Интервью Хиронаки (декабрь 2004)

http://www.ams.org/notices/200509/fea-hironaka.pdf

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Notices: Why did that problem seem significant to you?

Hironaka: I don’t know. It’s like a boy falling in love with a girl. It’s hard to say why. Afterward you can make all sorts of reasons. For instance, I studied quite a bit of abstract algebra, so anything that could be expressed in terms of algebra was interesting. But algebra itself is too abstract—it doesn’t catch your heart. This was a geometry problem, but not geometry per se. It was quite clear to me that you could not solve that kind of problem by geometric intuition. Oscar Zariski had already solved it for curves in one dimension, two dimensions, and even partly in three dimensions. So it was a question of higher dimensions. In the higher dimensions you cannot see everything, so you must have something, some tool, to guess or formulate things. And the tool was algebra, unquestionably algebra. That’s one reason the problem hit me. Also, I like basic things. Very clever people tend to jump to the new techniques: something is developing very fast, and you want to be on top of it; and if you are smart, you can be a top runner. But I am not so smart, so it is better that I start something where there are no techniques for the problem, and then I can just build step by step. But actually, it was not so hard. It turned out to be easier than I thought.

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Well, any problem that has many aspects and many facets that you must put together requires really immersing yourself into thinking about the whole problem. I remember that when I first called Zariski to say that I had solved the problem, he said, “You must have strong teeth.”

Notices: What does that mean?

Hironaka: I think he meant that I needed teeth for biting—the problem was tough, so you must really bite into it. But he was very kind. He was always encouraging me. He somehow had confidence in me. So I started writing and rewriting, writing and rewriting, and finally I finished.

<...>The resolution of singularities was done by hand. It doesn’t use large theory or techniques. <...> I was just making new definitions and working case by case.

Notices: So you were mostly making up your own theory as you went along, rather than using existing theories.

Hironaka: Yes, that’s my style, actually.

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Hironaka: Yes. Being away twenty years or so from the country and coming back, at least in the first few years, I had some difficulties. For instance, at one time I was chairman of the mathematics department at Harvard. When we discussed, for example, new appointments, everybody had a different opinion. One person would strongly recommend somebody, somebody else would recommend some other person, and we would discuss it. But in Japan it’s different. If you make a recommendation, then nobody says anything.

Notices: Nobody contradicts it?

Hironaka: No. In Japan it goes like this. Suppose Professor A recommends a young man, Dr. B. It is a fact that he recommended Dr. B: he presented the recommendation in a document. Why should he now insist on it? He thinks, “Let other people talk about it.” Then Professor A says something good about Professor C’s recommendation. If I take that seriously, then I have the wrong idea, and Professor A gets mad after the meeting. So I must listen very carefully and realize that he actually wanted support for his recommendation.

A Japanese person will insist on what he recommended or what he wants. But he doesn’t express it. Because if he expresses it and if it doesn’t come out in his way, then he has some kind of dishonor or disgrace. So you must be careful not to disgrace him and to guess what he really wants indirectly.

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Notices: Is there also a Japanese style of doing mathematics?

Hironaka: That’s hard to say. Mathematics is of course a science, but it also depends on personality. But certainly you see a difference in how people behave at conferences. If it’s only Japanese mathematicians, instead of making propaganda about their own ideas, usually they praise the ideas of others—and quite often without meaning it! You must get used to that kind of thing.

There is a cultural feature of Japanese people that affects not the product of doing mathematics but the way of doing mathematics. In some sense, it is similar to the Russian way. For example, Kyoshi Oka graduated from Kyoto University, and he didn’t publish for about ten years afterward, so he couldn’t get a job in a good university. Finally he got a job at Nara Women’s College. He was a bit crazy, but he was very original. I can see the same style and very high creativity in Mikio Sato and also to some extent in Kunihiko Kodaira. Kodaira went to the United States, so he became much more Western-style, but nonetheless his nature is like that. It is something to do with Japanese culture. This is a simplistic way to describe it, but usually in the Western world you try to express yourself, to show off in some way, to appear to be more than you are, and by doing so, you get more motivation and drive. And thanks to that, you reach a higher level of productivity and originality. That’s one way. But the Japanese way, at least the traditional way, is not like that. You don’t show off. You wait until somebody starts recognizing you. Even then, staying modest is considered a good, respectable feature. So not writing any papers for ten years—that’s nothing. The mathematician must believe in what he is doing, without showing off.

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Notices: Do you think mathematics is something that has an independent existence that mathematicians discover or that it is invented by humans?

Hironaka: I am not a historian, but, roughly speaking, after World War II, up to the 1960s and 1970s, mathematics was really by itself. It had a very strong motive to develop by itself through internal motivations and internal interests. For instance, Grothendieck is one person who lived by this principle. In the 1950s and 1960s we mathematicians looked down on people who talked about applications to the real world. If a mathematician started talking about applications, we would say, “Oh, he stopped being a mathematician; he has become an engineer,” even if he was doing important things. The first part of the twentieth century was a unique time, a phenomenal episode in the history of mathematics, with the field flourishing—at least we thought it was flourishing!—and being pure and independent of the world. This led to big progress, and mathematics changed quite a bit. Even then I remember some people saying that mathematicians are doing “abstract nonsense” or “pure nonsense”. But mathematicians didn’t think that way; they were doing pure mathematics. If somebody had asked, “How has your work helped the world or produced something you can use?” then I am sure around that time pure mathematicians would have said, “That’s a very stupid question! A very lowly question!”

Poincaré was a good mathematician from the point of view of pure mathematics. But at the same time, he emphasized the point that mathematics is nurtured by trying to understand physical phenomena. The very fast development of fundamental physics changed the world view quite a bit and gave a big push to mathematics. Look at Einstein’s work or quantum mechanics—mathematics was very useful there. But physics changed too. Nowadays, physics has changed from pure physics to much more applied. If you look at the Nobel Prize winners in physics, for instance, in the second half of the twentieth century, many are much more applied—they work on electricity, electronics, chemical applications, superconductivity, and that kind of thing. It’s much more the real world. I remember in the earlier part of the twentieth century, at least among my friends and teachers, there were many physicists interested in theoretical and fundamental physics. But that changed quite a bit. I think that had an effect on mathematics too.

Notices: If you look at the Fields Medalists, they are all as pure as ever.

Hironaka: That’s true! But I think the 1960s and 1970s were the peak of mathematics having an unquestionable raison d’être.