Gerd Faltings has won the 2026 Abel Prize
Peter Badge/Typos1
Gerd Faltings has won the 2026 Abel Prize, considered the Nobel prize of mathematics, for a groundbreaking proof which took mathematics by storm in 1983. His contributions helped 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 longstanding theorem first proposed by the Louis Mordell in 1922 which argues that increasingly complicated equations produce fewer solutions.
Faltings, who is based at the Max Planck Institute for Mathematics in Germany, says he was “honoured” when he found out the news, but was reserved about the impact of his achievements. “Somebody said, about climbing Mount Everest, it’s because it’s there and it was a problem,” says Faltings. “I solved [the Mordell conjecture], but in the end it doesn’t allow us to cure cancer or Alzheimer’s, it’s just extending our knowledge of things.”
The Mordell conjecture concerns Diophantine equations, a vast category which includes famous equations like a² + b² = c² from the Pythagorean theorem and aⁿ + bⁿ = cⁿ, which is at the centre 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 amount.
If these equations are rewritten with complex numbers, a kind of 2-dimensional number, and then plotted out as surfaces, like spheres or donuts, Mordell’s insight was that it is the number of holes the surface contains that determines how many solutions exist. Mordell intuited that for surfaces that had more holes than a donut, then there would only ever be a finite number of rational solutions, which are solutions using either whole numbers or fractions, but he couldn’t prove it.
When Faltings finally proved Mordell’s hunch more than six decades later, it surprised mathematicians not just for the result but in how he went about it. His proof combined ideas from seemingly disparate mathematical disciplines, like geometry and arithmetic. “It’s very short, it’s like a miracle,” says Akshay Venkatesh at the Institute for Advanced Study in Princeton. “It’s this paper of just 18 pages, and it intricately skips 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 may work out. “Sometimes I get ahead of people who try to prove everything right away, 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. “One thinks, how did he have the confidence to embark on this without knowing yet how these pieces are going to come together?”
Many of the conjectures that Faltings solved and the tools that he developed as part of his Mordell proof went on to form the foundations of some of the largest areas of mathematical research today, like p-adic Hodge theory, which examines the links between a shape’s curves and its structure, but using number systems quite unlike our own. He also directly influenced landmark developments in modern mathematics, such as paving the way for Andrew Wiles’ proof of Fermat’s Last Theorem, and mentoring Shinichi Mochizuki, the Japanese mathematician who controversially claims he has proven the abc conjecture.
Faltings says he did not intend to work on problems with such an outsize impact. “My idea has been, I shouldn’t look at what may make me famous and rich, but I try to find things which I like,” says Faltings. “Because if you work on things which you like, it’s more fun.”
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