![リガクル[RIGAKU-RU] Exploring Science](/ja/rigakuru/images/top/title_RIGAKURU.png)
A child precociously interested in math
3×5 means adding 3 five times. 5×3 means adding 5 three times. They seem different when we learn about them, yet the result is the same. Young Imai, still in elementary school, felt an indescribable sense of wonder. It was one reason the young Imai fell in love with mathematics.
“During my junior high school entrance exam prep at cram school, the teacher once asked us if we could prove whether there were infinitely many prime numbers. I kept thinking about the question for a while and using the problem-solving skills I had learned in class at the cram school, I managed to prove it in my own way that there are infinitely many prime numbers, though I am not sure if it was a proper proof. The experience of proving things using one’s own thinking made me realize, as much as a child of that age could, that mathematics is built on logic.”
Nevertheless, his interest in math was precocious indeed. By the time he was in junior high school, he was already studying high school material.
“It took nearly two hours by train to get to school from home, so I read high school math textbooks on the train.”
Because he was simply “enjoying the wonder” of mathematics.
By the way, from his third year of junior high through his third year of high school, Imai competed four years in a row in the International Mathematical Olympiad, winning a gold medal in his third year of high school.
Despite this, young Imai also developed an interest in physics, even if temporarily.
“During junior high and high school, I went through a phase wanting to understand the fundamentals, such as the beginning of the universe, and studied physics a bit. It was fascinating, but when I learned that studying physics in university required university-level mathematics, I decided to study math first, and in the end, I stuck with it (laughs). A world that is complete and can be completely explained by logic alone somehow suited me.”
And so, the young Imai enrolled in the Faculty of Science at the University of Tokyo.
The “Grand” Langlands Program
Now 41 years old, Imai became an associate professor at the early age of 28. He says that “studying mathematics is similar to practicing a craft,” and he is pursuing his research, the geometrization of the Langlands correspondence, enthusiastically. For many, this term may be unfamiliar, but it represents a “grand” ambition in mathematics, similar to the Grand Unified Theory in physics. Therefore, rather than merely practicing, Imai is embarking on an adventure in uncharted territory. To put it in gaming terms, Imai is the hero character continuing to challenge himself repeatedly.
“The Langlands Program posits a deep connection between two vastly different mathematical objects. One is called Galois representations, and the other is called automorphic representations. What connects these two is the Langlands correspondence.”
Imai explains it, but a comparison with the Grand Unified Theory in physics might make it clearer.
In physics, the four fundamental forces (strong force, weak force, electromagnetic force, and gravity) were once explained by separate theories. Grand Unification Theory (GUT) aims to unify these under a single theory. The weak and electromagnetic forces have already been unified under the Weinberg-Salam theory. Current research focuses on unifying it with the strong force. To understand Imai's “mathematical objects,” it might be helpful, though somewhat crude, to think of them like these four fundamental forces.
Alternatively, these mathematical objects are sometimes likened to continents. That is, within the world of mathematics, there are various fields, such as number theory, geometry, harmonic analysis, representation theory, and mathematical physics. These fields exist like separate continents. Mathematicians have pursued research independently on their respective continents, believing that even though they share the same mathematical world, there is no bridge connecting these continents. However, similar phenomena were found on these seemingly completely separate continents. In other words, it has become apparent that creatures thought to have evolved independently on each continent (in this case, the creatures are “Galois representations” and “automorphic representations”) appear to share mysterious commonalities. This suggests the existence of a deeper mathematical world extending beyond the boundaries of these continents. The evidence for this is the “Langlands correspondence,” and the “grand” bridge connecting the different continents is the Langlands Program.
The Langlands Program was born in 1967 from a lengthy letter written by Robert Langlands, then a 30-year-old unknown mathematician, to the renowned number theorist André Weil. Beginning with the humble opening, “If you do not wish to read it, please throw it in the trash,” this letter contained a surprising conjecture considered impossible at the time that there exists a correspondence between two objects belonging to different branches of mathematics. This was the starting point for everything. Robert Langlands later became a professor at the renowned Institute for Advanced Study at Princeton University, inheriting the very office once used by Einstein. This shows the significance of the Langlands Program for modern mathematics.
Toward something underneath the world of mathematics
Imai aims to tackle one of the challenges the Langlands Program faces. He seeks to uncover the deep connection between Galois representations and automorphic representations, mathematical “creatures” that seem to exist independently on separate “continents.”
Galois representations describe the behavior of Galois groups* in Galois theory, a subfield of number theory in mathematics. Automorphic representations, on the other hand, belong to the field of harmonic analysis, which deals with waves, frequencies, and other phenomena described by trigonometric functions. Number theory and harmonic analysis are considered entirely separate “continents,” but could there be a profound connection hidden between them? If so, what could it be? That is precisely what Imai seeks to uncover.
“We call the method of extracting information about a group's structure or properties a "representation." For example, in the case of the representation of the Galois group, the elements have corresponding automorphic representations. In other words, even though they are the representations of completely different groups, there is a one-to-one correspondence between them. This is the Langlands correspondence, which I have been researching using my own perspective.”
“Galois representations” are named after the French mathematician Évariste Galois, who laid the foundations of the theory. He died in 1832 (nearly two centuries ago) at the young age of 20, after losing a duel. The real drama was that he wrote down the theory that would be later named after him merely a few days before the duel, probably preparing to die. It was this Galois theory that proved there is no formula for solving equations of the fifth degree or higher. He was truly a genius.
When Imai says he is researching Langlands correspondence, it is certainly not as simple as just comparing two things side by side. It requires extremely complex and profound mathematical inquiry. The tools Imai employs for this are p-adic numbers and moduli spaces.
The distance between two numbers in the field of p-adic numbers is peculiar, measured by how many times it can be divided evenly by a prime number p. A moduli space is another special geometric space where individual geometric objects are represented by a single point. Further explanation is beyond the author's ability, but suffice it to say, Imai is navigating a path so challenging that advancing research on the Langlands correspondence requires employing a wide array of complex mathematical techniques and concepts.
“The geometricization of the Langlands correspondence within a number-theoretic framework, which I am currently working on, is a new, hot topic where no one yet knows what the final formulation should look like. So, I hope to contribute even a little. Langlands correspondence is a problem spanning a vast domain, so it is not something that one mathematician can explore alone and declare it unified. I believe it will be understood gradually, through the combined research of many people. I want to walk towards it, hoping that there is something unifying underneath.”
The Langlands Program also has significant connections to the physical world. This means that Imai's research contributes greatly not only to mathematics but also to the advancement of physics. In that sense, what Imai is walking toward has historical significance.
The widely expanding world of mathematics
The world of mathematics may appear to be a completed, perfect realm where nothing remains unknown. Imai says otherwise.
“The still undiscovered world of mathematics, I believe, expands widely, and our job is to gradually make it visible. How we formulate and grasp the objects of this world is crucial. If we do not make the proper formulation, no matter how hard we try, we will not gain any insight, even with the same mathematical objects. On the other hand, when we find a clever formulation, the problem itself becomes neatly formulized, and what is happening becomes clear and easy to understand. But such formulations do not appear suddenly. Progress comes gradually through trial and error, combining various methods and many people’s ideas.”
What lies at the end of that path? Imai says that even working mathematicians do not yet know.
“We do not truly know how widely the world of mathematics expands. There are still many elementary mathematical problems we do not understand, many that remain unsolved. So, personally, I believe there must still be a great deal of mathematics underneath that we need to understand, waiting to be discovered.”
In 1994, 300 years after the conjecture was made, mathematician Andrew Wiles proved Fermat's Last Theorem that there are no three positive integers x, y, and z that can satisfy the equation xⁿ + yⁿ = zⁿ for any integer value of n greater than 2. The Langlands Program played an important role in this proof. However, the modularity theorem, proposed even before Langlands by Japanese mathematicians Yutaka Taniyama and Goro Shimura, also played an important role and is said to have influenced the Langlands program itself. Many more mathematicians were involved in various ways, taking turns in the quest to prove Fermat's Last Theorem, until Wiles finally succeeded. In other words, as Imai explains, mathematics is something that progresses gradually through the combined efforts of many people.
Imai shares the following message with high school and university students interested in mathematics.
“Finding unexpected connections is what makes mathematics interesting. You may encounter abstract theories while studying and feel confused. But do not feel disheartened, and make sure to properly acquire the necessary skills. Once you develop good mathematical thinking, things you previously did not understand will become clear. Then you will be able to see relationships between different objects and explain interesting phenomena, which is a lot of fun. So, please do not give up if you do not understand or find something fun at a first pass. If you do, you will miss the chance to see something fascinating that lies just beyond.”
When asked what the happiest moment in his research career has been so far, Imai says it was when he finished his very first research project.
“It was during my master's program, around age 23. I had never done research before, and I had even doubted whether I was capable of it. Because of that, finishing the paper made me incredibly happy.”
The title of that paper was “On the moduli spaces of finite flat models,” and it was about the connection between Galois representations and moduli spaces.
“I was far happier in that moment when I solved the problem and got a result than when the paper was published.”
In his private life, he is a father to three children aged 11, 7, and 3. He is a kind father who helps his children with their math homework when they say they “do not get it."
*A group is a collection of numbers or operations that satisfy certain rules. For more details on groups, please refer to the interview with Yoshiki Oshima (Graduate School of Mathematical Sciences).
※Year of interview:2025
Interview/Text: OTA Minoru
Photography: KAIZUKA Junichi
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