Schock Prize to Mikio Sato

The 1997 Schock Prize in Mathematics to Mikio Sato

Professor Mikio Sato received the 1997 Rolf Schock Prize in Mathematics. The prize diploma was conferred by Princess Christina of Sweden at a ceremony on October 23, 1997, in the Konserthuset in Stockholm.

The presentation speech follows.

Princess Christina, President Dag, Dear Laureates, Ladies and Gentlemen,

Professor Mikio Sato receives the Schock Prize in Mathematics for his creation of the theory of hyperfunctions. He has been the driving force behind a world-leading group of researchers in algebraic analysis. His work in theoretical physics has increased our understanding of the divergences in quantum field theory.

Mikio Sato presented his theory of hyperfunctions in 1958 in a paper of only twenty seven pages. It was in Japanese. He also published two notes in English in the same year; they were only five pages each. A little later, in 1959 and 1960, he published a more extensive description of his theory in two papers in English totalling one hundred and six pages. For several years, that was all we could read from him on his theory. But he and his students worked, and now there is a huge literature on this field of mathematics written by him, his students, and many other mathematicians.

Mikio Sato was born in 1928. In 1952 he graduated from Tokyo University, where he majored in number theory, a rather different branch of mathematics, under the direction of Professor Sh^okichi Iyanaga. He spent some years of graduate study in physics, with Professor Shin'ichiro Tomonaga as advisor, who was to be awarded a Nobel Prize in Physics in 1965. In 1970 Sato became professor at RIMS, the Research Institute for Mathematical Sciences, at Kyoto University.

I studied theoretical philosophy at Stockholm University a little before and partly overlapping with Rolf Schock. Never during my time there did I imagine that I would once be trying to explain mathematical research that had been rewarded with his money. And yet this is exactly what I shall try to do now. I learnt about hyperfunctions from André Martineau and also, although very briefly, from Professor Sato himself (this happened on September 4, 1970, to be precise).

Calculus is a mathematical theory concerned with change. Its basic processes are differentiation and integration. I am well aware that these words do not belong to the general culture. But many particular instances of them do. We speak about the speed of a car measured in meters per second or miles per hour. We speak about its acceleration, measured in meters per second per second. We speak about inflation in the economy, measured in per cent per year. All these words represent change, and they are instances of the mathematical notion of a derivative. We sum our monthly salaries to an income per year; we talk about a country's gross national product. These are instances of the mathematical notion of integration.

For many years, since Newton and Leibniz, calculus has been concerned with gradual, smooth change. If a car moves along a road we can handle that and calculate its speed and acceleration. It moves perhaps an inch during a millisecond.

But the gradual, smooth changes are not the only changes in nature. There are also sudden changes: cars sometimes collide; there are explosions; there are earthquakes. Classical calculus could not handle that. New concepts were needed. Mathematicians called these new objects generalized functions. A function is something which represents any process in nature or technology, anything that can change.

Oliver Heaviside designed already in the nineteenth century a symbolic calculus to handle jumps in a variable, like a sudden increase in price. His calculus gave the right answers, but did not explain why. Paul Dirac, another Nobel Prize winner, introduced something with which one could calculate and which represents a sudden blow, like a collision of two cars or a lightning hitting an electric circuit.

These are examples of how scientists started to come to grips with the notion of sudden change.

In the 1940s, Laurent Schwartz created a theory of generalized functions called distributions. It gave us a means to describe these sudden changes. Sato learnt about this theory, but he was not satisfied with it. He wanted to approach the generalized functions in another way. The idea he came up with was to consider a function, a process, as a limit of phenomena in the complex domain. This means that the calculation is done, not on the original function but on a function of complex variables---one also says imaginary (imagined!) numbers. If we cannot do the calculations on a car in a collision, then we can invent a lot of cars that run outside the real world in a smooth way; they undergo a kind of soft collision, and we can do our calculations on these imaginary cars and then see what they represent in real life. This is the idea of a hyperfunction. It is said that it relies on an old idea in the Orient that phenomena in the real world are shadowed by phenomena in an imaginary world which lies outside the real world but infinitely close to it. And these imagined phenomena are more tractable than the phenomena of this world.

Mikio Sato went farther. A shock can come from different directions. It is not the same thing to be hit in the back as in the front. So we need to analyze the direction of shocks. This is done with microfunctions, the theory of which he presented in 1969.

I should also have said some words about Professor Sato's contributions to physics. He receives the Schock Prize also for that. But this is too difficult for me. Let me only say that also that part of his work is deeply appreciated by specialists.

Dear Professor Sato, on behalf of the Royal Swedish Academy of Sciences I extend to you our warm congratulations. Please step forward to receive the 1997 Rolf Schock Prize in Mathematics from the hands of Princess Christina.

Christer Kiselman

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