AI Startup Solves 4 Unsolved Math Problems
▼ Summary
– Mathematicians Chen and Gendron formulated an unsolved conjecture after encountering a roadblock in their algebraic geometry work.
– The startup Axiom’s AI tool, AxiomProver, successfully generated and verified a proof for this conjecture, finding a connection to a 19th-century numerical phenomenon.
– This proof is part of a series of solutions to previously unsolved math problems generated by Axiom’s AI, demonstrating the technology’s advancing mathematical capabilities.
– Axiom’s system combines large language models with a specialized prover trained to reason and verify proofs in a mathematical language, aiming for applications beyond pure math, like cybersecurity.
– The AI has also produced a complete, start-to-finish proof for another problem (Fel’s Conjecture), showing its ability to work autonomously on novel solutions.
Five years ago, mathematicians Dawei Chen and Quentin Gendron were grappling with a complex problem in algebraic geometry. Their work involved differentials, which are tools from calculus used to measure distances on curved surfaces. They hit a major obstacle when their argument relied on a peculiar number theory formula they couldn’t solve or justify. Ultimately, they published their idea as a conjecture, an unproven proposition, rather than a proven theorem.
Chen recently spent considerable time trying to get ChatGPT to solve this lingering problem, but the AI couldn’t crack it. The breakthrough came unexpectedly at a mathematics conference in Washington, D.C. There, Chen met Ken Ono, a prominent mathematician who had recently joined an AI startup called Axiom. After hearing about the problem, Ono used his company’s AI tool, named AxiomProver, to generate a complete proof by the very next morning. “Everything fell into place naturally after that,” Chen remarked. He collaborated with Axiom to document the proof, which is now available on the arXiv preprint server.
AxiomProver identified a link between the problem and a numerical phenomenon first examined in the 1800s. It then constructed and independently verified the proof. “What AxiomProver found was something that all the humans had missed,” Ono stated. This solution is among several unsolved math problems Axiom claims its system has addressed recently. While not tackling the field’s most famous puzzles, the AI has provided answers to questions that have perplexed specialists for years, showcasing the steady advancement of AI’s mathematical capabilities. Other researchers have also begun reporting success using AI tools to explore new concepts and resolve existing issues.
The methodologies Axiom is developing could have applications far beyond pure mathematics. For instance, similar techniques might be used to create software that is more resistant to specific cybersecurity threats. This would involve employing AI to verify that code is fundamentally reliable and secure. “Math is really the great test ground and sandbox for reality,” said Carina Hong, Axiom’s CEO. “We do believe that there are a lot of pretty important use cases of high commercial value.”
Axiom’s strategy merges large language models with its proprietary AxiomProver system, which is trained to logically reason through mathematical problems to reach verifiably correct solutions. The company notes that its AxiomSolver incorporates several significant advancements and newer techniques beyond similar concepts demonstrated by others in the field. Ono believes the AI-generated proof for the Chen-Gendron conjecture illustrates a meaningful new role for AI in assisting professional mathematicians. “This is a new paradigm for proving theorems,” he explained.
A key differentiator for Axiom’s system is its ability to verify proofs using a specialized mathematical language called Lean. This allows AxiomProver to develop genuinely novel solutions rather than simply searching through existing literature. Another new proof generated by the AI demonstrates this autonomous capability. It provides a solution to Fel’s Conjecture, a problem concerning syzygies, mathematical expressions where numbers align in algebra. Intriguingly, the conjecture involves formulas first discovered over a century ago in the notebook of the legendary mathematician Srinivasa Ramanujan. In this instance, AxiomProver didn’t just fill a gap; it devised the entire proof from start to finish, which has also been detailed in a paper on arXiv.
(Source: Wired)