One of the landmarks of Kyoto, the home of 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. The Japanese mathematician Masaki 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 led to significant advances in various fields. For example, Kashiwara succeeded in solving one of the problems posed by mathematician David Hilbert in the early 20th century and 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 recent press release from the Norwegian Academy of Sciences and Letters, which honored him with this year’s Abel Prize—one of the highest honors in mathematics.
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Kashiwara was born near Tokyo in 1947. He 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 the young Masaki enjoyed solving this problem using algebraic methods.
Here’s one example: Each crane and turtle respectively has two and four legs (y)—and both have one head (x). To calculate the number of cranes (k)and turtles (s), one must solve 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.
Kashiwara realized he enjoyed generalizing such questions. He excelled in school with his achievements. When he met the late mathematician Mikio Sato when he was Sato’s student 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. These can be used to describe the population of living organisms, the trajectory of the moon or the flow velocity of the Kamo River.
While differential equations can be written down quickly, they are much more difficult 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’s helpful to step back and examine the problem from a distance. In this case, the exact details may become blurred, but the general structure of the topic becomes visible. This approach is not only helpful for practical, everyday problems but can also be useful in mathematics.
A Japanese research group led by Sato pursued a similar 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, in physics, one studies a new particle by examining its interactions with other particles, the interplay of different equations should reveal 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. This allows general properties to be derived 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—it took a full 25 years 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 French colleague Pierre Schapira in a 2008 preprint paper that was 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, Kashiwara solved one of the most important problems in the field in 1980, a problem that Hilbert presented in his famous centenary address at the International Congress of Mathematicians in Paris in 1900. Among the 23 problems that Hilbert considered groundbreaking for 20th-century research, the 21st problem deals with differential equations. The German mathematician 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 this is indeed possible for certain types of surfaces—in these 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 Leipzig, Germany, and other experts used D-modules to evaluate quantum physical “path integrals.” These are used to calculate which processes take place in particle accelerators, for example, when 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 these is representation theory, which is used to describe symmetries. An object is considered symmetric 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 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 allow quantum groups to be represented by directed networks. This offers enormous advantages, allowing questions of representation theory to be answered through combinatorial considerations (arranging objects in a finite set), which are generally much simpler. These concepts have since proven their worth in both mathematics and physics.
“For over 50 years Masaki Kashiwara has reshaped and deeply enriched the fields of algebraic analysis and representation theory,” the Norwegian Academy of Sciences and Letters wrote in its recent press release. The mathematician has already been honored with numerous awards for all of this impressive research. This year’s Abel Prize, which honors a mathematician’s lifetime achievement, marks a culmination of what he has accomplished. The Abel Prize is modeled on the Nobel Prizes, which do not include mathematics, and comes with 7.5 million Norwegian kroner (approximately $710,000).
The 78-year-old does not seem to be thinking about retirement: he still regularly publishes new research findings and tries to enrich mathematics with new stepping stones.