One of the landmarks of Kyoto, the home of Japanese mathematician Masaki Kashiwara, is the Kamo River. At certain points, there are stepping stones that allow residents to cross the river away from the bridges. If you take a closer look at these stones, you can see how the water forms swirls and small eddies around them. Describing this flow of a liquid is not easy. You have to solve complicated equations that have been known for centuries but still pose many mysteries today: Do the equations always have a solution? How can they be calculated? And what properties do they have? It seems that mathematicians have reached a limit with the tools of their trade. To make progress, a new toolbox is needed. Kashiwara developed such a toolbox for similarly difficult questions in the 1970s.
Kashiwara introduced proven methods from algebra into analysis—the theory underlying calculus that explores functions, limits, and other concepts—and, together with his colleagues, founded an entirely new branch of mathematics: algebraic analysis. This move led to significant advances in various fields. For example, Kashiwara succeeded in solving one of the problems posed by German mathematician David Hilbert in the early part of the 20th century, and he developed new techniques that are now used in modern physics.
Kashiwara “has proved astonishing theorems with methods no one had imagined. He has been a true mathematical visionary,” read a press release from the Norwegian Academy of Science and Letters, which in March honored him with the 2025 Abel Prize—one of the highest honors in mathematics. Born near Tokyo in 1947, Kashiwara discovered his passion for mathematics at an early age through traditional Japanese puzzles known as tsurukamezan. These puzzles involve correctly calculating the number of cranes and turtles: Suppose x heads and y legs are visible. How many cranes and turtles are there? Kashiwara’s parents didn’t have much exposure to the abstract subject, but their son enjoyed solving this problem by using algebraic methods.
On supporting science journalism
If you're enjoying this article, consider supporting our award-winning journalism by subscribing. By purchasing a subscription you are helping to ensure the future of impactful stories about the discoveries and ideas shaping our world today.
Here’s one example: Each crane or turtle has two or four legs, respectively (y), and each has one head (x). Calculate the number of cranes (k) and turtles (s) by solving the following equations: 2k + 4s = y and k + s = x. For example, if 16 legs and five heads are visible, then there must be two cranes and three turtles.
Masaki Kashiwara, the 2025 Abel Prize laureate, holds his 1990 book, Sheaves on Manifolds.
Peter Badge/Typos1/The Abel Prize
Kashiwara realized he enjoyed generalizing such questions. He excelled in school. When he met the late mathematician Mikio Sato as one of Sato’s students at the University of Tokyo, he devoted himself to this type of problem-solving. Kashiwara was in the right place at the right time. Sato and his colleagues were then developing a completely new branch of mathematics that combines two distinct fields: analysis and algebra.
Nothing Stands Still
Kashiwara worked with his mentor on differential equations. In our world, everything is in motion; nothing remains permanently still. Even a gigantic mountain range such as the Himalayas grows or shrinks over time. Such changes can be expressed mathematically with the help of derivatives. All of physics is based on equations that contain derivatives, so-called differential equations. They can be used to describe the population of living organisms, the trajectory of the moon or the flow velocity of the Kamo River.
Differential equations can be written down quickly, but they take much longer to solve. In some special cases, the solution is known. In others, however, it is not even clear whether a problem can be solved at all. One of the most important unsolved problems in mathematics revolves around the question of whether the Navier-Stokes equations, which describe the flow behavior of fluids, always have a solution. Despite centuries of research in the field of analysis, many of the most pressing problems remain unsolved.
When you’re stuck on a problem, it sometimes helps to look at it from a different perspective. Often it is useful to step back and examine the problem from a distance. The exact details may become blurred, but the general structure of the topic could become visible. This approach not only helps with practical, everyday problems but also can be beneficial in mathematics.
A Japanese research group led by Sato pursued such an approach. The team wanted to examine differential equations from a different perspective. To do so, the researchers left the field of analysis and turned instead to algebra. Algebra is generally much more abstract: the focus is not necessarily the mathematical objects—in this case, the equations and their derivatives—but rather their behavior. Just as physicists study a new particle by examining its interactions with other particles, mathematicians can look at the interplay of different equations to uncover new insights. That is the idea underlying algebraic analysis.
So instead of picking out a specific differential equation and examining it in detail, Sato and his colleagues devoted themselves to an entire class of such equations. They also allowed the differential equations to move not only on a plane but also on curved surfaces—as if trying to describe a river on an oddly shaped planet. This approach may seem quite complex, but it actually opens up entirely new possibilities, allowing one to derive general properties for the class of differential equations under consideration that are not apparent for individual equations.
At the end of the 1960s, Sato organized a weekly seminar in which participants worked together to develop the concepts of the new theory. Among all the experts was Kashiwara, then a young student, who eagerly participated.
Into the Fast Lane with D-Modules
In 1970 Kashiwara began his master’s thesis under Sato. His task was to develop algebraic tools for investigating objects from analysis. Then only 23 years old, Kashiwara introduced so-called D-modules, which make it possible to extract valuable information from differential equations. D-modules can be used, for example, to determine whether the solutions to equations contain “singularities”—that is, whether there are regions where they assume infinite values. The modules can also be used to calculate how many solutions the equations have.
The results of Kashiwara’s master’s thesis shaped the emerging field of algebraic analysis. He had written his research in Japanese, however—25 years went by before it was translated into English and thus made accessible to a wider audience.
After graduating, Kashiwara went to Kyoto University, where he continued his collaboration with Sato and earned his doctorate. In doing so, he further developed the new methods he had established in his master’s thesis. “From 1970 to 1980, Kashiwara solved almost all the fundamental questions of D-module theory,” recalled his colleague Pierre Schapira, now an emeritus professor at Sorbonne University in Paris, in a 2008 preprint paper based on a 2007 talk. After completing his doctorate, Kashiwara accepted a position at Nagoya University, conducted research for a year at the Massachusetts Institute of Technology and then returned to Japan in 1978 to accept a professorship at Kyoto University.
With the help of D-modules, in 1980 Kashiwara solved one of the most important problems in the field, a problem Hilbert had presented in his centenary address at the International Congress of Mathematicians in Paris in 1900. Hilbert considered 23 problems to be groundbreaking for 20th-century research; the 21st problem deals with differential equations. Hilbert wanted to know whether it would always be possible to find a differential equation whose solution possessed singularities on a given curved surface. Kashiwara was able to prove that it is indeed possible for certain types of surfaces—in those cases, a suitable differential equation can be calculated.
D-modules have led to advances in many different areas of mathematics. But they are also proving helpful in physics. In 2023 mathematician Anna-Laura Sattelberger of the Max Planck Institute for Mathematics in the Sciences in Germany and other experts used D-modules to evaluate quantum physical “path integrals.” These integrals are used to calculate which processes take place in particle accelerators when, for instance, two protons collide, creating a series of new particles. The extremely complex integrals can be viewed as solutions of differential equations, which is why the methods of algebraic analysis can help to determine their properties.
On Symmetries and Quantum Groups
Kashiwara also had a significant influence on other areas of mathematics. One of them is representation theory, which is used to describe symmetries. An object is considered symmetrical if it looks the same after certain transformations (such as rotations or reflections). For example, an equilateral triangle can be rotated by multiples of 120 degrees without changing its shape. Representation theory enables experts to calculate symmetry transformations: What happens, for example, if you combine a 270-degree rotation with a reflection along the y-axis? Such questions can be answered particularly well if you represent the symmetry transformations by using matrices: the combination of transformations corresponds to the multiplication of the corresponding matrices.
Suitable representations cannot be found for all types of symmetries, however. In the course of his work, Kashiwara focused extensively on continuous symmetries, known in mathematics as Lie groups. He made significant progress in investigating their representations.
He also explored discrete “quantum groups” that are not continuous. Such discrete quantum groups play an important role in quantum physics. At the microscopic level, most quantities appear only in small pieces; the world appears to be quantized at the smallest scale. To describe the symmetries of quantized quantities, Kashiwara introduced the concept of crystal bases. These bases allow quantum groups to be represented by directed networks, which offers enormous advantages. Questions of representation theory can then be answered through combinatorial considerations (arranging objects in a finite set), which are generally much simpler. These concepts have since proved their worth in both mathematics and physics.
In its Abel Prize press release, the Norwegian Academy of Science and Letters quotes its president, Annelin Eriksen, as saying, “For over 50 years Masaki Kashiwara has reshaped and deeply enriched the fields of algebraic analysis and representation theory.” Kashiwara has already received numerous awards for all of this impressive research. The Abel, which honors a mathematician’s lifetime achievement, marks a culmination of his accomplishments. The prize is modeled on the Nobel Prizes, which do not include mathematics, and comes with 7.5 million Norwegian kroner (approximately $760,000).
Now nearly 80 years old, Kashiwara does not seem to be thinking about retirement: he still regularly publishes new research findings and tries to enrich mathematics with new stepping stones.
This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission.
