The birth of a new research field: From solving problems to creating them
What does it mean to introduce AI into computational algebra?
Solving equations is fundamental across many fields, from academic research to industrial applications. However, algorithms in computational algebra can be extremely slow. Depending on the specific setup, the amount of computation required can grow rapidly—even with just five to ten variables—making large-scale practical use difficult. I realized that conventional mathematical approaches might have their limits in terms of speeding up these algorithms. Simply increasing the number of variables while maintaining computational efficiency was becoming increasingly challenging.
For this reason, I began exploring the use of AI—particularly machine learning—to address the issue of processing speed in computational algebra. For example, if an AI system learns from pairs of equations and their solutions, it can begin to recognize the relationships between them and potentially develop strategies for solving equations more efficiently.
What do you see as the key novelty of this research?
The key idea is ‘creating problems.’ Machine learning requires large datasets for learning. In computational algebra, such datasets consist of pairs of equation problems and their solutions. In practice, however, preparing these datasets can be surprisingly difficult. To generate diverse, large-scale, and more efficient graining data, we propose reversing the conventional approach. Instead of only deriving solutions from given problems, I also explore the possibility of creating new problems from known solutions.
In traditional algebra research, the usual approach has been to first formulate a problem and then compute its solution. By contrast, the ‘backward’ approach—starting from a solution and constructing a problem—has received little attention, largely because there was no clear motivation to pursue it. However, the rise of machine learning has changed this situation. The need to generate large datasets for training has given new meaning to this backward approach. By creating problems from known solutions, it becomes possible to generate datasets more efficiently.
In fact, I feel that this line of research is opening new possibilities. Some mathematicians are already making new discoveries through this backward perspective. For me, this sense that afield can expand in unexpected ways is what makes research so exciting.
A chance encounter with my mentor—and a book
Can you tell us what led you to your current research topic?
When I first entered university, I didn’t have a clear idea of what I wanted to pursue. I was generally interested in mathematics and programming, but nothing specific. When I was assigned to a lab, I joined one that was working on optimization methods known as genetic algorithms, which are inspired by biological evolution. Around that time, a professor specializing in bioinformatics—who had just joined the lab—asked me, “Are you interested in algebra?”
By coincidence, I had just bought a book on algebra and had started reading it. I’m not usually the type to actively seek out math books, but I was drawn to the cover and picked up this one on Galois theory. I found it fascinating. It felt like a kind of coincidence—or even fate—and that experience ultimately led me to tackle my current research topic.
After completing my undergraduate studies, I moved to a computer vision lab for my master’s program. However, when I began my doctoral studies, I decided I wanted to return to algebra. I went back to my undergraduate lab and resumed my research in the field. During my undergraduate and master’s studies, my work primarily involved the use of algorithms for computation. In my doctoral research, however, I began combining algebra with algorithmic approaches, allowing me to explore algebra in greater depth.
*Computer vision: A field of technology that enables computers to interpret and understand visual information from images and video. For example, it includes research on facial recognition and object detection, such as identifying cars or other objects.
Your research topics seem quite different between undergraduate and master’s students. How do you see that?
At a fundamental level, I think they are connected through algebra. As an undergraduate, I studied linear algebra. Classical machine learning methods are largely based on linear algebra, so working in that area helped deepen my understanding of the subject.
Later, in my master’s program, I moved into computational algebra, which deals with nonlinear structures and can be seen as a more advanced area of algebra. While my research topics may have appeared quite different on the surface, I felt that linear and nonlinear algebra were connected within my own learning. In that sense, these experiences allowed me to develop a more comprehensive and integrated understanding of algebra as a whole.
Gathering Information Promptly, Pursuing Research Deliberately
AI is a fast-moving field. How do you keep up with the latest developments?
Developments in AI move so quickly that I am conscious of how much information I take in and how I keep up with it. In my lab seminars, three to four members are assigned each week, and each person presents three to four papers. I also regularly share papers through my lab’s messaging platform, which helps us stay up to date as a group.
Another important aspect is attending international conferences. In major AI conferences, all accepted papers are presented as posters, and around 5,000 papers may be accepted at a single conference. This allows me to absorb a tremendous amount of information in a short time. Because the posters are designed to be visually engaging, you can quickly grasp the overall idea of a study just by walking around. If something catches your attention, you can speak directly with the authors, which can sometimes lead to new collaborations.
In contrast, international conferences in computational algebra are much smaller, with around 70 accepted papers. Because you interact with the same researchers throughout the conference, it becomes easier to have in-depth discussions. This environment can be particularly conducive to developing collaborative research.
Based on the information you gather, how do you decide on your research topics?
One thing I keep in mind is choosing research topics that have a high barrier to entry for others. In my case, I draw on my background in algebra—which relatively few researchers specialize in—and work on projects that combine algebra with other fields. Because there is less direct competition, I can take the time to focus deeply on my research.
Another point I’m conscious of is avoiding work that competes on numerical performance. In areas like object recognition or image classification, where the goal is to achieve the highest accuracy, the field moves very quickly, and new methods are constantly emerging. Even if your work is state-of-the-art at one moment, it can be surpassed within a month. Instead of pursuing marginal improvements in performance, I try to focus on research that offers qualitatively different approaches and perspectives.
In my lab, each project is typically carried out by a single student or by myself. As a result, I try not to compete in terms of manpower. Instead, from an educational perspective, I focus on research topics that allow students to take ownership of their work and explore it in depth.
Research that sparks curiosity
How do you see the fusion of algebra and AI developing in the future?
So far, much of AI research has focused on improving benchmark performance. However, I believe the field is gradually shifting towards real-world applications. In the sciences, the trend is often referred to as ‘AI for Science,’ and I expect that computational algebra will also increasingly incorporate AI. Algebra existed long before computers, and with the advent of computing, the field of computational algebra emerged. Now, with the rise of AI, we may see the development of a new area—something like ‘AI Algebra.’ I am not sure what it will ultimately be called, but I hope to contribute to further expanding the field of algebra in this new direction.
In addition, I am working with my students to develop a library that will make it easier for mathematicians to learn and use AI. Our goal is to create an environment in which mathematicians themselves can actively expand the boundaries of their field.
I am also a member of the steering committee for Symbolic Computation and Machine Learning, a new research initiative focused on applying machine learning and deep learning to symbolic computation and computational algebra. Through this effort, I aim to help build a network that further integrates computational algebra and AI.
You believe that interaction and chance encounters are also important in AI research.
Absolutely. When people come together, new ideas emerge, and fields evolve through collaboration and exchange.
To foster that kind of interaction, it is important for one’s research topic and field to be engaging. I believe that qualities such as being interesting and stimulating intellectual curiosity are essential in research. In engineering and technology development, there are often many constraints related to real-world implementation, such as safety and accuracy. In AI in particular, goal-oriented thinking tends to take priority over curiosity. Of course, goal-oriented approaches are important. However, I believe that finding research genuinely interesting is what brings together people from diverse backgrounds and creates opportunities for meaningful exchange.
Having a background outside of informatics or AI can also be a source of uniqueness and lead to impactful research. To make the most of that, I think it’s important to publish your work and use it as a way to build connections with others.
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