On 20 May in Oslo, the prestigious Abel Prize, often dubbed the ‘Nobel Prize of Mathematics’, was awarded to Professor Masaki Kashiwara of Kyoto University. The Norwegian Academy of Science and Letters honoured him “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”.
The Abel Prize, established by the Norwegian government in 2001 and named after mathematician Niels Henrik Abel, recognizes extraordinary contributions to mathematics. It is awarded annually by the King of Norway and includes a monetary award of NOK 7.5 million. Last year it was won by Michel Talagrand for his research into probability theory and the extremes of randomness.
Kashiwara’s work, like that of many outstanding mathematicians, is about linking mathematical fields together. It involves using algebra as a tool to investigate geometry and symmetry. He has focused on using those ideas to find solutions for differential equations, which involve relationships between mathematical functions and their rates of change.
Early spark: from cranes and turtles to complex equations
Kashiwara’s passion for mathematics ignited during his school days in Japan, inspired by a traditional puzzle called Tsurukamezan. This problem involves determining the number of cranes and turtles based on the total count of heads and legs. Solving it sparked his interest in generalising methods to tackle a variety of problems, a curiosity that would shape his future contributions to mathematics.
Pioneering algebraic analysis and D-modules
Under the mentorship of Mikio Sato at the University of Tokyo, Kashiwara delved into a new field called algebraic analysis, which applies algebraic techniques to study differential equations. In 1970, at just 23, he completed a master’s thesis that laid the foundation for D-module theory: a framework that uses algebra to understand systems of linear partial differential equations. Although initially published only in Japanese, this work had a profound impact and was eventually translated into English due to its significance.
Bridging mathematics and physics: crystal bases
Kashiwara’s work extended into representation theory, where he introduced the concept of crystal bases. These structures simplify complex algebraic representations, making them more accessible and providing tools to study symmetries in quantum groups, a concept crucial in modern physics. His “grand loop argument” in this area is considered a masterpiece in mathematical proof.
The University of Birmingham’s David Craven described Kashiwara’s work as extremely abstract, far from direct real-world applications, and understanding even a simple summary of his work would require a PhD in mathematics as a bare minimum.
A legacy of inspiration and innovation
Over a career spanning more than five decades, Kashiwara has authored influential books and collaborated with over 70 mathematicians worldwide. His accolades include the Kyoto Prize, the Chern Medal, and Japan’s Order of the Sacred Treasure. Now he becomes the first Japanese mathematician to receive the Abel Prize, a testament to his groundbreaking work and its global impact.
“I am now working on the representation theory of quantum affine algebras and its related topics,” Kashiwara told New Scientist. “There is a nice conjecture: [the] ‘affine quiver conjecture’, but I still have no clue how to solve it.”
Kashiwara’s journey from solving elementary puzzles to transforming advanced mathematical theories exemplifies the profound impact of curiosity and dedication. His work continues to inspire mathematicians and scientists, bridging abstract concepts with real-world applications.
Find out more on the Abel Prize website.