Masaki Kashiwara awarded the 2025 Abel Prize

Biography of Masaki Kashiwara

With a career spanning more than fifty years and more than 300 publications and 70 collaborators, Masaki Kashiwara is an emblematic figure in mathematics, particularly in algebraic analysis and global analysis.

As part of his master's thesis in 1970, under the supervision of Mikio Sato, whom he met at the University of Tokyo, Masaki Kashiwara focused on linear partial differential equations. He thus developed a new approach to studying them: D-module theory. This theory has become a fundamental tool in many branches of mathematics.

His collaborations with France began a few years later. Between 1972 and 1973, he stayed at the University of Nice with Mikio Sato and Takahiro Kawai, at the invitation of researcher André Martineau. He would subsequently work in close collaboration with numerous mathematicians, including Michèle Vergne and Pierre Schapira, with whom he wrote the reference work “Sheaves on Manifolds” in 1990.

Testimony of Pierre Schapira, professor emeritus at the Institute of Mathematics of Jussieu-Paris Rive Gauche2

The work of Masaki Kashiwara is, at least initially, in the tradition of Mikio Sato: Algebraic Analysis and Microlocal Analysis.

Algebraic analysis consists of dealing with analysis problems with the tools and spirit of algebraic geometry. D-module theory is a good illustration of this. It was introduced by Kashiwara in his master's thesis in 1970 and he obtained the main results in the following 10 years (rationality of the zeros of the b-function, Riemann-Hilbert correspondence, Kazhdan-Lusztig conjecture (with J-L. Brylinski), etc.).

The microlocal point of view consists of interpreting various phenomena on a variety as the projection on the variety of phenomena living in the cotangent bundle, a space on which they are much easier to analyze. For example, the monograph by Sato-Kashiwara-Kawai, published in 1973, generically classifies microdifferential systems.

I started working with Masaki around 1978. In the early 80s we simultaneously came up with the idea of the microlocal sheaf theory, he in the complex setting via the “vanishing cycles” functor, I from a purely real perspective, motivated by hyperbolic equations. Our collaboration on this subject culminated in the publication of the book Sheaves on Manifolds in 1990. The idea is to apply Sato's vision to sheaves, and the micro-support of a sheaf is defined as the directions (in cotangent space) of non-propagation. Since the “characteristic variety” of a D-module is none other than the micro-support of the sheaf of its solutions (in the complex domain), the theory of linear systems of PDE largely becomes a sub-branch of microlocal sheaf theory.

But cotangent space has a very rich structure; it is a “symplectic manifold” and many mathematicians have realized that if the microlocal theory of sheaves uses symplectic geometry, then, conversely, it can be an extremely powerful tool for solving problems in this theory. This idea has led to an impressive number of publications, particularly in France under the impetus of Stéphane Guillermou, CNRS research fellow at the Laboratoire de Mathématiques Jean Leray1 .

Masaki Kashiwara's contribution to mathematics is not limited to D-modules or the microlocal theory of sheaves. In 1990, he discovered crystalline bases, which play a key role in the study of representations.

Masaki Kashiwara is a scientist who has profoundly influenced mathematics since the 1970s and whose mark will remain.

• 2CNRS/Sorbonne Université/Université Paris Cité
• 1CNRS/Université de Nantes