{"id":2169,"date":"2018-07-09T17:47:54","date_gmt":"2018-07-09T22:47:54","guid":{"rendered":"http:\/\/blogs.ams.org\/beyondreviews\/?p=2169"},"modified":"2018-07-09T17:47:54","modified_gmt":"2018-07-09T22:47:54","slug":"masaki-kashiwara-awarded-2018-kyoto-prize","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/beyondreviews\/2018\/07\/09\/masaki-kashiwara-awarded-2018-kyoto-prize\/","title":{"rendered":"Masaki Kashiwara awarded 2018 Kyoto Prize"},"content":{"rendered":"

Masaki Kashiwara<\/a>\u00a0has been awarded<\/a> the 2018 Kyoto Prize in\u00a0Basic Sciences<\/a>.\u00a0 The citation mentions in particular his impressive work on $\\scr D$-modules<\/a>.\u00a0 Kashiwara’s most cited paper in MathSciNet, however, is in representation theory:<\/p>\n

MR1115118<\/a>
\nKashiwara, M.(J-KYOT-R)
\nOn crystal bases of the Q-analogue of universal enveloping algebras.
\nDuke Math. J. 63 (1991), no. 2, 465\u2013516,<\/p>\n

which provides a canonical base for representations of the\u00a0quantized universal enveloping algebra $U_q(\\scr G)$ associated with a Kac-Moody Lie algebra.\u00a0 In most (maybe all) cases, these bases are the same as Lusztig’s canonical basis (see\u00a0MR1035415)<\/a>. The algebras have their origins in\u00a0exactly solvable models in statistical mechanics, and are important in representation theory for Lie groups and Lie algebras.\u00a0 Also in representation theory, Kashiwara and\u00a0Jean-Luc Brylinski<\/a> provided a solution to the Kazhdan<\/a>–Lusztig<\/a> conjecture<\/a> in their paper:<\/p>\n

MR0632980<\/a>
\nBrylinski, J.-L.; Kashiwara, M.
\nKazhdan-Lusztig conjecture and holonomic systems.
\nInvent. Math. 64 (1981), no. 3, 387\u2013410.<\/p>\n

Beilinson<\/a> and Bernstein<\/a> simultaneously proved the conjecture using similar methods, but with a slightly different take.\u00a0 \u00a0See\u00a0MR0610137<\/a>.<\/p>\n

Kashiwara has been good about writing books<\/a> and long survey articles explaining $\\scr D$-modules, microlocal analysis<\/a>, and related subjects.\u00a0 These are difficult subjects, involving ideas and techniques from several areas of mathematics.\u00a0 So books and surveys are very much appreciated.\u00a0 His little book<\/p>\n

MR1943036<\/a>
\nKashiwara, Masaki
\nD-modules and microlocal calculus<\/a>. (English summary)
\nTranslated 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<\/p>\n

is a great introduction to the subject of ${\\scr D}$-modules.\u00a0 \u00a0His book with Schapira<\/p>\n

MR1074006<\/a>
\nKashiwara, Masaki(J-KYOT-R); Schapira, Pierre<\/a>(F-PARIS13)
\nSheaves on manifolds.
\nWith 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<\/p>\n

is a standard reference\u00a0for microlocal analysis and $\\scr D$-modules.\u00a0 \u00a0His long survey article from the Katata Conference<\/p>\n

MR0420735<\/a>
\nSato, Mikio; Kawai, Takahiro; Kashiwara, Masaki
\nMicrofunctions and pseudo-differential equations. Hyperfunctions and pseudo-differential equations (Proc. Conf., Katata, 1971; dedicated to the memory of Andr\u00e9 Martineau), pp. 265\u2013529. Lecture Notes in Math., Vol. 287, Springer, Berlin, 1973.<\/p>\n

provided the canonical reference on that subject for a long time.<\/p>\n

Kashiwara is known for connecting deep ideas from algebraic geometry, homological algebra, and microlocal analysis.\u00a0 His work can be seen as an abstract approach to differential equations.\u00a0 If you look carefully at his papers, though, you see that his work is often grounded in very concrete examples.\u00a0 His famous paper [MR0485861<\/a>] with Kowata, Minemura, Okamoto, \u014cshima, and Tanaka solving the Helgason Conjecture writes out some very explicit calculations on the upper half-plane.\u00a0 \u00a0Part III<\/a> of his famous series of papers (Part I = MR0370665<\/a>; Part II = MR0511186<\/a>) on holonomic systems with regular singularities starts by considering a particular ODE.\u00a0 (It is not a simple ODE, but it is in one way very concrete.)\u00a0 I recall years ago asking him a question about holonomic systems and ${\\scr D}$-modules.\u00a0 It is too long ago for me to remember the exact question, but I do remember how he answered.\u00a0 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.\u00a0 Kashiwara thought a little, then said, “Let’s write down an ODE.”\u00a0 Momentarily, I was quite deflated.\u00a0 I thought my question was hard — and we’re writing down an ODE?\u00a0 But then Kashiwara modified the equation and demonstrated how it had picked up a property.\u00a0 Then he added some other complication.\u00a0 Then he put the equation on a Riemann surface, not the complex plane.\u00a0 Then it became harder again by some other tweak.\u00a0 Finally he pointed out that you could make a system of such equations, but in a way that they were really PDEs.\u00a0 And then he answered my question (whatever it was).<\/p>\n

<\/div>","protected":false},"excerpt":{"rendered":"

Masaki Kashiwara\u00a0has been awarded the 2018 Kyoto Prize in\u00a0Basic Sciences.\u00a0<\/p>\n

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