An 87-Year-Old Question
The problem at the heart of this decades-long quest is known as the Jacobian conjecture. It was first proposed in 1939 by the German mathematician Ott-Heinrich Keller. In simple terms, imagine a machine that takes a set of numbers as input and, using
a specific type of mathematical recipe called a polynomial transformation, produces a new set of numbers. The conjecture asked a fundamental question: if the machine’s recipe passes a certain simple test (its Jacobian determinant is a non-zero constant), can you always reverse the process? That is, given the output numbers, could you always work backward to find the original inputs? For decades, the answer was assumed to be 'yes,' but no one could prove it was true in every single case. Finding just one example where the machine passed the test but couldn't be reversed would be enough to break the entire conjecture.
Why It Mattered So Much
The Jacobian conjecture may sound abstract, but it sat on a prestigious list of unsolved problems for the 21st century compiled by the influential mathematician Stephen Smale. Its difficulty lies not in its complexity but in its deceptive simplicity. It’s easy to state, but proving it has been maddeningly difficult. Generations of algebraists and geometers have tried and failed to provide a definitive proof. The problem became a sort of mathematical Everest — a challenge that represented the limits of human understanding in a specific field of algebra. Its resistance to proof made it a famous, stubborn mystery. Mathematicians believed that solving it would unlock deeper insights into the nature of polynomial mappings, a cornerstone of many areas of science and engineering.
A Modern Breakthrough
The potential breakthrough came from an unexpected direction in July 2026. Levent Alpöge, a mathematician at Harvard University, decided to hunt for a counterexample with the help of an experimental AI model from Anthropic known as Fable. Instead of trying to construct an elaborate proof, their goal was to find a single, concrete example that violated the conjecture. Working together, they found one. The proposed counterexample is a remarkably compact formula — just 216 characters long — that describes a transformation that passes the initial test but appears to be irreversible. Alpöge announced the discovery on social media, cheekily noting that the AI had been working on it during the World Cup final. While the finding still needs to be formally verified by the global mathematics community, the fact that the counterexample is short and concrete means that other experts can check it relatively easily.
A New Era of Discovery?
This event is significant not just for potentially closing the book on an 87-year-old problem, but for what it says about the future of science. For a long time, the highest levels of mathematical reasoning were seen as a uniquely human domain, requiring intuition and creativity that machines lacked. This collaboration, however, showcases a new model: human expertise guiding the raw processing power of AI to navigate a maze of possibilities too vast for any person to check alone. The AI’s ability to find this needle in a mathematical haystack suggests that artificial intelligence could become a powerful partner for researchers. It points toward a future where human intuition directs AI exploration, allowing us to tackle complex problems that have remained out of reach for generations.
