A human problem: What is the proper role of AI in math research?

OpenAI recently announced that it tapped AI to solve the regularity problem of the Navier-Stokes equations, which describe the movement of liquids and gases. The news sparked controversy within the mathematical community, raising questions about the role of humans and AI tools in math research, as well as academic conduct, priorities and attribution.

Navier-Stokes is one of seven “Millennium Prize Problems” designated by a panel of leading mathematicians in 2000 as key problems to advance the field in the new century. The nonprofit Clay Mathematics Institute offered $1 million prize for the solution to each problem.

Levon Nurbekyan, Emory assistant professor of mathematics, is among those pondering the fallout from OpenAI’s announcement, even as he uses AI himself as a research tool. 

“The technology is very serious and we need to think carefully about how we use it and its impact on the profession and society,” he says.

Nurbekyan led a Research Experience for Undergraduates (REU) at Emory this past summer, which included 11 undergraduate students from across the country, mentored by faculty and postdoctoral fellows in the Department of Mathematics. The REU program, funded by the National Science Foundation, aims to have undergraduates tackle frontier research problems, to help prepare them for careers in academia and industry. The theme for this year’s REU cohort: AI for discovering new mathematics. 

“We need people with high-level math skills who also are skilled at using and interacting with AI in responsible and useful ways,” Nurbekyan says. “This new technology builds directly on the centuries of efforts and achievements by mathematicians. It is absolutely essential to be part of the conversation of where this technology leads the professional community.”

In the following Q&A, Nurbekyan talks about his hopes, and concerns, for the field of mathematics in the AI era.
 

What is your research focus?

I have a wide set of interests that span both theoretical and applied mathematics. To summarize, I study optimal control theory, calculus of variations, partial differential equations, game theory and the interplay between these fields and machine learning. 

It goes both ways. You use machine learning tools to address core questions in these fields. Research questions in these fields can, in turn, be used to improve core machine-learning algorithms. Math has this property of universality that allows for a lot of cross fertilization.

For example, partial differential equations that characterize heat transfer are related to equations that are used to price financial instruments, like derivatives. Generative and large language models are seemingly unrelated to game theory and fluid dynamics, but when you look at the equations underlying them, you see that they are very closely related.

More recently I’ve also become interested in the applications of AI technology in mathematical reasoning.

What sparked your interest in that topic?

Towards the end of 2024, AI systems started doing more impressive things in the space of abstract mathematical reasoning, beyond just doing numerical computations. And this year, there was a series of impressive achievements by frontier AI models cracking some of the most challenging open problems. For example, OpenAI made a splash in May by solving the 80-year-old Planer Unit Distance Problem.

I would say that OpenAI’s use of its internal model to solve the Navier-Stokes problem is an even more significant development, not just because it’s a difficult problem. It’s also a problem that guided parts of the mathematics community in their choice of research questions and activities.

Why was Navier-Stokes selected as a key problem?

Navier-Stokes partial differential equations are used to model the motion of liquids and gases. The models already show remarkable accuracy in real-world applications, such as in designing airplanes, modeling blood flow and weather forecasting. While the models work good enough to do the job in some applications, there is always a gap between reality and a model.

Mathematicians try to understand the behavior and limitations of these models, including how faithfully they describe the real world. It’s not just about knowing the answer to this problem. It’s about greater precision and a deeper understanding of both mathematics and the world it describes.

Applications are, and have always been, an important driver for research in mathematics, but there is also this strange phenomenon that when you’re thinking about hard math problems, you’re growing your understanding of math. And you may come up with tools that can be super useful elsewhere.

For instance, mathematicians developed ideas about complex numbers from the 16th through the 19th centuries, initially in connection with solving polynomial equations, long before their modern and seemingly unrelated applications could have been anticipated. When quantum theory arrived, researchers realized that complex numbers were essential for describing quantum reality. This is not a unique case by any means. There are numerous examples throughout history. 

In a similar spirit, the Navier-Stokes problem was included among the Millennium Prize Problems with the expectation that progress on such a fundamental question could stimulate developments in partial differential equations and beyond.

What are the upsides to using AI in mathematics research?

AI can greatly accelerate your research. It gives you more power. 

For example, [think about] doing literature searches to learn about the history of a problem. Typically, that’s a tedious, time-consuming task. AI systems are pretty good at finding information, summarizing it and explaining it in a contextually relevant way. They hallucinate references sometimes. You need to double-check everything but, nevertheless, AI can accelerate the search process, help take notes and create initial drafts.

AI can also help in the process of creating prototypes and doing initial experiments on ideas. Instead of spending weeks heading in a futile direction, you can get to a more promising direction quicker.

And AI can serve as a good sounding board to bounce ideas back and forth, like a research collaborator. 

I use AI in all these ways. I double-check all the sources and make sure that everyone who is due credit gets it. And, of course, I acknowledge use of the technology.

What are some of the downsides?

AI often removes the social aspect of doing research. Part of the joy of being a mathematician, just like in many professions, is sharing your ideas and your appreciation for the subject with your peers. It’s a human endeavor of trying to understand something better together.

There is a sort of hive-mind aspect to it. The collective mathematics community is an ecosystem carrying knowledge. The way you create, preserve and transform this knowledge is by writing and sharing papers, but also by talking about it. In a paper you’re trying to express your ideas in a compact way and you’re not including your failed attempts or how you arrived at the solution. In conversations at conferences, there’s lots of exchanges of information like this. And you may meet someone that you have great chemistry with and, chances are, you’re going to produce great stuff with that person. 

Another downside of relying too much on AI could be that it removes the natural friction, or slow process, of working hard to understand something. It could be like fast food, in some sense. It’s like if a professor tells you a solution without you first trying to solve a problem yourself. That takes away some of the friction that helps you to learn a new skill, understand a new concept at a deep level, and internalize that concept so that you will never forget it.

Also, an AI output might reduce something to a “yes/no” type of answer without revealing the thought process leading to it. You might miss an opportunity to learn something useful, perhaps a very interesting idea that emerged during the reasoning process but was ultimately discarded because it did not work for the problem at hand.

What is the best way to balance human input and AI?

Mathematicians are actively debating this question. No one knows the right answer or what AI will be capable of tomorrow. There is no consensus that AI is universally good and that it is ushering in a golden age. There is also no consensus that it is universally bad.

AI is changing the way we do math. What is our role now as mathematicians? How do we measure output? How do we attribute credit? When someone uses AI to solve a problem is the person who merely clicked the button the author of the paper? Should this person be credited for clicking the button? These are big, unanswered questions.

It’s important for mathematicians, and for all researchers, to be involved in the conversations about the role of AI. We need to bring our expertise and experience to these discussions so that different viewpoints are grounded in an accurate understanding of the technology and its implications. It’s important that we work together to arrive at informed decisions that are a net benefit for mankind. I’m hopeful that we can do that.

Has using AI changed how you feel about math?

I’ve always loved math and I still do. I just enjoy the process. I started out more in the theoretical side but then realized that I also enjoy seeing applications. It’s almost like magic. You imagine something in your head and then something gets built in the real world and you see that it works like you imagined it would.

For me, AI has so far been mostly positive. I can work faster and still produce high-quality work. I’m making a conscious effort to try to isolate the important and meaningful parts of the research I do from AI so that I learn something new and I improve my understanding of the subject on a deeper level to share with others. At the end of the day, I want to grow as a mathematician, not just maximize the papers I produce.

I go to conferences and I regularly talk to colleagues. The sense of community is very important to me. Even though AI is changing the way we do math, I don’t think it will replace people. 

Fundamentally, mathematicians will still be necessary to internalize, interpret and think of ways to apply the fruits of mathematical research for the benefit of society. It’s a natural alignment that comes from mathematicians being part of that society themselves, even if their role in producing raw mathematical output is modified or reimagined.

What advice do you have for students?

Stay curious and always work to improve your understanding of math by internalizing and thinking deeply about mathematical concepts. AI can serve as a useful tool to help you do that.