Artificial intelligence (AI) is based entirely on mathematics and uses algorithms for both learning and prediction. But can we improve AI to solve difficult mathematical problems? This was one of the key questions posed at a recent event hosted by the American Institute of Mathematics (AIM). AIM is a Mathematical Sciences Research Institute within the Richard N. Merkin Center for Pure and Applied Mathematics at the California Institute of Technology, supported by the National Science Foundation.
At the end of May, 40 academic mathematicians and representatives from AI industry partners gathered at AIM’s headquarters in Caltech Hall for a four-day workshop on AI and integers, the branch of mathematics that studies the properties of whole numbers. Participants came from across Canada and the United Kingdom to participate in morning presentations, afternoon discussion groups, and evening social activities.
The presentation included a demonstration of AI for those less familiar with the platform, as well as a recollection of past experiences using AI for mathematical challenges. For example, Andy Booker, a professor of pure mathematics at the University of Bristol in the UK, discussed an approach he discovered working with Claude Code, an AI-powered coding system, to study a central problem in number theory: the L-function inverse problem.
The afternoon of the first day of the workshop was focused on creating a large list of benchmark problems that participants would like the AI to tackle.
“Problems vary in difficulty, ranging from the impossibly difficult problems that have puzzled number theorists for generations to interesting problems that are worth tackling but require some effort,” says Michael Rubinstein, a mathematics professor at the University of Waterloo in Ontario, Canada, and one of the five co-organizers of the event.
Starting the event by soliciting a list of issues is a hallmark of AIM workshops. Attendees then spend most of their time discussing the issues in breakout groups. After the event, the issue will be listed on the organization’s website as an open-access resource for anyone to explore.
“A typical AIM workshop focuses deeply on one mathematical problem, with different groups looking at different angles or very fine-grained subproblems,” said Alex Meiburg, co-organizer of the workshop and research engineer at Harmonic, an AI research lab focused on developing mathematical superintelligence. “This is about AI and number theory more broadly, and is intended for all of us to learn and discover what AI can do for mathematics today and how we hope it will help us in the next year or two.”
For Myburgh, who is also a postdoctoral fellow at the Perimeter Institute, an independent research center for fundamental theoretical physics in Waterloo, Canada, some of the biggest takeaways from the event were not directly related to how AI can solve math problems, but about the different ways the technology can help mathematicians.
“What we found is that AI can really speed up the experimentation and brainstorming part of math,” he says. For example, Myburgh says AI is so good at coding that it can reduce work that would take a mathematician weeks to just 30 minutes. “We are entering a big wave of rapid experimentation and investigation.”
Rubinstein agrees that AI can help speed up mathematical discovery. He says Frontier’s large language models (highly specialized types of AI such as Anthropic’s Claude and OpenAI’s ChatGPT) have gotten much better in the last year at proposing approaches to mathematical problems and solving them abstractly.
“The best models are trained on the entire corpus of human knowledge and are also very capable of connecting methods from one area of mathematics to another,” Rubinstein added.
Although many of the problems discussed at the workshop have no current application, many topics in number theory are so fundamental in nature that they end up being useful in ways no one expected much later, Myburgh says. Modern cryptography is a prime example.
“All this workshop taught us a lot about how AI can address problems, and we hope that the skills we developed in this workshop will be applied to other ‘classically hard’ problems in more applications,” he says. “I’m optimistic that AI will make a lot of mathematics more interesting and appealing to a variety of people.”
Michele Tarquini, a physics graduate student, attended the workshop to learn about emerging problems in mathematics that AI could help advance and to better understand how mathematicians are thinking about the potential role of AI in mathematics.
“I have always thought that AI has a huge, in some ways, largely untapped potential to advance theoretical fields such as mathematics and theoretical physics,” Tarquini says. “It was really interesting to hear other perspectives and see how many mathematicians are becoming increasingly interested in the potential of AI in mathematics.”
At the same time, he points out that using AI to solve difficult mathematical problems is not immediately easy. Finding the right tool for a particular problem remains difficult, and it takes time to understand whether a particular approach will help. Nevertheless, Tarquini said participants discussed many issues that could be well-suited to AI.
”Developing tools that can help AI solve difficult mathematical problems will likely have a profound impact beyond mathematics itself, Tarquini says. “Such tools have the potential to impact many other areas of AI and lead to applications in a wide range of fields that impact broader society.”
Sergei Gukov, John D. MacArthur Professor of Theoretical Physics and Mathematics and AIM Executive Director, said he is excited that AIM hosted the first AI for Math workshop and hopes to plan more.
“The rapid pace of AI development is beginning to change the way mathematical research is done, and many valuable lessons were learned during the event,” said Gukov, who is also director of the Markin Center. “I was grateful and excited to have participants from several industry partners, including Anthropic and Harmonic, join us for the workshop. This is the beginning of what I hope will be a lasting collaboration.”
