Artificial intelligence (AI) is based entirely on mathematics, using algorithms both to learn and make predictions. But can AI be improved to help solve difficult mathematical problems? That was one of the major questions posed at a recent event hosted by the American Institute of Mathematics (AIM), a mathematical sciences research institute supported by the National Science Foundation and housed in the Richard N. Merkin Center for Pure and Applied Mathematics at Caltech.
At the end of May, 40 mathematicians from academia and representatives from AI industry partners gathered at AIM's home in Caltech Hall for a four-day workshop on AI and number theory, a branch of math that studies the properties of integers, or whole numbers. Participants came from across the country, Canada, and the United Kingdom to engage in morning presentations, afternoon discussion groups, and evening social activities.
Presentations included demonstrations of AI for those less familiar with the platforms and recollections by some of the scholars of their previous experiences using AI for mathematical challenges. For example, Andy Booker, a professor of pure mathematics at the University of Bristol in England, discussed an approach he uncovered in collaboration with an AI-operated coding system, Claude Code, for studying the "Converse Problem for L-functions," a central problem in number theory.
On the first day of the workshops, the afternoon was focused on making a big list of benchmark problems that participants wanted to see AI try to tackle.
"The problems vary in difficulty, ranging from impossibly difficult problems that have stumped number theorists for generations to interesting questions that would be worth working on but that would require some effort," says Michael Rubinstein, a professor of mathematics at the University of Waterloo in Ontario, Canada, and one of five co-organizers of the event.
Kicking off events by soliciting a list of problems is a hallmark of AIM workshops; attendees then spend much of the time discussing the problems in breakout groups. After the event, the problems are listed on the organization's website as an open-access resource for anyone to explore.
"The typical AIM workshop focuses deeply on one mathematical problem, with different groups looking at different angles or very fine subproblems," says Alex Meiburg, a co-organizer of the workshop and a research engineer at Harmonic, an AI research lab focused on developing mathematical superintelligence. "This one was more broadly about AI and number theory, with a goal that we all learn and discover a bit about what AI can do for math today and how we hope it can start to help us in the next year or two."
For Meiburg, who is also a postdoctoral research fellow at the Perimeter Institute, an independent research center in foundational theoretical physics located in Waterloo, Canada, some of the biggest takeaways from the event were not directly related to how AI can solve math problems but more about the different ways the technology can help mathematicians.
"What we found is that AI can really speed up the experimenting or brainstorming part of math," he says. For example, Meiburg says AI is really good at coding, reducing what might take a mathematician a couple of weeks of work to just half an hour. "We're entering a big wave where rapid experimentation and investigation is possible."
Rubinstein agrees that AI will help speed up mathematical discovery. He says the frontier large language models (a highly specialized type of AI, such as Anthropic's Claude and OpenAI's ChatGPT) have become very good in the last year at suggesting approaches to math problems and at working them out abstractly.
"The best models have been trained on the entire corpus of human knowledge, and they have also become extremely capable at connecting methods from one area to another area of mathematics," Rubinstein adds.
While many of the problems discussed at the workshops are not close to any application right now, Meiburg says a lot of topics in number theory end up being useful much later in ways no one could expect since they're so basic in nature. Modern cryptography is the classic example.
"This workshop all taught us a lot about how AI can tackle problems, and I expect that the skills we developed at this workshop will be applied to other 'classically hard' problems with more applications," he says. "I'm optimistic that AI can help us make a lot of math more interesting and engaging to different people."
Michele Tarquini, a graduate student in physics, attended the workshop to learn about new problems in mathematics where AI might be able to help make progress and to better understand how mathematicians are thinking about the potential role of AI in math.
"I came in thinking that AI has great and, in some sense, still largely unexplored potential for making progress in theoretical disciplines such as mathematics and theoretical physics," Tarquini says. "It was very interesting to hear other perspectives as well and to see that many mathematicians are becoming increasingly interested in the possibilities of AI for math."
At the same time, he notes that making progress on hard mathematical problems using AI is neither immediate nor easy. Finding the right tools for a specific problem is still difficult, and it takes time to understand whether a given approach is useful. Nonetheless, Tarquini says participants discussed many problems that could potentially be a good fit for AI.
"I believe that developing tools capable of helping solve hard mathematical problems with AI would likely have a significant impact beyond mathematics itself," Tarquini says. "Such tools could influence many other areas of AI and lead to applications in a wide range of fields that affect society more broadly."
Sergei Gukov, the John D. MacArthur Professor of Theoretical Physics and Mathematics and executive director of AIM, says he is thrilled that AIM has hosted its first AI for Math workshop and is eager to plan more.
"The rapid pace of AI development is starting to reshape how we do mathematical research, and many valuable lessons emerged during the event," says Gukov, who is also director of the Merkin Center. "I was grateful and excited to see attendees from several industry partners—such as Anthropic and Harmonic—at the workshops. This is the start of what I hope will be lasting collaborations."
Katie Neith ([email protected])
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