## Meeting mathematics “for real” in college

Associate Professor Hidetaka Sakai, the recipient of the 2019 MSJ Analysis Prize, is one of the leading researchers in the Japanese mathematical community. He was described as “an internationally esteemed, global leader in the study of Painlevé equations” when he received the award. Sakai himself, however, is not so sure, saying, "I don't remember why I decided to specialize in mathematics, I just somehow ended up doing so.” When asked if he liked elementary mathematics when he was a child, he replies, "I don't remember much about that either, but I suppose I did. I was not good at memorizing, so I did not do well on exams," he laughs and tilts his head in contemplation. “It was in college (Kyoto University) that I met mathematics “for real,” and it was something completely different from what I had seen before. It seemed very new to me.” That is when he realized the depth, mystery, and philosophical aspects of the “universe” called mathematics… and he was captivated.

“I think that there are stories in mathematics. You can create your own story by proposing an idea that you find interesting. That was what made me want to do this as a profession. When I decided to go to graduate school, I did not think much about finding a job or anything like that. Looking back, I think it was a daring choice that I made without thinking about the future (laughs), but that is how much fun I had studying mathematics.”

In Sakai's own words, the “fascination” is fun.

“Maybe it is more like a game. A game you can play forever."

The story of Sakai and the Rubik’s Cube when he was a junior in college is a prime example of such “fun.”

“When I was a child, I bought a Rubik's Cube, which came with a guide so that anyone could solve it. But I didn't want to look at the guide, so in the end, I gave up without ever solving the Rubik's cube. Then I learned in a college lecture about group theory that if a single operation has only a finite number of states and you repeat the same operation, you will sooner or later return to the original state. At that moment, it struck me that I could solve the Rubik's Cube using this principle. That day, when I got home, I fell into a trance. I kept playing with the Rubik's Cube until, at last, I managed to solve it."

From the time when he was a university student to this day, mathematics has meant "fun and fascination" above all else for Sakai.

## Finding "fascination": the Painlevé equations

Sakai considers walking his hobby and pondering various issues, he often followed the path called the “Philosopher’s Walk” near Kyoto University when he was a student there.

“I always ruminated over trivial things at the time,” Sakai laughs, “but I think many mathematicians like walking as it often helps deepen one's thinking,” he adds.

The Philosopher's Walk is a charming two-kilometer path along a narrow canal between Ginkaku-ji Temple and Nanzen-ji Temple. It is called the Philosopher’s Walk because Kitaro Nishida, a philosopher from Kyoto University, loved to walk along this path. Incidentally, two mathematicians from Kyoto University whom Sakai admires may have also taken strolls in contemplation.

“There are many mathematicians of all ages and origins whom I respect and love, but if I had to name just a few, I would have to say Professor Michio Jimbo (professor emeritus both at Kyoto University and at the University of Tokyo), who was teaching at Kyoto University when I was there, and Professor Mikio Sato (1928-2023), who was his professor.”

Mikio Sato had already retired when Sakai enrolled at Kyoto University, but he continues to be a monumental presence for Sakai.

“Professor Sato is a mathematician renowned for building the theory of hyperfunctions. I am very much influenced by him, and I wish I could do the kind of mathematics he did. His vision was vast, but mine is still too narrow to produce results," says Sakai modestly. However, Sakai has already made various world-class "discoveries," earning him the aforementioned MSJ Analysis Prize. The subject of his research is Painlevé equations.

The Painlevé equations are differential equations (more precisely, nonlinear ordinary differential equations) discovered around 1900 by Paul Painlevé (1863-1933), a mathematician who later served as prime minister of France. The six types of equations, though according to Sakai, eight types would be more appropriate, share the so-called “Painlevé property.” Immediately after their discovery, the equations were studied and analyzed extensively by many mathematicians. However, a mere decade or so later, they disappeared from the mathematical world because no researcher could make any advancements or find practical applications. That is how Sakai describes the history of the equations in an academic book he authored.

“The work done by Painlevé and others at the beginning of the twentieth century was, as Poincaré so aptly noted, a lone island in the high seas, away from the continent of mathematics.”

In some sense, the Painlevé equations were "heretical," far removed from the mathematical trends of the early 20th century. Sakai continues, however, as follows.

“The story of the Painlevé equations seems to stand alone and apart from the larger story (author's note: current trends in modern mathematics). Now, mathematicians are beginning to pause and look for new directions. With the powerful tools of mathematics, they are beginning to think about what they really want to know.

There is no way to know where the gold is buried, but it has been said for a long time that of all the non-obvious things, the simplest are the most interesting.”

Sakai found in the Painlevé equations the main motivation he had for mathematics: “fun.”

## The fruits of many years of labor

It was in the field of physics where the Painlevé equations, which some mathematicians had called mysterious, came back into the spotlight. In the 1970s, it was discovered that the correlation function of the two-dimensional Ising model (a statistical mechanics model that describes phase transitions) could be written in terms of the solution of the Painlevé equations. This was a major event that came as a surprise to the mathematical community at the time. For the first time in half a century, interest in the Painlevé equations was rekindled.

“It's incredibly strange. The Painlevé equations are very mathematical, entirely mental constructions. Only later do we find out that they play an important role in solving a critical function of real-world statistical physics. It is strange, as is often the case with the relationship between mathematics and physics"

Sakai's motivation for his research was to understand the Painlevé equations. The idea he came up with became the subject of his doctoral dissertation.

“Curious to see what I could find out, in my doctoral dissertation, I began to work on how special functions (author's note: useful and unique functions), which are subjects of analysis, could be associated with the geometry of surfaces. I realized I could use geometry, algebra, and various other fields. Although the subjects are differential equations called the Painlevé equations, I also found out that, when looking at surfaces in general, difference equations arose that could be treated in the same framework. Of course, a lot of research had already been done before. My contribution, in the form of my doctoral dissertation, was a complete picture of the Painlevé equations, taking into account the discrete Painlevé equations, the extensions of the Painlevé equations.”

The 19th century saw the discoveries of special functions such as elliptic and hypergeometric functions, which were useful in various fields outside of mathematics. In fact, the Painlevé equations were born of such world-expanding ambitions. The solution to a differential equation is not a number but a function, so research on differential equations means discovering new functions. Sakai's inspiration is not different. By analyzing the Painlevé equations, he aims to discover previously unknown special functions. To achieve this, he had to step outside the field of analysis. His doctoral thesis was the first step in taking a fresh look at the Painlevé equations through other fields, such as algebra or geometry.

“I am now trying to investigate higher dimensions, imagining how the equations would behave in four dimensions,” he says.

The theory in algebraic geometry that geometrically classifies Painlevé equations, including discrete Painlevé equations, is now called “Sakai theory.” Sakai is continuing to push the limits of what can be known, with the Painlevé equations in focus. His research has garnered attention from overseas as well.

## Looking for wonders where nobody else is looking

Still, one wonders: does Sakai ever get stuck in his research or think about quitting mathematics?

“Not really,” he says casually and explains how he is different from the stereotypical image of a mathematician, sitting at his desk, looking troubled while thinking deeply, as he is trying to figure out the answer to a difficult problem.

“I do not solve problems very often. Quite the contrary: I am more interested in coming up with puzzles to solve. “I found this problem…can someone solve it?” I come up with ideas that are more in the vein of “it would be nice to see where this leads.” I like looking for wonders lurking in places where nobody else is looking.”

That said, mathematics seems to be an already complete, perfect world where anything new or mysterious is hard to find.

“There are still many things that we do not understand. This has been true, especially since computers were introduced in the second half of the 20th century. Mathematicians could perform calculations that had previously been done by hand using machines and many mysterious phenomena that had been invisible before began to appear. As a consequence, the field of mathematics has expanded tremendously. There are still many things that we do not completely understand.”

While Sakai and other mathematicians’ research has shed considerable light on Painlevé equations, many unknowns remain. That is why Sakai continues his research.

“We have just succeeded in creating a theory for the classification of four-dimensional Painlevé equations, so our results have not been thoroughly analyzed yet by other mathematicians. There is a lot of research being done overseas, and all sorts of problems and puzzles keep arising.”

Sakai says this with “too much fun” written on his face.

Mathematics provides the base for computer science as well. As such, it is perhaps one of the most popular disciplines among the youth today. What does Sakai, whose main motivators are “fun” and “fascination,” think about mathematics for practical use?

“I actually think it would be fascinating if we discovered ideas that had practical uses in fields like cryptography, and thus had an impact on society. If I ever came up with such an idea, I would pursue it further. As long as I find it interesting. But I haven't come up with such ideas.”

The reason why Sakai does not do research that has a direct connection to the real world is that he has not come up with ideas that have such implications.

“I can only do the kind of math that I can do. I am not that smart (laughs). But I find it fun to find puzzles and solve them. And that makes life fun.”

Sakai says he hopes that young people will also find the "fun" in mathematics.

“Although a certain theorem says that quintic equations are unsolvable when I tell young students that if they use complex analysis they can solve those equations, some students’ faces light up. I hope that by conveying such wonderful and fascinating ideas, I can attract many young people to study mathematics.”

In his free time, Sakai loves to sing karaoke, but the only new songs he knows are the Pokemon songs that his young daughter taught him. He still takes walks. So today, too, Sakai will probably spend hours strolling around, thinking about Painlevé equations. Although the path he walks along now is not the Philosopher’s Walk, but the bustling Yamate Street and Meguro River in the heart of Tokyo.

※Year of interview:2023

Interview/Text: Minoru Ota

Photography: Junichi Kaizuka