Mathematics
Source reports covering Mathematics. Stories will appear when analysis and community reactions are ready.
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Source reports
Will it be a golden age or a dark age for mathematics (and science)?
opinion
17 min read
Francesco Maggi argues that large language models could accelerate mathematical and scientific discovery but could also concentrate knowledge and weaken independent human research. He cites LLM-generated work including a finite-time blow-up solution for the forced Navier–Stokes equations and a proposed proof of the quasi-Riemann Hypothesis, while stressing that mathematicians must independently scrutinize, interpret, and assimilate such results. Lean, a proof assistant, can formally verify proofs, but important machine-generated Lean proofs may run to hundreds of thousands of lines and remain difficult to translate into understandable natural language. Maggi says theorem proving alone does not define mathematical progress: mathematicians also create new concepts and formal languages through “mathematization,” as exemplified by Newton, Riemann, Einstein, Shannon, and Grothendieck. He argues that authorship, citation, peer review, confidentiality, and the evaluation and training of researchers require new standards when LLMs contribute to research. He also warns that proprietary frontier systems could give a few companies disproportionate control over scientific priorities and capabilities. Maggi calls for coordinated, publicly governed scientific LLM infrastructure, researcher access to frontier systems, disclosure and reproducibility standards for LLM-generated scholarship, confidentiality protections, and research into effects on education and training. He argues these measures could preserve an open, intellectually independent scientific culture while using LLMs to support theory-building across mathematics and other sciences.
Mathematics in the age of AI
research
1 min read
“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 of Lean verified mathematics
research
3 min read
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.
What sort of maths are LLMs good at?
opinion
28 min read
OpenAI has solved ten significant problems in mathematics and theoretical computer science, including the construction of a non-sofic group and a proof regarding the superexponential growth of the multicolour Ramsey number. These achievements demonstrate the capabilities of large language models (LLMs) in tackling complex mathematical challenges.
Terence Tao: Mathematics in the Age of AI [pdf]
research
1 min read
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 Dark Night of Mathematics
opinion
7 min read
Recent developments in large language models (LLMs) have generated counterexamples to several long-standing mathematical conjectures. This has sparked varied reactions among mathematicians and enthusiasts, leading to discussions about the implications for the field of mathematics.
Fields Medals 2026
news
1 min read
The Fields Medal is awarded for exceptional mathematical achievement, recognizing both past work and future potential. For 2026, the prize funds are provided by the University of Toronto and supported by the Fields Institute, with a recipient recognized for contributions to partial differential equations and the Boltzmann equation.
A digestion of the Jacobian conjecture counterexample
research
12 min read
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.
Claude Fable produced a counterexample to the Jacobian Conjecture
research
1 min read
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).
Mathematicians issue warning as AI rapidly gains ground
news
2 min read
Sixteen mathematicians have issued the Leiden Declaration on Artificial Intelligence and Mathematics, expressing concerns about the rapid advancement of AI in their field. The document serves as both a warning and a call to action regarding the implications of AI for mathematical research and practice.
Amateur armed with ChatGPT solves an Erdős problem
news
4 min read
Liam Price, a 23-year-old without advanced mathematics training, solved a 60-year-old problem previously unsolved by mathematicians using ChatGPT Pro. Recent advancements in artificial intelligence have led to solutions for several conjectures known as Erdős problems.
Google's Aletheia AI Agent Autonomously Solves 6/10 Novel FirstProof Math Problems
research
1 min read
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.
Mathematicians issue a major challenge to AI—show us your work
news
3 min read
Top mathematicians have issued a challenge to AI systems to solve a set of unsolved mathematical problems within a week, as part of an initiative called “First Proof.” These problems are newly formulated and are not present in any large language model's training data.