# Masaki Kashiwara awarded 2018 Kyoto Prize

Masaki Kashiwara has been awarded the 2018 Kyoto Prize in Basic SciencesThe citation mentions in particular his impressive work on $\scr D$-modules.  Kashiwara’s most cited paper in MathSciNet, however, is in representation theory:

MR1115118
Kashiwara, M.(J-KYOT-R)
On crystal bases of the Q-analogue of universal enveloping algebras.
Duke Math. J. 63 (1991), no. 2, 465–516,

which provides a canonical base for representations of the quantized universal enveloping algebra $U_q(\scr G)$ associated with a Kac-Moody Lie algebra.  In most (maybe all) cases, these bases are the same as Lusztig’s canonical basis (see MR1035415). The algebras have their origins in exactly solvable models in statistical mechanics, and are important in representation theory for Lie groups and Lie algebras.  Also in representation theory, Kashiwara and Jean-Luc Brylinski provided a solution to the KazhdanLusztig conjecture in their paper:

MR0632980
Brylinski, J.-L.; Kashiwara, M.
Kazhdan-Lusztig conjecture and holonomic systems.
Invent. Math. 64 (1981), no. 3, 387–410.

Beilinson and Bernstein simultaneously proved the conjecture using similar methods, but with a slightly different take.   See MR0610137.

Kashiwara has been good about writing books and long survey articles explaining $\scr D$-modules, microlocal analysis, and related subjects.  These are difficult subjects, involving ideas and techniques from several areas of mathematics.  So books and surveys are very much appreciated.  His little book

MR1943036
Kashiwara, Masaki
D-modules and microlocal calculus. (English summary)
Translated from the 2000 Japanese original by Mutsumi Saito. Translations of Mathematical Monographs, 217. Iwanami Series in Modern Mathematics. American Mathematical Society, Providence, RI, 2003. xvi+254 pp. ISBN: 0-8218-2766-9

is a great introduction to the subject of ${\scr D}$-modules.   His book with Schapira

MR1074006
Kashiwara, Masaki(J-KYOT-R); Schapira, Pierre(F-PARIS13)
Sheaves on manifolds.
With a chapter in French by Christian Houzel. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 292. Springer-Verlag, Berlin, 1990. x+512 pp. ISBN: 3-540-51861-4

is a standard reference for microlocal analysis and $\scr D$-modules.   His long survey article from the Katata Conference

MR0420735
Sato, Mikio; Kawai, Takahiro; Kashiwara, Masaki
Microfunctions and pseudo-differential equations. Hyperfunctions and pseudo-differential equations (Proc. Conf., Katata, 1971; dedicated to the memory of André Martineau), pp. 265–529. Lecture Notes in Math., Vol. 287, Springer, Berlin, 1973.

provided the canonical reference on that subject for a long time.

Kashiwara is known for connecting deep ideas from algebraic geometry, homological algebra, and microlocal analysis.  His work can be seen as an abstract approach to differential equations.  If you look carefully at his papers, though, you see that his work is often grounded in very concrete examples.  His famous paper [MR0485861] with Kowata, Minemura, Okamoto, Ōshima, and Tanaka solving the Helgason Conjecture writes out some very explicit calculations on the upper half-plane.   Part III of his famous series of papers (Part I = MR0370665; Part II = MR0511186) on holonomic systems with regular singularities starts by considering a particular ODE.  (It is not a simple ODE, but it is in one way very concrete.)  I recall years ago asking him a question about holonomic systems and ${\scr D}$-modules.  It is too long ago for me to remember the exact question, but I do remember how he answered.  I had already talked to my former thesis advisor about the question, who suggested I talk to Kashiwara — and he just happened to be visiting at the time.  Kashiwara thought a little, then said, “Let’s write down an ODE.”  Momentarily, I was quite deflated.  I thought my question was hard — and we’re writing down an ODE?  But then Kashiwara modified the equation and demonstrated how it had picked up a property.  Then he added some other complication.  Then he put the equation on a Riemann surface, not the complex plane.  Then it became harder again by some other tweak.  Finally he pointed out that you could make a system of such equations, but in a way that they were really PDEs.  And then he answered my question (whatever it was).