Dr. Kashiwara established the theory of D-modules, thereby playing a decisive role in the creation and development of algebraic analysis. His numerous achievements—including the establishment of the Riemann-Hilbert correspondence, its application to representation theory, and construction of crystal basis theory—have exerted great influence on various fields of mathematics and contributed strongly to their development.
Dr. Masaki Kashiwara has numerous outstanding achievements including the theory of D-modules, which forms the core of algebraic analysis. Developing this theory from its foundation, he applied it across various fields of modern mathematics.
Algebraic analysis is the study of objects in analysis such as differential equations by using the methods of modern algebra. In his earlier studies, Dr. Kashiwara, jointly with Dr. Mikio Sato and Dr. Takahiro Kawai, completed the classification theory of systems of linear partial differential equations (1). In algebraic analysis, systems of linear differential equations are studied as modules over a ring D of differential operators, that is, as D-modules. Establishing the basic theory of D-modules all alone, Dr. Kashiwara laid the foundation for its subsequent development by proving many important theorems such as finite dimensionality of solutions of holonomic systems (2).
One of his most remarkable achievements is the construction of the Riemann-Hilbert correspondence. To each linear differential equation is associated the concept of monodromy group, a topological datum which measures the multi-valuedness of its solutions. The Riemann-Hilbert problem asks the converse, namely whether there exists a linear differential equation which has a given monodromy group. It had been solved affirmatively in dimension one. The case of higher dimensions had been a longstanding issue, to which Dr. Kashiwara provided an ideal answer in the form of a one-to-one correspondence between regular holonomic D-modules and constructible sheaves (3, 4). This is a beautiful synthesis of geometry, algebra and analysis, and its influence extends beyond the respective fields. Dr. Kashiwara and his collaborator were one of the two research groups who solved the Kazhdan-Lusztig conjecture as an important application of the Riemann-Hilbert correspondence to representation theory (5). Furthermore, the collaborative research on the extension to infinite dimensional Lie algebras (6, 7) became a key step in completing the Lusztig program on representations of algebraic groups in positive characteristic.
Theory of crystal bases of quantum groups is another important achievement of Dr. Kashiwara in representation theory. Quantum groups are a deformation of Lie algebras by a parameter q. Dr. Kashiwara found that a significant simplification occurs in the limit where q becomes 0 and introduced crystal bases at q = 0 (8). Encoding essential information about representations of quantum groups, crystal bases have become a powerful tool in representation theory, combinatorics, and integrable systems. Dr. Kashiwara further showed that crystal bases uniquely extend to global crystal bases defined for an arbitrary value of q (9), which turned out to coincide with the canonical bases introduced by Dr. Lusztig from a completely different point of view.
Dr. Kashiwara has made a wide range of contributions, often collaborating with many coauthors. Some of his outstanding achievements include developing micro local analysis of sheaves (10, 11) and constructing micro localization of symplectic manifolds (12). Even today, he continues to contribute to his field with important results such as the extension of the Riemann-Hilbert correspondence to irregular singularities (13) and categorification of representations of quantum groups (14).
The theory of D-modules has spread to other fields such as number theory, shaping one of the trends in modern mathematics. Dr. Kashiwara’s truly original work sustained over almost half a century is expected to remain influential to the development of mathematical sciences.
(1) Sato M et al. (1973) Microfunctions and pseudo-differential equations. In Lecture Notes in Math. 287 (Springer-Verlag, Berlin-Heidelberg-New York): 265–529.
(2) Kashiwara M (1975) On the maximally overdetermined system of linear differential equations, I. Publ RIMS, Kyoto Univ 10: 563–579.
(3) Kashiwara M (1980) Faisceaux constructibles et systèmes holonomes d’équations aux dérivées partielles linéaires à points singuliers réguliers. In Séminaire Goulaouic-Schwartz, 1979–80, Exposé 19 (École Polytechnique, Palaiseau).
(4) Kashiwara M (1984) The Riemann-Hilbert problem for holonomic systems. Publ RIMS, Kyoto Univ 20: 319–365.
(5) Brylinski J-L & Kashiwara M (1981) Kazhdan-Lusztig conjecture and holonomic systems. Invent Math 64: 387–410.
(6) Kashiwara M & Tanisaki T (1995) Kazhdan-Lusztig conjecture for affine Lie algebras with negative level. Duke Math J 77: 21–62.
(7) Kashiwara M & Tanisaki T (1996) Kazhdan-Lusztig conjecture for affine Lie algebras with negative level Ⅱ: Nonintegral case. Duke Math J 84: 771–813.
(8) Kashiwara M (1990) Crystalizing the q-analogue of universal enveloping algebras. Comm Math Phys, 133: 249–260.
(9) Kashiwara M (1991) On crystal bases of the q-analogue of universal enveloping algebras. Duke Math J 63: 465–516.
(10) Kashiwara M & Schapira P (1985) Microlocal study of sheaves. Astérisque 128.
(11) Kashiwara M (1985) Index theorem for constructible sheaves. Astérisque 130: 193–209.
(12) Kashiwara M & Rouquier R (2008) Microlocalization of rational Cherednik algebras. Duke Math J 144: 525–573.
(13) D’Agnolo A & Kashiwara M (2016) Riemann-Hilbert correspondence for holonomic D-modules, Publ Math-Paris 123: 69–197.
(14) Kang S-J & Kashiwara M (2012) Categorification of highest weight modules via Khovanov-Lauda-Rouquier algebras. Invent Math 190: 699–742.
Two years after matriculating at the University of Tokyo in 1965, I joined that university’s Mathematics Department but I did not have the slightest inclination to become a mathematician at that time. It was only my encounter with Dr. Mikio Sato, in 1968, that ultimately led me to pursue a career in mathematics. Dr. Hikosaburo Komatsu (still in his 30s) had just returned from the U.S., and he and Dr. Sato were holding weekly algebraic analysis seminars. At the strong recommendation of Dr. Takahiro Kawai, who was my senior by a year, I began attending those seminars and that was where I got to know Dr. Sato. That chance meeting opened the door to my subsequent career as a researcher in mathematics, specifically algebraic analysis.
In fact, Dr. Sato was the founder of the field of algebraic analysis. Today, we mathematicians call a variable quantity a function, and the act of researching a function is referred to as mathematical analysis, whereas algebra involves study in which numbers and their mathematical operations (sums and products) are expanded beyond ordinary numbers for the purposes of research. Then, algebraic analysis employs algebra to elucidate essential qualities that lie deep within mathematical analysis.
In 1969, when I began attending his seminars, Dr. Sato presented the idea of micro local analysis, which makes it possible to handle non-smooth functions algebraically. This approach would subsequently spread to mathematical fields beyond mathematical analysis in various ways. Since then, my main work has been to establish a technique for connecting geometry and algebra through utilization of mathematical analysis.
After completing my master’s program in 1971, I took up an assistant position at Kyoto University’s Research Institute for Mathematical Sciences (RIMS), where I spent several years enthusiastically establishing micro local analysis with Drs. Sato and Kawai, who had also transferred there. Having the chance to engage in mathematics with those two eminent figures represented a giant leap forward in my life as a mathematician. From them, I learned the joy of conducting mathematical research. Thereafter, I went on to achieve many things, including the establishment of the Riemann-Hilbert correspondence and discovery of crystal bases, and it was my encounter with those two pioneers that laid the foundations for all of my subsequent mathematical research.