Gerd Faltings has won the 2026 Abel Prize
Peter Badge/Typos1
Gerd Faltings has been awarded the 2026 Abel Prize, often described as the Nobel Prize for mathematics, for a transformative proof that revolutionized the field in 1983. His work has been pivotal in shaping the field of arithmetic geometry, a cornerstone of modern mathematics.
Faltings’ most notable accomplishment, which also earned him the prestigious Fields Medal in 1986, was solving the Mordell conjecture. This longstanding theorem, introduced by Louis Mordell in 1922, posits that more complex equations tend to have fewer solutions.
Based at the Max Planck Institute for Mathematics in Germany, Faltings expressed his honor at receiving the news but remained modest about the significance of his work. “Someone said, about climbing Mount Everest, it’s because it’s there and it was a problem,” Faltings remarked. “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 deals with Diophantine equations, a broad class that includes well-known equations such as a² + b² = c² from the Pythagorean theorem and aⁿ + bⁿ = cⁿ, central to Fermat’s Last Theorem. Mordell sought to determine which of these equations, in their more generalized forms, possess infinitely many solutions and which have only a finite number.
When these equations are expressed using complex numbers, which are two-dimensional, and visualized as surfaces like spheres or donuts, Mordell’s intuition was that the number of holes in the surface dictates the number of solutions. He hypothesized that surfaces with more holes than a donut would have only a finite number of rational solutions, which are solutions using whole numbers or fractions, but he was unable to prove it.
Faltings’ eventual proof of Mordell’s insight, over six decades later, astonished mathematicians both for its outcome and its innovative approach. His proof seamlessly integrated concepts from distinct mathematical fields, such as geometry and arithmetic. “It’s very short, it’s like a miracle,” commented Akshay Venkatesh from 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 attributes his success to his comfort with uncertainty and willingness to explore ideas that might not be immediately verifiable but that he senses could be fruitful. “Sometimes I get ahead of people who try to prove everything right away, but sometimes I also go astray,” he explained.
“One of the impressive things about his argument is that it covers so much, and the pieces have to fit together,” observed Venkatesh. “One thinks, how did he have the confidence to embark on this without knowing yet how these pieces are going to come together?”
The conjectures Faltings resolved and the methodologies he developed in proving the Mordell conjecture have laid the groundwork for significant areas of mathematical research, such as p-adic Hodge theory, which studies the correlation between a shape’s curves and its structure using unconventional number systems. His work also directly influenced major developments in mathematics, including Andrew Wiles’ proof of Fermat’s Last Theorem, and he has mentored Shinichi Mochizuki, the Japanese mathematician who controversially claims to have solved the abc conjecture.
Faltings maintains that his goal was never to work on problems for fame or fortune. “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,” he stated. “Because if you work on things which you like, it’s more fun.”
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