AI cracks Jacobian conjecture and unsettles mathematicians

An AI model has resolved the Jacobian conjecture, a problem rooted in Ott-Heinrich Keller’s work since 1939 and centered on when mathematical maps can be reversed from outputs to inputs. The result, announced by Anthropic employee Levant Alpöge, produced a counterexample whose Jacobian determinant holds steady at −2 everywhere while sending three different starting points to the same destination.

The result follows a rapid run of model-driven math advances since mid-2025, including systems solving five of six problems at the International Mathematical Olympiad and an OpenAI model disproving an 80-year-old Erdős conjecture on combinatorial geometry in May. In June, 16 researchers from 15 universities issued the Leiden Declaration on Artificial Intelligence and Mathematics, calling for guardrails around transparency, attribution, and peer review.

Mathematicians are split between excitement and concern. Kevin Buzzard of Imperial College London called it a big day but said current models often provide the “how” without the “why,” while University of Chicago mathematician Akhil Mathew said verified answers still need a coherent story. Buzzard’s Lean proof-checking language had already checked the result, underscoring how proof-writing models paired with machine verification could erode one of the field’s remaining human advantages.

The shift is landing as the profession faces pressure from funding cuts and shrinking doctoral admissions. Federal funding for mathematics research has fallen roughly 72% under the Trump administration’s cuts to the National Science Foundation, while PhD admissions at top research universities are down 15% this fall. Buzzard argued that humans may still matter most in choosing the right questions, a skill he said machines handle poorly.