“Mathematics in the age of AI” examines how the mathematical community might respond if artificial-intelligence tools become capable of research-level mathematical work. The essay treats the arrival of those capabilities as a hypothesis rather than debating whether or when they will emerge. The work shifts attention from AI capability to the goals and values of mathematical research. It uses mathematics’ problem-solving component as a case study for considering what researchers value beyond the production of solutions. The essay is based on a public lecture delivered at the 2026 International Congress of Mathematicians and was submitted to arXiv on August 17, 2026.

Palomar, a registry for Lean-verified mathematics incubated by the Lean FRO and ICARM, has opened for submissions. It records fixed GitHub repository snapshots containing formal Lean proofs of old or new mathematical results, including work produced by humans, AI systems, or both. Terence Tao serves on its scientific advisory board alongside Jeremy Avigad, Matthew Ballard, Jaume de Dios, Nestor Guillen, Bryna Kra, Kim Morrison, Ravi Vakil, and Akshay Venkatesh. Each submission must include a challenge file that states the claimed results in human-readable Lean, a solution module containing proofs, and a formalization.yaml metadata file with an informal description and disclosures. Palomar mechanically uses Lean Comparator to verify that the solution typechecks and proves the challenge file’s statements. A large language model separately assesses whether the informal description appears to match those formal statements, while the registry also checks minimal repository standards. Registration does not constitute peer review for novelty, interest, or mathematical accuracy. Tao reported successfully submitting his recent Lean formalization of the proof of Sendov’s conjecture as a test and plans to submit older formalizations. AI agents can assist with submission mechanics, though Palomar recommends human review.

Terence Tao explores the intersection of mathematics and artificial intelligence, emphasizing how AI can enhance mathematical research and problem-solving. He discusses the implications of AI for the future of mathematical discovery and education.

The Jacobian conjecture posits that a polynomial map in complex variables with a non-zero constant Jacobian is invertible with a polynomial inverse. Local invertibility is equivalent to the Jacobian being non-zero, following from the inverse function theorem.

The Jacobian conjecture is stated to be false, with a specific mapping from \(\mathbb{C}^3\) to \(\mathbb{C}^3\) shown to have a Jacobian determinant of -2. The mapping sends the points (0, 0, -1/4), (1, -3/2, 13/2), and (-1, 3/2, 13/2) to (-1/4, 0, 0).

Aletheia, powered by Gemini 3 Deep Think, autonomously solved 6 out of 10 problems in the FirstProof challenge. Expert assessments confirmed the accuracy of Aletheia's solutions for the problems completed.