Axiom Math, a company situated in Palo Alto, California, has unveiled a complimentary new AI resource for mathematicians, crafted to identify mathematical patterns that may unveil answers to enduring challenges.
This instrument, termed Axplorer, is a revamped version of a previous model called PatternBoost that François Charton, who is now a research scientist at Axiom, helped develop in 2024 during his tenure at Meta. While PatternBoost utilized a supercomputer, Axplorer operates on a Mac Pro.
The goal is to empower users with the capabilities of PatternBoost, which was instrumental in solving a challenging math conundrum recognized as the Turán four-cycles problem, making it accessible to anyone capable of installing Axplorer on their respective computers.
Recently, the US Defense Advanced Research Projects Agency initiated a program named expMath—an abbreviation for Exponentiating Mathematics—to motivate mathematicians to create and leverage AI resources. Axiom considers itself a contributor to that effort.
According to Charton, breakthroughs in mathematics have monumental implications across technology. In particular, novel mathematics is vital for advancements in computer science, from developing next-gen AI to enhancing internet security.
Many achievements with AI tools have focused on resolving existing issues. Yet solving problems is not the sole function of mathematicians, states Axiom Math’s founder and CEO, Carina Hong. She notes that mathematics is exploratory and experimental.
MIT Technology Review interviewed Charton and Hong last week in an exclusive video discussion about their innovative tool and the potential impact of AI on mathematics.
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In recent months, several mathematicians have employed LLMs, such as OpenAI’s GPT-5, to derive answers for unresolved issues, especially problems posed by the 20th-century mathematician Paul Erdős, who left behind numerous challenges upon his passing.
However, Charton is skeptical about those triumphs. “There are countless problems that remain unsolved merely because nobody has examined them, and it’s straightforward to uncover a few treasures that you can tackle,” he states. He aims to focus on the more formidable challenges—“the significant problems that have been thoroughly researched and have been tackled by prominent figures.”
The Turán four-cycles problem that PatternBoost addressed is one such challenge, according to Charton. (This problem holds considerable importance in graph theory, a mathematical field employed to study complex networks like social media, supply chains, and search engine rankings. Picture a page dotted with points. The challenge involves determining how to connect as many of those points as possible without forming loops that involve four consecutive points.) Axiom Math claims it has utilized Axplorer to match or exceed the best-known outcomes for two additional major problems in graph theory as well.
“LLMs excel if your objective is a derivative of something previously accomplished,” remarks Charton. “This isn’t unexpected—LLMs are trained on all existing data. But one might argue that LLMs are conservative. They tend to recycle established concepts.”
Nevertheless, numerous mathematical challenges demand fresh ideas or insights that have never been encountered before. Often, such insights arise from recognizing patterns that were previously unnoticed. These discoveries can lead to entirely new branches of mathematics.
PatternBoost was created to assist mathematicians in identifying new patterns. Provide the tool with an example, and it will generate similar instances. You can select the intriguing ones and feed them back in. The tool then produces more like those, and this process continues.
It’s akin to Google DeepMind’s AlphaEvolve, a system that utilizes an LLM to devise innovative solutions for challenges. AlphaEvolve retains the best suggestions and prompts the LLM to enhance them.
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Researchers have effectively used both AlphaEvolve and PatternBoost to uncover new solutions for longstanding mathematical issues. The drawback is that those tools operate on extensive GPU clusters and are not accessible to the majority of mathematicians.
Mathematicians are enthusiastic about AlphaEvolve, according to Charton. “However, it’s restricted—you must have access. You need to request the DeepMind representative to input your problem on your behalf.”
When Charton resolved the Turán problem with PatternBoost, he was still affiliated with Meta. “I literally had thousands, sometimes tens of thousands, of machines at my disposal,” he explains. “It operated for three weeks. It was an embarrassingly brute-force approach.”
According to the Axiom Math team, Axplorer is considerably faster and more efficient. Charton notes that it took Axplorer a mere 2.5 hours to replicate the Turán result achieved by PatternBoost. Moreover, it runs on a single machine.
Geordie Williamson, a mathematician from the University of Sydney, who collaborated on PatternBoost with Charton, has yet to experiment with Axplorer. Nonetheless, he is eager to observe how mathematicians will utilize it. (Williamson still occasionally partners with Charton on academic endeavors but indicates he is not otherwise affiliated with Axiom Math.)
Williamson asserts that Axiom Math has made multiple enhancements to PatternBoost that (in theory) render Axplorer applicable to a broader spectrum of mathematical issues. “It remains to be seen how substantial these improvements are,” he adds.
“We find ourselves in an unusual period right now, where numerous companies present tools they wish us to utilize,” Williamson states. “I would suggest that mathematicians are slightly inundated by the possibilities. The effect of having yet another tool like this remains unclear to me.”
Hong acknowledges that a multitude of AI tools are currently being marketed to mathematicians. Some even necessitate mathematicians to train their own neural networks. This is a detraction for her, as she is a mathematician herself. Instead, Axplorer provides a guided experience, taking you through your desired process step by step, she mentions.
The Axplorer source code is open to the public and accessible via GitHub. Hong is hopeful that students and researchers will employ the instrument to create sample solutions and counterexamples for the problems they are addressing, thereby accelerating mathematical discoveries.
Williamson welcomes innovative tools and states that he frequently uses LLMs. However, he does not believe mathematicians should abandon traditional methods just yet. “In my biased perspective, PatternBoost is a wonderful concept, but it is certainly not a cure-all,” he remarks. “I would be keen for us not to overlook more practical strategies.”
