Gerd Faltings has won the Abel Prize 2026
Peter mark/typo1
Gerd Faltings has won the 2026 Abel Prize, considered the Nobel Prize in mathematics, for a ground-breaking proof that took mathematics by storm in 1983. His contribution helped to establish one of the most important fields in modern mathematics, arithmetic geometry.
The crowning achievement of Faltings, who also won the prestigious Fields Medal in 1986 for the same work, was proving the Mordell Conjecture, a long-standing theorem first proposed by Louis Mordell in 1922 that asserts that increasingly complicated equations yield fewer solutions.
Faltings, who is based at the Max Planck Institute for Mathematics in Germany, says he was “honored” when he found out the news but was reserved about the impact of his achievements. “Someone said that about climbing Mount Everest, it’s because it’s there and it was a problem,” Faltings says. “I solved (the Mordell conjecture), but in the end it doesn’t allow us to cure cancer or Alzheimer’s, it just expands our knowledge of things.”
The Mordell conjecture applies to Diophantine equations, a huge category that includes famous equations such as a² + b² = c² from the Pythagorean theorem and aⁿ + bⁿ = cⁿ, which is at the center of Fermat’s famous last theorem. Mordell wanted to understand which of these Diophantine equations, in their more general form, have infinitely many solutions, and which have only a finite number.
If these equations are rewritten with complex numbers, a kind of 2-dimensional number, and then plotted as surfaces, like spheres or doughnuts, Mordell’s insight was that it is the number of holes the surface contains that determines how many solutions there are. Mordell believed that for surfaces that had more holes than a donut, there would only always be a finite number of rational solutions, which are solutions that use either whole numbers or fractions, but he could not prove it.
When Faltings finally proved Mordell’s hunch more than six decades later, it surprised mathematicians, not only in the result, but in how he went about it. His proof combined ideas from seemingly different mathematical disciplines, such as geometry and arithmetic. “It’s very short, it’s like a miracle,” says Akshay Venkatesh of the Institute for Advanced Study in Princeton. “There is this article in only 18 pages, and it jumps intricately between different techniques and different intuitions.”
Faltings credits his success to being comfortable with uncertainty, and taking risks on ideas that may not be proven but that he has a hunch might work out. “Sometimes I get in front of people who try to prove everything at once, but sometimes I also go astray,” says Faltings.
“One of the impressive things about his argument is that it covers so much, and the pieces have to fit together,” says Venkatesh. “You think, how did he have the confidence to take this on without knowing yet how these pieces are going to fit together?”
Many of the conjectures that Faltings solved and the tools he developed as part of the Mordell proof formed the basis of some of the largest areas of mathematical research today, such as p-adic Hodge theory, which examines the connections between a shape’s curves and its structure, but uses number systems quite unlike our own. He also directly influenced the development of modern mathematicssuch as paving the way for Andrew Wiles’ proof of Fermat’s Last Theorem, and mentoring Shinichi Mochizuki, the Japanese mathematician who controversially claims to have proved the abc conjecture.
Faltings says he did not intend to work on issues with such a large impact. “My idea has been that I shouldn’t look at what can make me famous and rich, but I try to find things that I like,” says Faltings. “Because if you’re working on things you like, it’s more fun.”
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