Memories of Mikio Sato (1928–2023)

Mikio Sato (born April 18, 1928, dec. January 9, 2023) was a mathematician of great depth and originality. Initiator of the theory of hyperfunctions and $D$-modules, he made influential contributions in diverse areas of mathematics ranging from analysis, number theory to mathematical physics.

Figure 1.

Mikio Sato, Nice, France 1970.

He earned his degree in 1963 at the University of Tokyo. After holding appointments as professor at Osaka University and the University of Tokyo, in 1970 he moved to the Research Institute for Mathematical Sciences (RIMS), Kyoto University, where he served as director from 1987 to 1991. He was awarded the Asahi Prize in 1969, the Japan Academy Prize in 1976, the Fujiwara Prize in 1987, the Rolf Schock Prize in 1997 and the Wolf Prize in 2003. He was also chosen a Person of Cultural Merit by the Japanese government in 1984.

Mikio Sato, a Visionary of Mathematics

By Pierre Schapira

Mikio⁠¹ Sato’s passing on January 9, 2023 was very sad news for all of us who had the chance to meet him and share his vision of mathematics. Sato’s vision was radically new and revolutionary, too much so to be immediately understood by the mathematical community, in particular, by the analysts. Sato’s aim in developing the theory of “algebraic analysis” was to treat problems of analysis using the tools of algebraic geometry. However, Sato never made any special effort to propagate his ideas widely.

¹

This paper is a modified version of a text that already appeared in the Notices of the AMS, February 2007 after a first publication in French, in La Gazette des Mathématiciens 97 (2003) on the occasion of Sato’s reception of the 2002/2003 Wolf Prize. The part concerned with number theory of these publications had benefited from the scientific comments of Jean-Benoît Bost and Antoine Chambert-Loir, and also Pierre Colmez for the last version. I warmly thank all of them.

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Sato did not write a lot, did not communicate easily, and attended very few meetings, but he invented an influential new way of doing analysis, and opened a new horizon in mathematics: the microlocal approach. Thanks to Sato, we understand that many phenomena which appear on a manifold are in fact the projection on the manifold of things living in the cotangent bundle to the manifold. Being “local” becomes, in some sense, global with respect to this projection.

Mikio Sato also created a school, “the Kyoto school,” among whom Masaki Kashiwara, Takahiro Kawai, Tetsuji Miwa, and Michio Jimbo should be mentioned.

Born in 1928,⁠² Sato became known in mathematics in 1959–1960 with his theory of hyperfunctions. Indeed, his studies had been seriously disrupted by the war, particularly by the bombing of Tokyo. After his family home was burned down, he had to work as a coal delivery man and later as a school teacher. At age 29 he became an assistant at the University of Tokyo. He studied mathematics and physics, on his own.

²

See 1 for more details about Sato’s life.

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To understand the originality of Sato’s theory of hyperfunctions, one has to place it in the mathematical landscape of the time. Mathematical analysis from the 1950s to the 1970s was under the domination of functional analysis, marked by the success of the theory of distributions. People were essentially looking for existence theorems for linear partial differential equations (LPDE) and most of the proofs were reduced to finding “the right functional space,” to obtain some a priori estimate and apply the Hahn–Banach theorem.

It was in this environment that Sato defined hyperfunctions 2 in 1959–1960 as boundary values of holomorphic functions, a discovery which allowed him to obtain a position at the University of Tokyo thanks to the clever patronage of Professor Iyanaga, an exceptionally open-minded person and a great friend of French culture. Next, Sato spent two years in the USA, in Princeton, where he unsuccessfully tried to convince André Weil of the relevance of his cohomological approach to analysis.

Sato’s method was radically new, in no way using the notion of limit. His hyperfunctions are not limits of functions in any sense of the word, and the space of hyperfunctions has no natural topology other than the trivial one. For his construction, Sato invented local cohomology in parallel with Alexander Grothendieck. This was truly a revolutionary vision of analysis. And, besides its evident originality, Sato’s approach had deep implications since it naturally led to microlocal analysis.

The theory of LPDE with variable coefficients was at its early beginnings in the years 1965–1970 and under the shock of Hans Lewy’s example which showed that a very simple first order linear equation $(-\sqrt {-1}\partial _1 +\partial _2-2(x_1+\sqrt {-1}x_2)\partial _3)u=v$ had no solution, even a local solution, in the space of distributions.⁠³ The fact that an equation had no solution was quite disturbing at that time. People thought it was a defect of the theory, that the spaces one had considered were too small to admit solutions. Of course, often just the opposite is true and one finds that the occurrence of a cohomological obstruction heralds interesting phenomena: the lack of a solution is the demonstration of some deep and hidden geometrical phenomena. In the case of the Hans Lewy equation, the hidden geometry is “microlocal” and this equation is microlocally equivalent to an induced Cauchy–Riemann equation on a real hypersurface of the complex space.

³

The slightly simpler equation $(\partial _1 +\sqrt {-1}x_1\partial _2)u=v$ does not have any solution in the space of germs at the origin of distributions in $\mathbb{R}^2$ either, nor even in the space of germs of hyperfunctions.

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Figure 2.

Mikio Sato and Pierre Schapira, ICM, Nice 1970.

In mathematics, as in physics, in order to treat phenomena in a given (affine) space, one is naturally led to compute in the dual space. One method to pass from a vector space to its dual, which is most commonly used in analysis, is the Fourier transform. This transform, being not of local nature, is not easily adapted to calculus on manifolds. By contrast, Sato’s method is perfectly suited to manifolds. If $M$ is a real analytic manifold, $X$ a complexification of $M$, what plays the role of the dual space is now the conormal bundle $T^*_MX$ to $M$ in the cotangent bundle $T^*X$, something local on $M$. (Note that $T^*_MX$ is isomorphic to $\sqrt {-1}T^*M$.) In order to pass from $M$ to $T^*_MX$, Sato constructed a key tool of sheaf theory, the microlocalization functor $\mu _M(\cdot )$, the “Fourier–Sato” transform of the specialization functor $\nu _M(\cdot )$. This is how Sato defines the analytic wave front set of hyperfunctions (in particular, of distributions), a closed conic subset of the cotangent bundle, and he shows that if a hyperfunction $u$ is a solution of the equation $Pu=0$, then its wave front set is contained in the intersection with $T^*_MX$ of the characteristic variety of the operator $P$ 3. It is then clear (but it was not so clear at the time) that if you want to understand what happens on a real manifold, you better look at what happens on a complex neighborhood of the manifold.

Of course, at this time other mathematicians (especially Lars Hörmander) and physicists (e.g., Daniel Iagolnitzer) had the intuition that the cotangent bundle was the natural space for analysis, and in fact this intuition arose much earlier, in particular in the work of Jacques Hadamard, Fritz John, and Jean Leray. Indeed, pseudo-differential operators did exist before the wave front set.

In 1973, Sato and his two students, M. Kashiwara and T. Kawai, published a treatise 4 on the microlocal analysis of LPDE. Certainly this work had a considerable impact, although most analysts didn’t understand a single word. Hörmander and his school then adapted the classical Fourier transform to these new ideas, leading to the now popular theory of Fourier-integral operators (see for example 5).

The microlocalization functor is the starting point of microlocal analysis but it is also at the origin of the microlocal theory of sheaves, due to Kashiwara and the author 6. This theory associates to a sheaf on a real manifold $M$ its microsupport, a closed conic subset of the cotangent bundle $T^*M$ and allows one to treat sheaves “microlocally” in $T^*M$. The theory of systems of LPDE becomes essentially sheaf theory, the only analytic ingredient being the Cauchy-Kowalevsky theorem. One of the deepest results of 4 was the involutivity theorem which asserts that the characteristic variety of a microdifferential system (in particular, of a $\mathcal{D}$-module, see below) is co-isotropic. Similarly the microsupport of a sheaf is proved to be co-isotropic, which makes a link between sheaf theory and symplectic topology that is the origin of numerous important results.

Also note that by a fair return of things, microlocal analysis, through microlocal sheaf theory, appeared quite recently in algebraic geometry, under the impulse of Sasha Beilinson 7.

Already in the 1960s, Sato had the intuition of $\mathcal{D}$-module theory, of holonomic systems and of the $b$-function (the so-called Bernstein–Sato $b$-function). He gave a series of talks on these topics at Tokyo University but had to stop for lack of combatants. His ideas were reconsidered and systematically developed by Masaki Kashiwara in his 1969 thesis.⁠⁴ As its name indicates, a $\mathcal{D}$-module is a module over the sheaf of rings $\mathcal{D}$ of differential operators, and a module over a ring essentially means “a system of linear equations”⁠⁵ with coefficients in this ring. The task is now to treat (general) systems of LPDE. This theory, which also simultaneously appeared in the more algebraic framework developed by Joseph Bernstein, a student of Israel Gelfand, quickly had considerable success in several branches of mathematics. In 1970–1980, Kashiwara obtained almost all the fundamental results of the theory, in particular those concerned with holonomic modules, such as his constructibility theorem, his index theorem for holomorphic solutions of holonomic modules, the proof of the rationality of the zeroes of the $b$-function, and his proof of the (regular) Riemann-Hilbert correspondence.

⁴

See 8 for an overview of Kashiwara’s work, a part of which was deeply influenced by Sato’s ideas.

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⁵

According to Mikio Sato (personal communication), at the origin of this idea is the mathematician and philosopher of the 17th century, E. W. von Tschirnhaus.

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The mathematical landscape of 1970–1980 had thus considerably changed. Not only did one treat equations with variable coefficients, but one treated systems of such equations and moreover one worked microlocally, that is, in the cotangent bundle, the phase space of the physicists. But there were two schools in the world: the $C^\infty$ school, in the continuation of classical analysis and headed by Hörmander who developed the calculus of Fourier integral operators,⁠⁶ and the analytic school that Sato established, which was almost nonexistent outside Japan and France.

⁶

Many names should be quoted at this point, in particular those of Viktor Maslov and Vladimir Arnold.

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France was a strategic place to receive Sato’s ideas since they are based on, or parallel to, those of both Jean Leray and Alexander Grothendieck. Like Leray, Sato understood that singularities have to be sought in the complex domain, even for the understanding of real phenomena. Sato’s algebraic analysis is based on sheaf theory, a theory invented by Leray in 1944 when he was a prisoner of war, clarified by Henri Cartan, and made extraordinarily efficient by Grothendieck and his formalism of derived categories and the six opérations.

Sato, who was motivated by physics as usual, then tackled the analysis of the $S$-matrix in light of microlocal analysis. With his two new students, M. Jimbo and T. Miwa 9, he explicitly constructed the solution of the $n$-points function of the Ising model in dimension $2$ using Schlesinger’s classical theory of isomonodromic deformations of ordinary differential equations. This naturally led him to the study of KdV-type nonlinear equations. In 1981, he and his wife Yasuko Sato (see 10 and 11) interpreted the solutions of the KP-hierarchies as points of an infinite Grasmannian manifold and introduced his famous $\tau$-function. These results would be applied to other classes of equations and would have a great impact in mathematical physics in the study of integrable systems and field theory in dimension $2$.

In parallel with his work in analysis and in mathematical physics, Sato obtained remarkable results in group theory and in number theory. He introduced the theory of prehomogeneous vector spaces, that is, of linear representations of complex reductive groups with a dense orbit. The important case where the complement of this orbit is a hypersurface gives good examples of $b$-functions (see 1213).

In 1962, using a construction of auxiliary (Kuga-Sato) varieties, Sato also discovered how to deduce the Ramanujan conjecture on the coefficients of the modular form $\Delta$ from Weil’s conjectures concerning the number of solutions of polynomial equations on finite fields. His ideas allowed Michio Kuga and Goro Shimura to treat the case of compact quotients of the Poincaré half-space and one had to wait another ten years for Pierre Deligne to definitely prove that Weil’s conjectures imply Ramanujan and Petersson’s conjecture.

Mikio Sato shared the Wolf Prize with John Tate in 2002/03. They also share a famous conjecture in number theory concerning the repartition of Frobenius angles. Let $P$ be a degree-3 polynomial with integer coefficients and simple roots. Hasse has shown that for any prime $p$ which does not divide the discriminant of $P$, the number of solutions of the congruence $y^2=P(x)\pmod p$ is like $p-a_p$, with $\mathopen{|} a_p \mathclose{|}\leq 2\sqrt p$. When writing $a_p= 2\sqrt p \cos \theta _p$ with $0\leq \theta _p\leq \pi$, the Sato–Tate conjecture predicts that these angles $\theta _p$ are distributed with the probability density $(2/\pi )\sin ^2\theta$ (in absence of complex multiplication). Note that Tate was led to this conjecture by the study of algebraic cycles and Sato by computing numerical data.

Sato’s most recent works are essentially unpublished (see however 14) and have been presented in seminars attended only by a small group of people. They treat an algebraic approach of nonlinear systems of PDE, in particular of holonomic systems, of which theta functions are examples of solutions!

Looking back, 50 years later, we realize that Sato’s approach to mathematics is not so different from that of Grothendieck, that Sato did have the incredible temerity to treat analysis as algebraic geometry and was also able to build the algebraic and geometric tools adapted to his problems.

His influence on mathematics is, and will remain, considerable.

Pierre Schapira is a professor emeritus at the Sorbonne Université, CNRS IMJ-PRG, France. His email address is pierre.schapira@imj-prg.fr.

Personal Reminiscences of the Late Professor Mikio Sato

By Takahiro Kawai and Yoshitsugu Takei

Concerning the concrete description of Sato’s works (including those in number theory) we refer the reader to the “Commentary” of the collected papers of M. Sato 15, and here we note some impressive episodes which lie behind them.

1) In 1973 in Nice, one of us (T. Kawai) was waiting for a friend to come to bring him to the airport. Suddenly Sato appeared and urged Kawai to take a taxi. As a result, Kawai was just in time for the flight. It was surprising as Sato was notorious for being rather loose in time: He himself said he had been amazed to learn the word “punctual.”

2) In 1974, Kawai asked Sato whether he should begin the study of nonlinear differential equations or study the analytic $S$-matrix theory. His response was very clear: “You might study nonlinear differential equations at any time you would wish to do, but you should study the $S$-matrix theory before 30 years of age.” It was a critically useful suggestion in Kawai’s career.

Figure 3.

Sato’s family at the coast of the Mediterranean Sea near CIRM, Marseille, France, 1991.

3) These facts show how carefully Sato observed the surroundings. We guess it was tightly tied up with the fact that Sato had once been a night-school teacher to support his family.

4) At the same time, Sato always enjoyed discussions with young people: As the late Shintani said, “How to use Sato? Bring one to him, then he will give you back ten.”

5) The most remarkable example was the “Sato-Komatsu seminar” in 1968 and 1969. It was a part of a very well-prepared project of the late Professor Komatsu to call Sato back to the theory of partial differential equations from number theory. It included an explanation of his results to Sato in New York in 1966 and his introductory lectures on the theory of hyperfunctions in 1967. The historically marvelous result of this seminar was Sato’s creation of microfunction theory in 1969. We tremble to imagine what if Komatsu had not organized the seminar.

6) Parenthetically we note that [Kawai-Takei; Adv. Math. 80 (1990), 110–133] is a nice successor to F. Suzuki’s talk in the seminar on first order equations.

Figure 4.

At the party celebrating Kawai’s 60th birthday in 2005: Sato, Kawai and his wife, and Kashiwara (from the left) for the first, and Pham and Sato (from the left) for the second.

7) Apparently Sato recalled the importance of the Sato-Komatsu seminar. When he was inspired by F. Pham’s talk at the Delphi symposium in 1987 and became confident that the singular perturbation theory fortified with the Borel resummation was important in analysis, he urged the surrounding people to study this subject with him. His enthusiasm for this trial was very impressive particularly in view of the fact that he was busy as the director of the Research Institute for Mathematical Sciences of Kyoto University.

8) Fortunately his enthusiasm found a nice resonator: T. Aoki figured out Stokes curves with the help of a computer. In parenthesis, we feel nostalgic to see the first figure which consists of many tiny arrows indicating the vector fields.

Figure 5.

At the party celebrating Komatsu’s 80th birthday in 2015: Komatsu, Kataoka, and Sato (from the left).

9) A great person has a broad view of life: Sato left the study of the singular perturbation theory to young students when he felt they had begun to study the novel subject, and he tried to attack new issues. “Papers are like decorations,” he once said. We refer the reader to 16 for the later development of the theory.

Many thanks for all your support and advice, Mentor Sato!

Takahiro Kawai is a professor emeritus at RIMS, Kyoto University.

Yoshitsugu Takei is a professor in the Science and Engineering Department of Mathematical Sciences at Doshisha University. His email address is ytakei@mail.doshisha.ac.jp.

Mikio Sato, My Mentor

By Masaki Kashiwara

I first saw Professor Mikio Sato when I was a senior in the University of Tokyo in 1968. He was then starting to construct the theory of microlocal analysis.

He was the main speaker of the seminar on algebraic analysis. This seminar was organized by Professor Hikosaburo Komatsu (1935–2022) in order to understand the work of Sato. It was a weekly seminar just started in 1968. Sato gave numerous talks explaining his theory. One of his main ideas was to consider a real manifold inside a complex manifold (a complex neighborhood). This already appeared in his theory of hyperfunctions. However he expanded this idea to a great extent. Hyperfunctions are obtained as sums of boundary values of holomorphic functions defined in a tubular neighborhood (in a complex neighborhood). We can think that a hyperfunction has milder singularities if these tubular neighborhoods are bigger. It was the starting point of his theory of microlocal analysis. Through these considerations, he found that those generalized functions live in the cotangent bundle. Of course we can study functions locally on a real manifold. We can study functions on the cotangent bundle much more precisely. This is the essence of microlocal analysis.

In order to construct the theory, he had to start from the beginning to establish a basic theory by hand. For example, he constructed the theory of relative cohomologies of sheaves in his own way in order to introduce microfunctions. When he constructed microfunctions, he introduced the notion of the Fourier-Sato transform of sheaves. The original Fourier transform is a correspondence between functions on a vector space and ones on the dual vector space. Sato found a correspondence between sheaves on a vector space and ones on the dual vector space, which is now called the Fourier-Sato transform.

He also noticed the importance of $D$-modules in the study of linear partial differential equations. Especially, he introduced the notion of maximally overdetermined systems (of linear partial differential equations), which is now known under the name of “holonomic $D$-modules.” These new ideas were presented in the Algebraic Analysis Seminar. I was very impressed by the overflowing ideas of Sato, and I decided to study algebraic analysis when I became a master course student.

Figure 6.

Mikio Sato, Katata conference, 1971.

In 1970, Professor Sato moved to Research Institute for Mathematical Sciences (RIMS), Kyoto University. In the next year, I followed him to RIMS. I worked with him and Takahiro Kawai to complete the theory of microlocal analysis. Inspired by the work of Maslov and Egorov, Sato introduced the notion of microdifferential operators which live on the cotangent bundle and act on microfunctions. Moreover, he discovered that microfunctions are invariant by symplectic transformation on the cotangent bundle. This opened a new understanding in analysis.

Figure 7.

Mikio Sato at RIMS Public Lecture, Kyoto 1976.

It is very sad that Professor Sato passed away on January 9, 2023. However, he remains my mentor and his marvelous work will remain forever.

Masaki Kashiwara is a professor at the Research Institute for Mathematical Sciences at Kyoto University. His email address is masaki@kurims.kyoto-u.ac.jp.

Mikio Sato and the Creation of the Tau Function

By Barry M. McCoy

I first met Professor Sato in 1980 at a week-long workshop sponsored by Erhart Seminar Training in San Francisco where Sato and his two junior collaborators, Michio Jimbo and Tetsuji Miwa, reported on their recent work which united the mathematical theory of isomonodromic deformation theory with the physics of the statistical mechanics of the Ising model. This was made possible by the invention of the tau function.

The journey to the invention of the tau function begins with the 1964 paper of John Myers 17 who discovered that the scattering of electromagnetic radiation by a finite conducting strip can be described with the use of a particular Painlevé III function. This was the first time which Painlevé functions had appeared in Physics.

The next step was published in 1976 by Wu, McCoy, Tracy, and Barouch 18 who found that Myers’ work could be adapted to make an exact computation of the two point correlation function of the two-dimensional Ising model in the continuum limit near the critical temperature. Remarkably, exactly the same Painlevé III function found by Myers was needed for the construction.

This was the background for the invention of the tau function. In a series of six remarkable papers from 1978 to 1980, Sato and his collaborators Miwa, Jimbo, and Mori 9,19 put these previous computations into the vastly more general mathematical setting of isomonodromic deformation of first order systems of $m \times m$ ordinary differential equations with $n$ singular points $a_j$

$$\begin{equation*} \frac{dY(z)}{dz}=\left\{\sum _{j=1}^n\sum _{k=0}^{r_k}\frac{A_{j,k}}{(z-a_j)^{k+1}} -\sum _{k=1}^{r_\infty }A_{\infty ,k}z^{k-1}\right\}Y(z). \end{equation*}$$

The study of these isomonodromic problems was first posed by Riemann and is one of Hilbert’s 24 problems. In the Fuchsian case where $r_k=r_{\infty }=0$, the conditions on the $A_j$ for the deformations of the singular points $a_j$ to be isomonodromic (the Schlesinger equations) were known by the beginning of the 20th century. The great advance made by Sato et al was the discovery in paper 2 of 9 of a closed one form which can be extended to all the Fuchsian systems

$$\begin{equation*} \omega =\frac{1}{2}\sum _{j\neq k} \operatorname {Tr} A_j(a)A_k(a) \operatorname {d} \ln (a_j-a_k) \end{equation*}$$

where the $A_j(a)$ satisfy Schlesinger’s equations and the eigenvalues of $A_j$ are all distinct with no integer differences. From this, the tau function is defined as

$$\begin{equation*} \operatorname {d} \operatorname {ln} \tau =\omega . \end{equation*}$$

This is generalized to non Fuchsian cases in paper 3 of 9 and 19. Furthermore, in 19 it was shown that this one form is a Hamiltonian which characterizes the tau function.

For a particular $2\times 2$ linear system, this tau function was then shown to characterize the scaled two point function of the Ising model and was computed from deformation theory in terms of a Painlevé V function which is equivalent to the Painlevé III function of 18. Even more impressive is the result obtained soon after by Jimbo and Miwa 20 that the diagonal two point function of the Ising model $C(N,N)$ on the lattice is the tau function of a $2 \times 2$ linear problem with four Fuchsian singularites which gives a Painlevé VI function.

Following the discoveries of 9 Jimbo, Miwa, Mori, and Sato 19 found that correlation functions for the impenetrable Bose gas in one dimension are tau functions of another $2\times 2$ system which characterizes Painlevé V functions. Subsequently tau functions have revolutionized the study of random matrix theory by expressing distributions of eigenvalues of random matrices in terms of Painlevé functions. More recently, further applications of tau functions to the Ising model have been made by Witte [Nonlinearity 29 (2016) 131–160] who found that the row correlation $C(0,N)$ is the tau function of a $2\times 2$ system with 6 singularities.

In the years following the discovery of the tau function, there have been many mathematical studies of the one forms $\omega$ for $m \times m$ systems with $m>2$. In 1984, Myers [Physica D 11 (1984) 51–89] studied the wave scattering from a broken corner by use of the exact same methods he previously used for the strip to obtain a solution which involves a $2 \times 2$ matrix version of the Painlevé III equation found in 17, which is based on a $4 \times 4$ linear system.

Subsequent to the Erhart Seminar Training workshop, I had the privilege of meeting Professor Sato many times at the Research Institute for Mathematical Sciences in Kyoto and attending his seminars. I am convinced that the applications of Sato’s tau function to physics have not yet been fully realized.

Barry M. McCoy is a Distinguished Professor Emeritus at the CNYang Institute for Theoretical Physics at StonyBrook University. His email address is mccoy@max2.physics.sunysb.edu.

In Memoriam of Mikio Sato

By Craig Tracy

I spoke with Professor Sato only a couple of times so my comments here necessarily lack first-hand personal stories. What I can do is highlight some of Professor Sato’s profound work in integrable systems and statistical physics.

In the 1970s, T. T. Wu, B. M. McCoy, E. Barouch, and I 1821 were involved in the analysis of the two-point correlation function, $\langle \sigma _{0 0}\sigma _{M N}\rangle$, of the two-dimensional, zero field Ising model in the massive scaling limit.⁠⁷ Using work of J. Myers 17 on scattering from a finite strip we derived the following

⁷

When the vertical and horizontal interactions are equal, the massive scaling limit is $R=\sqrt {M^2+N^2}\rightarrow \infty$, $\xi (T)\rightarrow \infty$ with $R/\xi (T)\rightarrow r$, $r>0$. Here $\xi (T)$ is the correlation length, $T$ is temperature, and $T_c$ the critical temperature where $\xi (T)\rightarrow \infty$ as $T\rightarrow T_c$.

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$$\begin{equation} \frac{\langle \sigma _{0 0}\sigma _{M N} \rangle }{\mathcal{M}_T^2} \longrightarrow \left\{\sinh (\frac{1}{2}\psi (r))\atop \cosh (\frac{1}{2}\psi (r))\right\} \times \exp \left[ \frac{1}{4} \int _r^\infty \left(\sinh ^2\psi (s)-(\frac{d\psi (s)}{ds})^2\right) s \, ds\right] {\tag{1}} \end{equation}$$

where the upper (lower) equation holds for $T\rightarrow T_c$ from above (below) and $\mathcal{M}_T$ is a normalization factor.⁠⁸ The function $\psi$ satisfies the differential equation in the scaling variable $r$

⁸

For $T<T_c$, $\mathcal{M}_T$ is the spontaneous magnetization, see 18 for details.

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$$\begin{equation} \frac{d^2 \psi }{dr^2}+\frac{1}{r} \frac{d\psi }{dr}= \frac{1}{2} \sinh (2\psi ){\tag{2}} \end{equation}$$

with boundary condition $\psi (r)\sim \frac{2}{\pi }K_0(r)$ as $r\rightarrow \infty$. The substitution $\eta (r)=e^{-\psi (r)}$ transforms 2 into a Painlevé III equation.

In the period 1978–1981, Sato, Miwa, and Jimbo 9 vastly extended the 2-point analysis to $n$-point functions, $\langle \sigma _{a_1}\cdots \sigma _{a_n}\rangle$. Introducing monodromy preserving deformations of the 2D Euclidean Dirac equation, they derived a set of deformation equations (function of the points $a_j, j=1,\ldots , n$). When $n=2$, the SMJ deformation equations reduce to the Painlevé III equation; thus “explaining” its appearance in the WMTB result. We remark that the analysis of the SMJ deformation equations is still open, e.g., connection formulas. For $n=2$ the connection problem is solved in 2122.

Sato together with Jimbo, Miwa, and Môri 19 studied $\tau (a)\coloneq \det (1-K)$ where $K$ is the integral operator with kernel

$$\begin{equation*} \frac{1}{\pi } \frac{\sin (x-y)}{x-y} \chi _I(y). \end{equation*}$$

Here $I = \bigcup _{j=1}^n(a_{2j-1},a_{2j})$ and $\chi _I(y)$ is the characteristic function of the set $I$. In random matrix theory, $\tau (a)$ is the probability no eigenvalue of the rescaled GUE lies in $I$. A deformation theory for the Fredholm integral equation is derived 19; and for the special case of $I=(-t/2,t/2)$, the deformation equations reduce to Painlevé V. This is the first appearance of Painlevé transcendents in random matrix theory.

The notion of a $\tau$-function is now a central object in integrable systems thanks to the work of Professor Sato and his colleagues Miwa and Jimbo.

Craig Tracy is a Distinguished Professor Emeritus in the department of mathematics at UCDavis. His email address is tracy@math.ucdavis.edu.

Memories from 2019 and 1980

By Tetsuji Miwa

Figure 8.

Hiroko and Masaki Kashiwara, Mikio and Yasuko Sato, Setsuko Miwa (from the left), Kyoto 2019.

My last meeting with Professor Mikio Sato was in November 2019, at Hakusa-Sonso, a Japanese restaurant near Ginkakuji. A photo is at hand, taken at a luncheon meeting. I sang a song of Mikis Theodrakis in the meeting.

In the summer of 1980, during the Soliton Meeting in Crete, Greece, one day was spent in an excursion up to the mountains and down to the beach, where an evening concert was held. There were several huge speakers, which blasted songs including some by Mikis Theodrakis. I was there in the first line of the audience with my family, and Mikio and Yasuko Sato and Michio Jimbo.

Tetsuji Miwa is a professor emeritus at Kyoto University. His email address is sanlunz442@gmail.com.

Professor Mikio Sato: Personal Recollections

By Michio Jimbo

1. In 1973, I was an undergraduate student at Tokyo, wondering whether to pursue my graduate study in Tokyo or Kyoto. Upon hearing my hesitation, Professor Sato kindly offered me an opportunity to visit him in Kyoto. This was quite a surprise, because I had never met him except at the oral examination earlier that summer. He took pains to explain to me in informal language what he had been doing with Kawai and Kashiwara, i.e., microlocal analysis. I did not understand much, but Sato’s enthusiasm and vision left a strong impression on me.

Some time after that, I had a chance to chat with Daisuke Fujiwara, whose lectures I was attending. When I mentioned my meeting with Sato, he said “If I were you I would go to Kyoto without a moment’s hesitation.” Realizing how much Sato was esteemed by young mathematicians, I followed Fujiwara’s advice.

2. I was extremely lucky to be able to join Sato’s research project, together with Miwa, during the years 1975–1980. This was the time when he had finished the monumental work 4 with Kawai and Kashiwara, and his interests were turned to problems in theoretical physics. The outcome of his research during this period, “holonomic quantum fields,” is summarized in the commentary on Sato’s collected papers 15.

Sato aimed at applying microlocal analysis to the study of S matrices and Green’s functions in quantum field theory. In order to gain insight into the problem, he spent some time looking for a simple nontrivial model where computations can be done explicitly. This led him to study the scaling limit of the Ising model.

Then he learned about the work of Wu, McCoy, Tracy, and Barouch 18 which gives the two point correlation functions in terms of a Painlevé transcendent. He soon recognized the role of monodromy-preserving deformation of linear differential equations behind the Ising model. The relevant deformation theory was formulated smoothly, but it remained unclear how to relate it to the calculation of spin-spin correlators. The missing key was found when he came up with the idea of inserting fermions into the correlators. (Though this may sound commonplace today, one should bear in mind that it was long before the advent of conformal field theory.)

Figure 9.

Tetsuji Miwa, Hermann Flaschka, Mikio Sato, and Jimbo (from the left). Clarkson College, USA, August 1979.

During the course, I was able to witness the way his thoughts shaped and developed. It was the most valuable occasion for me.

3. When I talked to Sato in private, he was always very kind and willing to explain on the blackboard. If it was over a meal he would start writing formulas on the back of a chopstick bag. I then started to feel that the ideas behind his work are very natural and even intuitively clear.

By all accounts, his lectures and conference talks were energetic and fascinating. He would start talking about the big picture, often meander into various related subjects, and eventually come to the point—only when the time was up.

Figure 10.

Mikio Sato and Yasuko Sato. Aiguille du Midi, France, September 1979.

4. Reading mathematical literature, Sato liked concisely written books and papers containing succinct statements, rather than those giving lengthy explanations including all the technical details. When proofs became necessary it was easier for him to produce them on his own. Since his youth, one of his favorite readings had been the Encyclopedia of Mathematics edited by the Japan Mathematical Society. He carefully chose genuinely important papers on a subject and carried copies with him. One such example was Lax’s paper on the Lax pair for soliton equations.

Figure 11.

Leon Takhtajan, Alexander Zamolodchikov, and Alexander Belavin (front from the left); Jimbo, Tetsuji Miwa, Mikio Sato and his son Nobuo (back from the left). Kyoto, October 1988.

5. Sato got interested in soliton theory because of its many structural similarities with holonomic quantum fields. He was particularly impressed by Hirota’s original method for solving various soliton equations.

Most soliton equations, such as the KdV and Toda equations, have an infinite number of mutually commuting flows, constituting hierarchies of higher order equations. Around 1979–1980, Mikio and Yasuko Sato performed a thorough-going study of Hirota’s bilinear equations. At a workshop at RIMS, they presented a list of all Hirota equations for KdV, KP, and other hierarchies to rather high orders, and reported that their numbers are given by partition numbers. It took them more than 1000 hours to carry through the computation with a pocket calculator. (One has to recall again that this was the time long before personal computers became available.) It was a mystery to everyone else why they were going through these elaborate computations.

The answer was revealed in Sato’s talk in January 1981. He explained that the space of solutions to the KP hierarchy forms an infinite-dimensional Grassmannian, and Hirota’s equations are the Plücker relations defining this manifold. This beautiful result, obtained after a long inductive process, is reminiscent of Euler’s work.

Sato once made the following comment about Euler: At first glance, his work might seem a collection of miscellaneous results. However I cannot but think that all was coherent in himself—that he had been led by something clearly visible to his own mind. His work is marvelous precisely for that reason. I feel that he was really a great master, greater than is usually considered.

It seems to me that these words best describe Sato’s own mathematics.

Michio Jimbo is a professor emeritus at Rikkyo University. His email address is jimbomm@rikkyo.ac.jp.