faltings is a towering figure in arithmetic geometry,” Holden states. “His ideas and results have significantly reshaped the field.”

Mathematics has evolved greatly since Faltings made his pivotal contributions. He expresses no desire to compete with the current wave of mathematicians tackling major open problems. “Nowadays, it seems that any interesting problem attracts a large number of people,” he comments. “I’m somewhat relieved that I don’t need to compete with them.”

As for his reaction to this crowning achievement, Faltings remains composed, adhering to the traditionally stoic demeanor 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.”

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