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Mikio Sato, a Visionary of Mathematics, Volume 54, Number 2

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Mikio Sato, a Visionary of Mathematics, Volume 54, Number 2 Mikio Sato, a Visionary of Mathematics Pierre Schapira Like singularities in mathematics and physics, ideas People were essentially looking for existence propagate, and the speed of propagation depends theorems for linear partial differential equations highly on the energy put into promoting them. (PDE), and most of the proofs were reduced to Mikio Sato did not spend a great deal of time or finding “the right functional space”, to prove some energy popularizing his ideas. We can hope that a priori estimate and apply the Hahn-Banach his receipt of the 2002/2003 Wolf Prize1 will help theorem. make better known his deep work, which is perhaps It was in this environment that Mikio Sato too original to be immediately accepted. Sato does defined hyperfunctions in 1959–1960 as boundary not write a lot, does not communicate easily, and values of holomorphic functions, a discovery that attends very few meetings. But he invented a new allowed him to obtain a position at Tokyo Univer- way of doing analysis, “Algebraic Analysis”, and sity thanks to the clever patronage of Shokichi created a school, “the Kyoto school”. Iyanaga, an exceptionally open-minded person 2 Born in 1928 , Sato became known in mathe- and a great friend of French culture. Next, Sato matics only in 1959–1960 with his theory of spent two years in the USA, in Princeton, where he hyperfunctions. Indeed, his studies had been unsuccessfully tried to convince André Weil of the seriously disrupted by the war, particularly by relevance of his cohomological approach to the bombing of Tokyo. After his family home analysis. burned down, he had to work as a coal delivery man Sato’s method was radically new, in no way and later as a school teacher. At age 29 he became using the notion of limit. His hyperfunctions are an assistant professor at Tokyo University. He stud- not limits of functions in any sense of the word, ied mathematics and physics, on his own. and the space of hyperfunctions has no natural To understand the originality of Sato’s theory topology other than the trivial one. For his con- of hyperfunctions, one has to place it in the struction, Sato invented local cohomology in mathematical landscape of the time. Mathematical parallel with Grothendieck. This was truly a revo- analysis from the 1950s to the 1970s was under lutionary vision of analysis. the domination of functional analysis, marked But besides its evident originality, Sato’s ap- by the success of the theory of distributions. proach had deep implications since it naturally led Pierre Schapira is professor of mathematics at the Uni- to microlocal analysis, as I will try to explain. versité Pierre et Marie Curie, Institut de Mathématiques, The theory of linear PDE with variable coeffi- Paris, France. His contact addresses are: schapira@ cients was in its early beginnings in the years math.jussieu.fr. and http://www.math.jussieu. 1965–1970 and under the shock of Hans Lewy’s fr/~schapira/. example√ showing that the first√ order linear equa- 1 Translation of a paper appearing in La Gazette des tion (− −1∂1 + ∂2 − 2(x1 + −1x2)∂3)u = v had Mathématiciens 97 (2003) on the occasion of Sato’s re- no solution, even a local solution, in the space of ceiving the 2002/2003 Wolf Prize. distributions3. The fact that an equation had no 2 In 1990, Sato gave an interview to Emmanuel Andronikof √ 3 who unfortunately passed away in 1994. I have made use The slightly simpler equation (∂1 + −1x1∂2)u = v does of his notes, which were edited by A. D’Agnolo. I also have not have any solution in the space of germs at the origin benefited from the scientific comments of J-B. Bost and of distributions in R2 either, nor even in the space of A. Chambert-Loir, whom I warmly thank. germs of hyperfunctions. FEBRUARY 2007 NOTICES OF THE AMS 243 Indeed, pseudo-differential operators did exist before the wave front set. But Sato was the first to make the objects of analysis, such as distributions, live in the cotangent space, and for that purpose he constructed a key tool of sheaf theory, the microlocalization functor, that is, the “Fourier- Sato” transform of the specialization functor. This is also the origin of the microlocal theory of sheaves of [3]. In 1973 Sato and his two students, M. Kashiwara and T. Kawai, published a treatise on the microlocal analysis of PDE [8]. Certainly this work had a considerable impact, although most an- alysts did not understand a single word. Hörman- der and his school then adapted the classical Fourier transform to these new ideas, leading to the Mikio Sato (left) with Pierre Schapira, around 1972. now popular theory of Fourier-integral operators. solution was quite disturbing at that time. People Already in the 1960s, Sato had the intuition of D-module theory, of holonomic systems, and of the thought that it was a defect of the theory, that the b-function (the so-called Bernstein-Sato b-func- spaces one had considered were too small to admit tion). He gave a series of talks on these topics at the solutions. Of course, often just the opposite Tokyo University but had to stop for lack of com- is true and one finds that the occurrence of a batants. His ideas were reconsidered and system- cohomological obstruction heralds interesting atically developed by Masaki Kashiwara in his 1969 phenomena: the lack of a solution is the demon- thesis ([1], [2]). As its name indicates, a D-module stration of some deep and hidden geometrical is a module over the (sheaf of) ring(s) D of differ- phenomena. In the case of the Hans Lewy equation, ential operators, and a module over a ring essen- the hidden geometry is “microlocal”, and this tially means “a system of linear equations” with equation is microlocally equivalent to an induced coefficients in this ring. The task is now to treat Cauchy-Riemann equation on a real hypersurface (general) systems of linear PDE. This theory, which of the complex space. also simultaneously appeared in Moscow in a more In mathematics, as in physics, in order to treat algebraic framework developed by Gelfand’s stu- phenomena in a given (affine) space, one is natu- dent J. Bernstein, quickly had considerable suc- rally led to compute in the dual space. One way, cess in several branches of mathematics. In 1970– the most commonly used in analysis, is via the 1980, Kashiwara obtained almost all the funda- Fourier transform. This transform, far from being mental results of the theory, in particular those of a local nature, is not easily adapted to calculus concerned with holonomic modules, such as his on manifolds. By contrast, Sato’s method is per- constructibility theorem, his index theorem for fectly suited for this case: you can complexify a real holomorphic solutions of holonomic modules, analytic manifold and, instead of looking at the the proof of the rationality of the zeroes of the behavior at infinity of the Fourier transform, you b-function, and his theory of regular holonomic look “where the boundary values come from”. In modules. technical terms, one regards the cotangent bundle √ The mathematical landscape of 1970–1980 had (more precisely, −1-times the cotangent bundle) thus considerably changed. Not only did one treat as the conormal bundle to the real space in the com- equations with variable coefficients, but one treated plex space. This is how Sato defines the analytic systems of such equations and moreover one wave front set of hyperfunctions (in particular, of worked microlocally, that is, in the cotangent bun- distributions), a closed conic subset of the cotan- dle, the phase space of the physicists. But there gent bundle, and he shows that if a hyperfunction were two schools in the world: the C∞ school u is a solution of the equation Pu =0, then its issuing from classical analysis and headed by wave front set is contained in the real part of the Hörmander, who developed the calculus of Fourier characteristic variety of the operator P. This is the integral operators4, and the analytic school that starting point of microlocal analysis, invented by Sato established, which was almost nonexistent Sato, a kind of revolution in analysis. outside Japan and France. Of course, at this time other mathematicians France was a strategic place to receive Sato’s (especially L. Hörmander) and physicists (e.g., ideas since they are based on those of both Jean D. Iagolnitzer) had the intuition that the cotangent Leray and Alexandre Grothendieck. Like Leray, Sato space was the natural space for analysis, and in fact this intuition arose much earlier (in the work of 4Many names should be quoted at this point, in particu- J. Hadamard, F. John, and J. Leray, in particular). lar those of V. Maslov and Yu. Egorov. 244 NOTICES OF THE AMS VOLUME 54, NUMBER 2 understood that singularities have to be sought in studying algebraic cycles and Sato by computing the complex domain, even for the understanding numerical data. of real phenomena. Sato’s algebraic analysis is Sato’s most recent works are essentially un- based on sheaf theory, a theory invented by Leray published and have been presented in seminars in 1944 when he was a prisoner of war, clarified attended only by a small group of people. They treat by Cartan, and made extraordinarily efficient by an algebraic approach to nonlinear systems of PDE, Grothendieck and his formalism of derived cate- in particular holonomic systems, of which theta gories and the six opérations. functions are examples of solutions! Sato, motivated by physics as usual, then Looking back, forty years later, we realize that tackled the analysis of the S-matrix in light of Sato’s approach to mathematics is not so different microlocal analysis.
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