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Fifty years of Mathematics

with Masaki Kashiwara

Pierre Schapira

Sorbonne Université, Paris, France

ICM 2018Rio de Janeiro, August 1st

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Before Kashiwara

Masaki Kashiwara was a student of Mikio Sato and the storybegins long ago, in the late fifties, when Sato created a newbranch of mathematics, now called “Algebraic Analysis” bypublishing his papers on hyperfunction theory and developedhis vision of analysis and linear partial differential equations(LPDE) in a series of lectures at Tokyo University.

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Hyperfunctions on a real analytic manifold M are cohomologyclasses supported by M of the sheaf OX of holomorphicfunctions on a complexification X of M .

In these old times, trying to understand real phenomena bycomplexifying a real manifold and looking at what happens inthe complex domain was a totally new idea. The use ofcohomology of sheaves in analysis was definitely revolutionary.

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Figure : 佐藤幹夫 and 柏原正樹

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Kashiwara’s thesis and D-module theory

Then came Masaki Kashiwara. In his master’s thesis, TokyoUniversity, 1970, Kashiwara establishes the foundations ofanalytic D-module theory, a theory which is now afundamental tool in many branches of mathematics, fromnumber theory to mathematical physics. (Soon after andindependently, Joseph Bernstein developed a similar theory inthe algebraic setting.)

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On a complex manifold X , one has the non commutative sheafof rings DX of holomorphic partial differential operators. A leftcoherent DX -module M is locally represented (non uniquely)by an exact sequence

DN1X·A0−→ DN0X −→M −→ 0.

In other words, M is locally the cokernel of a matrix ofdifferential operators acting on the right on DX .This is the intrinsic way to formulate what is a finite system ofLPDE with finitely many unknowns.

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In his thesis, Kashiwara gives the main algebraic properties ofthe sheaf of rings DX (coherency, homological dimension,etc.).

He defines the characteristic variety char(M) of a coherentDX -module M, a closed conic complex analytic subset of thecotangent bundle T ∗X .

He defines the operations on D-modules and extends theclassical Cauchy-Kowalevski theorem to general systems ofLPDE.

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The SKK paper

The seventies are, for the analysts, the time of microlocalanalysis, initiated by Mikio Sato with the analytic wave frontset. In the famous paper, known as [SKK], by Mikio Sato,Takahiro Kawai and Masaki Kashiwara, the sheaf of rings ofmicrodifferential operators is constructed on the cotangentbundle T ∗X and two fundamental results are obtained.

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(i) The involutivity of characteristics of (micro)-differentialsystems. This fundamental question had, around 1970, only apartial answer (Quillen, Guillemin and Sternberg). Later, in1981, Gabber will give a purely algebraic proof.

(ii) The microlocal classification at generic points of anysystem of microdifferential equations on a real manifold. Theseare proved to be equivalent, after a so-called quantized contacttransform, to a combination of a partial De Rham system, apartial Dolbeault system and a Hans Lewy’s type system.

The SKK paper had an enormous influence on the analysis ofpartial differential equations (see in particular the work ofHörmander and Sjöstrand).

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The Riemann-Hilbert correspondence

Since the characteristic variety of a coherent D-module isinvolutive, it is natural to look at the extreme case, when thisvariety is Lagrangian. One calls such systems “holonomic”.

They are the higher dimensional version of classical ordinarydifferential equations (ODE). Among ODE, there is a class ofparticular interest, called the class of Fuchsian equations orequations with regular singularities.

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Roughly speaking, the classical Riemann-Hilbert (R-H)correspondence is based on the following question: given afinite set of points on the Riemann sphere and, at each point,an invertible matrix of complex numbers, does it exist a uniqueFuchsian ODE whose singularities are the given points andsuch that the monodromy of its holomorphic solutions are thegiven matrices.A partial answer is given by Pierre Deligne in 1970 who treatsregular connections.

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From 1975 to 1980, Masaki Kashiwara gives a preciseformulation of the R-H correspondence in any dimension andproves it.

In 1975 he proves that the contravariant functor Sol , which toa holonomic DX -module M associates the complex of itsholomorphic solutions, takes its values in the derived categoryof C-constructible sheaves.At the same time, he proves that the complex one obtainssatisfies what will be called five years later by Beilinson,Bernstein, Deligne and Gabber, the perversity conditions.

Notice that the idea of perverse sheaves comes from LPDE!

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Then Kashiwara introduced (around 1978, with Oshima andKawai) the notion of regular holonomic module and proved in1980 the R-H correspondence, namely an equivalence ofcategories between the derived category of regular holonomicD-modules and the derived category of C-constructiblesheaves.For that purpose, he introduced the functor Thom oftempered cohomology.

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Microlocal sheaf theory

From 1982 to 1990, Kashiwara and myself introduced anddeveloped the microlocal theory of sheaves.

This theory emerged from a joint paper in which we solved theCauchy problem for hyperfunction solutions of hyperbolicD-modules. Indeed, the basic idea is that of microsupport ofsheaves which gives a precise meaning to the concept ofpropagation for sheaves on real manifolds.

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The microsupport of sheaves is, in some sense, a real analogueof the characteristic variety of coherent D-modules on complexmanifolds and the functorial properties of the microsupport arevery similar to those of the characteristic variety of D-modules.Moreover, the microsupport of the complex Sol(M) ofholomorphic solutions of a coherent D-module M is nothingbut the characteristic variety of M. This is essentially acorollary of the Cauchy-Kowalewski theorem, in its preciseform due to Petrowski-Leray.

Moreover, and this is one of the main results of the theory, themicrosupport is involutive.

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Microlocal sheaf theory has found applications in other fieldsof Mathematics.

• First, in representation theory, as we shall see soon.• In symplectic topology and knot theory, essentially with

Tamarkin, Nadler, Zaslow, and also Guillermou, Shende,Treumann and many others.

• Recently, Alexander Beilinson adapted the definition ofthe microsupport of sheaves to arithmetic geometry andthe theory is now developing, in particular by TakeshiSaito.

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Other major contributions

In this presentation, I have preferred to concentrate on oneaspect of Kashiwara’s work, namely Algebraic Analysis, ratherthan to compile a catalog of all his work, and even in that case20 minutes would not have been enough.

Let’s remember, however, that Masaki Kashiwara obtainedfundamental results in many other fields of Mathematics.

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Representation theory• Kashiwara proves with Michèle Vergne a fundamental

result on the Campbell-Hausdorff formula (1978). Thereis currently a lot of activity stemming from this paper.

• With Jean-Luc Brylinski, he solves the Kazhdan-Lusztigconjecture on infinite-dimensional representations ofsimple Lie algebras (1981) (a result obtainedsimultaneously by Beilinson-Bernstein).

• These results were generalized to Kac-Moody algebras byKashiwara with Toshiyuki Tanisaki (1990).

• He reinterprets the Harish-Chandra theory in terms ofD-module theory and obtains by this method importanttheorems on semi-simple Lie algebras with Ryoshi Hottain 1984 and on real reductive groups with WilfriedSchmid (1994).

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Quantum groups

In 1990, inspired by solvable lattice models, Kashiwaraintroduces the groundbreaking notion of crystal bases ofquantum groups.He developed this idea with collaborators, in particularSeok-Jin Kang, Myungho Kim and Se-jin Oh.

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Mathematical Physics

Masaki Kashiwara has also obtained important results inMathematical Physics.

• In collaboration with T. Kawai and H. Stapp, he appliedthe theory of holonomic D-modules to the study ofFeynman integrals (1977).

• With Etsuro Date, Michio Jimbo and Tetsuji Miwa(1981–82), he found links between hierarchies of solitonequations and representations of infinite dimensional Liealgebras, a subject initiated by Mikio Sato and YasukoSato.

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Recent work

• Kashiwara and myself have developed the theory ofDeformation Quantization (DQ) modules, obtaining inparticular an analogue of the Grauert direct imagetheorem and a theorem of perversity for holonomicDQ-modules (2006–2012).

• Kashiwara and Raphaël Rouquier apply the theory ofDQ-modules to construct a microlocalization of rationalCherednik algebras (2008).

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• Kashiwara and Kari Vilonen have solved the so-calledcodimension 3 conjecture (2015), a holonomic version ofthe classical Siu-Trautmann theorem.

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• Kashiwara and myself have introduced and developed thetheory of indsheaves and, as a particular case, sheaves forthe subanalytic topology (2001). This allows one todefine the indsheaf of tempered holomorphic functions ona complex manifold and we showed that the complex Sol tof tempered holomorphic solutions of an irregularholonomic D-module contains much more informationsthan the usual complex Sol .

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• On the other hand, around 2010, Takuro Mochizuki madea breakthrough in the theory of irregular holonomicD-modules (with contributions of C. Sabbah andK-S Kedlaya). Based on this work, and with the tool ofSol t , Kashiwara, in collaboration with A. D’Agnolo,obtained a kind of irregular R-H correspondence (2016).

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Conclusion

Kashiwara’s contribution to mathematics is really astonishingand it should be mentioned that his influence is not only due tohis published work, but also to many informal talks. Importantsubjects such as second microlocalization, complex quantizedcontact transformations, the famous“watermelon theorem”,etc. were initiated by him, although not published. Kashiwarais an invaluable source of inspiration for many people.

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Figure : Masaki Kashiwara at IHES, 2015

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