This year, the Abel Prize — the field’s highest honor — has been awarded to Masaki Kashiwara, prolific Japanese mathematician whose work has quietly reshaped how we understand some of the most complex equations in existence.
The Norwegian Academy of Science and Letters announced the award “for his fundamental contributions to algebraic analysis and representation theory, in particular the development of the theory of D-modules and the discovery of crystal bases.”
But what does that mean?
From Cranes and Turtles to Quantum Symmetries
Kashiwara’s fascination with mathematics began with a childhood puzzle known in Japan as Tsurukamezan — a riddle involving figuring out the number of cranes and turtles based on the number of heads and legs. It was here, as a boy solving riddles at school, that he discovered the joy of generalization — of not just solving one problem, but building a method to solve all similar problems.
This joy started drawing him towards mathematics.
Born in 1947 in Yūki, a small city near Tokyo, Kashiwara’s path soon led him to the University of Tokyo, where he studied under the visionary mathematician Mikio Sato. Sato had pioneered a bold idea: use algebra — not traditional calculus — to understand the behavior of differential equations. These equations are the mathematical tools used to describe change across everything from heat and sound to gravity and quantum fields.
This new field became known as algebraic analysis, and it’s also what Kashiwara focused on early in his career. For his master’s thesis, Kashiwara laid its foundations with a theory that would eventually ripple through modern mathematics: the D-module theory.
In simplest terms, D-modules offer a way to study linear partial differential equations through the lens of algebra. “This demonstrated early on the power of algebraic methods in tackling problems of an analytic nature,” the Abel committee noted. That thesis, written in Japanese and unpublished in English for 25 years, became a seminal work across the globe.
Moving on to more ethereal things
By the 1980s, Kashiwara’s work took aim at problems that had stymied mathematicians for generations. Chief among them was Hilbert’s 21st problem, better known today as the Riemann–Hilbert correspondence. At its heart, it is about whether every set of prescribed behaviors for a function — called monodromies — can arise from a differential equation with certain properties.
As the 20th century drew to a close, Kashiwara pivoted toward a new frontier: quantum groups. These structures, emerging from physics and statistical mechanics, called for entirely new mathematical tools.
Quantum groups emerged from efforts to understand symmetries in quantum field theory and statistical mechanics. As nothing quantum can ever be simple or straightforward, these groups differ by having a parameter that reflects all these quantum oddities. They are, in a sense, “twisted” versions of non-quantum groups. Despite their abstract nature, quantum groups provide powerful tools for studying things like subatomic particles. In other words, these groups can help us understand the very building blocks of the universe.
Here, Kashiwara introduced crystal bases, which capture the behavior of representations — the mathematical expression of symmetries — in a way that bridges the fields of algebra and combinatorics. Imagine turning a complex equation into a crystal: rigid, symmetrical, and strikingly simple. The technique, known as the grand loop argument, is still considered a tour de force in mathematical proof. That’s the principle of what Kashiwara’s method did.
Not many people truly understand Kashiwara’s contributions
If by now, you feel fascinated but also intimidated and lost, well, it would take a PhD-level equivalent just to start to grasp what Kashiwara contributed.
He’s an extremely prolific mathematician who has worked with more than 70 collaborators; he’s contributed groundbreaking theories for over 50 years, and even now, his work is at the forefront of contemporary mathematics. Though many of his achievements are highly abstract, their reach stretches far beyond academia. His tools help physicists describe quantum systems and inspire mathematicians studying everything from string theory to knot theory.
His influence has not gone unnoticed. Kashiwara has received Japan’s Order of the Sacred Treasure, the Kyoto Prize, the Chern Medal, and dozens of other accolades. In 2023 and 2024, he was still publishing award-winning work, including new extensions of the Riemann–Hilbert correspondence.
In honoring him with the 2025 Abel Prize, the Academy called Kashiwara’s work a “spectacular” journey across mathematics — from abstract algebra to concrete breakthroughs in analysis, geometry, and symmetry. For a man who began with puzzles about turtles and cranes, Kashiwara has shown the world that sometimes, the simplest questions lead to the most profound answers
The Abel Prize, established by the Norwegian government in 2001 and named after 19th-century mathematician Niels Henrik Abel, is awarded annually by the King of Norway to recognize extraordinary contributions to the field of mathematics. Often regarded as the closest equivalent to a Nobel Prize in the discipline, the Abel Prize honors both lifetime achievements and groundbreaking work that has transformed mathematical understanding. Previous laureates include pioneers such as Karen Uhlenbeck, the first woman to receive the prize for her work in geometric analysis and gauge theory; Andrew Wiles, who famously proved Fermat’s Last Theorem; and Michel Talagrand, honored last year for advancing probability theory and the study of randomness.
