傳播數學知識．促進數學教育

Interview with Prof. Yoichi Miyaoka

**Interview Editorial Consultant:** Tai-Ping Liu

**Interviewers:** Ching-Li Chai (CLC), Jungkai Chen (JKC)

**Interviewee:** Yoichi Miyaoka (YM)

**Date:** November 23rd, 2012

**Venue:** Institute of Mathematics, Academia Sinica

Yoichi Miyaoka was born in 1949. He got his PhD in 1977 in the University of Tokyo. He was a professor in Kyoto University from 1993 to 2001 and currently is a professor in the University of Tokyo. In 2011 he was the President of the Mathematical Society of Japan. His main research is about algebraic geometry; his important work related to the proof of the Bogomolov-Miyaoka-Yau inequality which was published in 1977.

CLC: There is a ritual for people being interviewed to talk about their formative years. Some people believe that mathematicians are born. There are two views but it is interesting that people are different.

YM: In my case, I didn’t consider about being a mathematician until I was 20 years old. Originally I was interested in physics and astronomy. I liked various kinds of sciences very much. When I entered University of Tokyo, I was interested in becoming a physicist but I found that I had no talents in experiments. Experiments are time consuming and every experiment I did was a failure. So I converted to mathematics. When I went to the math department, my first interest was in partial differential equations. However, I found my teacher Kodaira, who had come back to Japan in 1968. I was in the math department in 1970 and I asked him to be my advisor. That was how I became a mathematician

CLC: What’s your experience working with Kodaira?

YM: His lectures were very clear and actually he spoke in a soft and low voice and what he wrote on the blackboard naturally became a textbook. Everything was very clear, it was very easy to follow.

JKC: So at the same time were Iitaka and Kawamata there?

YM: Yes. Kawamata is 3 years younger than I am. Actually, when I went to graduate school, Kodaira was elected to be the Dean. So he was too busy to supervise us and Ueno supervised us. When Ueno went to Germany, Iitaka took charge of the seminars. For the first seminar, the textbook used was “Complex manifolds” by Morrow and Kodaira; that was for seniors. When we went to graduate school, the first material used was the second part of “Théorie de Hodge” by Pierre Deligne as well as several original papers including Clemens’s proof on non-rationality of cubic three-folds.

JKC: So you got your degree from Tokyo University and then spent a couple of years in Germany.

YM: When I graduate from the master course, I hope to become some sort of assistant professor which was usual in that period. But I failed to find any position for one and half years after finishing the master course; it was very disappointing. Eventually, I found the position in Tokyo Metropolitan University, and in 1977 I finished the PhD thesis on the so called Miyaoka inequality. Four months after obtaining the PhD, I went to Germany to stay for 2 years. It was a very good tradition that Japanese universities used to allow young researchers to be outside of Japan for usually two years and sometimes three years. For example, Mori stayed at Harvard for three years.

CLC: What was your experience when you did your PhD thesis? Usually it is a good story. Did you find the problem yourself?

YM: Actually, Miles Reid visited Japan via Soviet Union in 1976, and he brought me a paper of Bogomolov. It was very hard to follow his argument so I believed that there must be an easier or simpler proof for it. I found a simpler proof but that proof gave much better results.

JKC: And that turns out to be the so called Miyaoka-Yau inequalities?

YM: Originally, Bogomolov’s result was that $c_1^2$ was less than four times $c_2$. Essentially that follows from semistability of rank two vector bundles. But actually he used the so called the de Franchis theorem, which follows from the fact that every global 1-form (or global r-form in general) on a projective variety is d-closed. And, if you worked out the d-closedness, then you can prove any line bundle contained in a cotangent bundle cannot be very big. That means, for example in the surface case, a line bundle contained in the cotangent bundle has non-positive self-intersection. From that follows naturally the Miyaoka inequality. Of course, Yau’s inequality is, in some sense, much better, but his proof is difficult and much more complicated. It contends that if the equality holds that universal cover is ball-like. So in that case it is much better, but my argument is much easier and elementary.

JKC: From your point of view, how do you compare the algebraic method and the analytic method?

YM: In general, the algebraic method is simpler and easier to follow, but in some cases, when the results involve purely analytic aspects (for example, information about the universal covering), the differential geometric method is much more natural in some sense.

CLC: Was there an eureka moment? André Weil once said that every mathematician worthy of his salt has such a moment. When you work on a something and got stuck for a long time, you thinking about it, then at certain point you had that revelation. After that one got addicted to the sensation, and tried very hard to repeat that experience. I am sure there is such a moment for you.

YM: It was a very happy moment. Actually, I worked very hard to understand Bogomolov’s argument and tried to improve it for several months. When I was working on that, I stayed up for maybe 72 hours without a break; then I realized just one thing: we should consider branched covering of surfaces and everything become a union of sections. Then it was very, very trivial. Then I completed the proof maybe in just 6 hours. So it was a very exciting experience.

CLC: Was there a particular moment during those 72 hours?

YM: The 72 hours seemed to be nothing now. It was a very good feeling to work so hard for some time.

JKC: About 25 years ago, you proved the abundance conjecture in dimension three, which is a very important step in the minimal model program. How do you think of the recent development of algebraic geometry, especially the minimal model program?

YM: At least Birkar-Cascini-Hacon-McKernan has achieved a milestone. Especially they proved that canonical ring is always finitely regenerated. So it settles maybe half of the Mori program in some sense. But of course we need to generalize the abundance theorem in dimension four or more. I have no idea how to prove that but the direction should be that we should find some methods to construct global forms, not necessarily the highest weight forms but maybe the intermediate differential forms. Hopefully there should be two methods. One maybe differential geometry but I don’t know how to use it. Another is the refinement of Miyaoka-Yau inequality. In the three dimensional case, we needed an inequality involving only $c_1$ and $c_2$. But there should be a very precise inequality involving the higher Chern classes in general.

JKC: Involving the and $c_3$ and $c_4$.

YM: By Matsusaka’s Big Theorem and Masaki Maruyama’s results on semistable vector bundles, if you fix $c_1$ and $c_2$ (the first and second Chern classes) and assume some sort of stability then higher Chern classes are bounded. But we don’t know what the correct formulation of such explicit bound is. So we should seriously try to formulate such inequalities in higher Chern classes. For instance if $c_1$ is zero and $c_2$ is zero and (the vector bundle is) stable, then every higher Chern classes necessarily vanish. But if $c_1$ is zero and $c_2$ is positive, then I don’t know what the higher Chern classes should be. If you have such an inequality for example in the 4-dimensional case and you prove some very precise inequality involving $c_4$, then we can expect something.

JKC: So you expect that some sort of inequalities between Chern classes will be very useful?

YM: Yes, I think so, but unfortunately, I have no formulation which is definitely true.

JKC: How about the other approach proposed by Shokurov, using some sort of ACC conjecture?

YM: Right, I am not sure that the ACC conjecture itself could prove everything. I am not quite sure but maybe his idea is very effective in certain cases. But anyway, if you can prove that there is at least just one section, then the inductive treatment will be very effective. It is very good to have some inductive argument or to have some good varieties, but I don’t know how to construct such good varieties.

JKC: So, you think that the minimal model program is still the mainstream of algebraic geometry? How do you see the Gromov-Witten theory, that kind of geometry?

YM: It’s a very interesting problem. I was also interested in proving the existence of rational curves in every Calabi-Yau three-fold, but it was actually very difficult. My original idea was to use symplectic geometry and deformation theory by extending analytic deformation theory to the category of symplectic geometry. It is very easy to construct the pseudo-holomorphic $S^2$ immersed in symplectic manifolds. If you can smoothly deform a general symplectic structure on the Calabi-Yau back to the original complex structure and if you can deform such an immersed pseudo-holomorphic $S^2$ back to a holomorphic curve, then you should have a projective rational curve. But there is a very delicate point that is caused by the orientation problem. Suppose there is one pair and you deform it. Then it can suddenly vanish because of orientation problem: negative orientation. That was the problem.

JKC: So this is another approach to proving the existence of rational curves which is quite different from Mori’s method.

YM: Yes, in the general Calabi-Yau case, maybe. It is more difficult than Mori’s method. Of course there might be some different geometric method to find them via harmonic maps, but technically it should be very, very difficult.

CLC: I know very little about higher dimension algebraic geometry. But as a graduate student, I was really fascinated by Mori’s proof using reduction to characteristic p method to prove results in characteristic 0.

YM: That is actually astonishing. An amazing method.

JKC: You are the president of Mathematical Society of Japan(MSJ), how do you consider the role of the MSJ to mathematical society in East Asia?

YM: Of course as you know, mathematics in East Asia is developing very quickly, especially in mainland China, Taiwan and Korea. They all produce very many good mathematicians right now. But of course, Japan has the longest history of mathematics and we should be very happy if we could help East Asia develop mathematics by means of joint seminars, joint conferences and so on. Now, mainland China is a very big country and it might be very good to have Korea, Taiwan and Japan to form a coalition to make a good counter balance in some sense.

CLC: The history of the development of mathematics in Japan is fascinating; it rose to a very high standard in a relatively short amount of time.

YM: Actually it took quite some time. Japanese modern mathematics begins in 1870s, but of course, the first original achievement was made by Teiji Takagi in the 1920s. So it took fifty years. Takagi’s work was somehow isolated and the 2nd generation started their original works in the 1940s during World War II (WWII). That second generation is represented by Kodaira, Ito and so on. So it took seventy years. In the case of applied sciences, for example, applied chemistry or medical sciences, it took much a shorter time for catching up. For medical sciences, Japan made high level medical advancements in the early 1900s. So it took only 30 years to obtain a high standard in applied sciences, but in mathematics it took a lot longer, and so was theoretical physics.

JKC: What do you consider as the main different between theoretical and applied sciences?

YM: People can easily understand the importance of applied sciences, technology and engineering. People don’t understand the true meaning of pure, theoretical sciences. You need good students to apply to math departments to get good mathematicians. But if people don’t understand the importance of fundamental sciences, they will tend to go for example to the medical school or the engineering school, not the school of sciences.

JKC: It seems that Japanese students did not often go abroad in that time.

YM: Actually, in the 1950 and 60s, most bright students went to United States after graduating from the master course, or as soon as they got positions of assistants. Actually, it was standard in Japan to let young researchers to stay abroad for one or two years, since maybe 1880s.

JKC: And this was supported by the government.

YM: Yes, but of course after WWII Japan was so poor that they were financed by grants from American foundations.

JKC: What is the situation nowadays?

YM: Nowadays, young researchers are reluctant to go abroad. Japan is too comfortable. It is not very good. They should go abroad and make friends out there. You need some contact with the international community. They need international experience.

JKC: Does the governments still support this kind of program?

YM: They encourage people but it is very hard to force them to go. On top of that, now we have more teaching duties than before. Because of the financial crisis and the budget deficit of the Japanese government, they cut the number of teachers. So there are more teaching duties. As a result, young people cannot go and stay for a long time. Young people are busier now. Furthermore, it is now very difficult to find jobs in Japan. That means you have to stay as a post-doctorate longer since they have to apply for positions at many universities if they want to stay in Japan.

CLC: There could be a question of balance between these two approaches. After the war there was a whole generation of algebraic geometers and number theorists, such as Igusa, Iwasawa and Shimura and many others, who went to the US. The impact of their work in number theory is tremendous. If they stayed in Japan, their impact would be…?

YM: The impact would be smaller perhaps. Maybe it was very good for them to have stayed in the US, and they had more stimulation. If they had stayed in Japan, they would have been too happy in teaching. Goro Shimura could still stay very active, but some of them would have been too happy in Japan to teach good students.

CLC: Shimura as you have just mentioned is still doing absolutely original work. The ideas they brought out are still guiding the development of the theory in many ways. So is Kodaira.

YM: After he returned to Japan, he became the Dean of the Science Faculty and he was too busy to write papers. His last paper was written in 1972. But maybe if he had stayed in the US his last paper could be written in 1976 or maybe five to six years later.

JKC: Yes, but I think the impact of Kodaira to the new generation of Japanese mathematicians is still great, from the view of two decades later.

YM: His existence absolutely was a guidance for us. He regularly presided seminars on Saturdays. Many, many people went to these seminars. Actually he didn’t ask many questions; he didn’t make many remarks. But instead of him, Iitaka, Ueno, Tetsuji Shioda and Eiji Horikawa made many serious questions and remarks for young researchers. Kodaira would only say ‘Ah, it’s very interesting.’ But it was indeed very encouraging to have him and his short comments at these seminars.

JKC: So how do you compare Kodaira’s contribution in research and in educating the young generation?

YM: It’s very important to teach as well as to do research. But maybe, for Kodaira, for research at least, maybe US should be better. But the fact was that Kodaira didn’t like US style very much because his English was not very good. His writing was very good, but he had some difficulty to express himself in conversation. Another factor, perhaps trivial, is that he didn’t drive.

CLC: He didn’t drive! During those years, if you did not drive. You definitely had problems.

YM: But of course, for example, Shimura is completely the opposite case.

CLC: I don’t really know him. Just had the opportunity to interact with him but it’s not in any significant ways.

YM: By the way Shimura knows very well the Chinese literature, including some long novels, he reads many Chinese novels in original text.

CLC: You seem to be also in that category of scholars.

YM: No, I can only read the very classical ones but not novels.

JKC: How do you think about the interaction and coalition between Japan and Taiwan?

YM: We have regular exchange of members to participate in annual meetings. But anyway, next year (2013) we will have the Asian Mathematical Conference. So that will be some sort of start for the East Asian Union of Mathematicians perhaps. At least South East Asia has a Southeast Asia Mathematical Society.

JKC: You mean by Singapore and others.

YM: Yes, maybe we should have East Asian Union of Mathematicians for China, Taiwan, Korea, Japan and Hong Kong. It must be a good one, because we have so many mathematicians.

CLC: What do you think about the educational issue on cultivating the interest in mathematics among school students and encouraging the study of mathematics at the university level before they choose the profession?

YM: Fortunately, we have many young students who are really interested in mathematics. For example, at our university, maybe there are four or five very good students who are really interested in mathematics. For example, some of them may have read EGA at high school.

CLC: At high school? This is a punishment!

YM: Yes, sometimes it happens. But the problem is that we have very few girls who want to study mathematics. I don’t know why that is the case, but maybe one of the problems is that high school teachers tend to encourage girls to go to medical school or pharmacy or biosciences, not to mathematics or physics.

JKC: What’s the reason for this?

YM: Maybe they have some image of girls’ occupations or professions. But that is not correct because mathematics is a very good profession for girls. In mathematics we don’t have experiments. You can study mathematics at home or even in a hospital. If you are raising babies, you can still study and do mathematical researches. But if you are doing serious experiments, then you cannot do anything when you are raising babies. So mathematics is actually a very good profession for girls. But we have very few of them in our department.

CLC: I never realized that there is this gender gap in Japan. I don’t know that much about the situation here, but certainly from my past experiences, there were quite a number of women mathematicians from Taiwan.

YM: Jean-Pierre Bourguignon told me that maybe 30 % of mathematics students were girls in France. In Italy, more than half are girls. How about in America?

CLC: In the US I think the situation is getting better. Gender gaps certainly still exist and the situation was not good. There were certainly more women in mathematics here in Taiwan in the 1970s compared to the US after the late 1990s or even the beginning of this century. Still there is a problem. I find it quite fascinating that your high school students are studying mathematics at a very high level. Do they have assistance from faculty or do they somehow do it on their own?

YM: All on their own, yes. For example, Takeshi Saito, he was one of them.

CLC: At least one has to know the existence of EGA. So, I guess books helped. I don’t know how they found them.

YM: There are some special high schools which have very good libraries and maybe some teachers know some serious mathematics perhaps. I am not from such schools.

CLC: Oh I see, are they elite schools?

YM: Right, several private high schools or, more precisely, unions of middle schools and high schools. They teach for six years. For example, at a high school called Musashi, they subscribe to Annals of Mathematics.

CLC: They must have very good faculty. I presume they also allow a lot of freedom to the students. I don’t know if there is anything similar in Taiwan. In US there are, but I don’t know many of them.

YM: Of course, there are only a few of them in Japan.

JKC: In high schools, they put so much effort on the exam instead of the knowledge itself.

CLC: In US of course at Stuyvesant and also at Bronx High, many of their science faculties have PhDs from the very top universities. I presume it is the same in Japan.

YM: Actually, although there are many mathematics departments in Japan, serious mathematicians are produced by only a few math departments. Maybe 95% to 99% of them are produced by only ten departments, for example, Tokyo University, Waseda University and some others.

CLC: Ten! That is a very good number.

YM: But it depends on the definition of serious mathematicians.

CLC: I am not sure about the number in Taiwan.

YM: We have many math departments, maybe hundreds of math departments.

CLC: Many hundreds I assume.

YM: Not so many because most small colleges do not have mathematics department at all.

CLC: Does that mean that therefore, their students wouldn’t have much access to mathematics?

YM: They teach only humanities and so on or just the easy things. The total number of universities and colleges in Japan is about 700s, but most of them are originally junior colleges.

JKC: So how many serious mathematicians?

YM: At least I have to warn you it could not be a very rigorous estimate. The Mathematical Society of Japan has about 5000 members and 3000 of them have positions at academy. So, if my definition is very loose, then the number of serious mathematicians is about 3000.

CLC: That’s a good number.

YM: If ‘serious mathematician’ is meant in a more rigorous way, namely if it means researchers making serious researches right now, then the number is maybe between 500 to 1000.

CLC: To me, that still sounds very good.

JKC: That is a lot more than we have here

YM: Just because Japan has more populations about five to six times.

JKC: But somehow the number of mathematicians is more than five times then what we have here.

CLC: Maybe students from high schools they were much too constrained and guided in some way by the entrance exams system. It may or may not be true but people believe that has some influences on doing creative work later.

YM: Entrance exam is always a problem in Japan. But the entrance exam is not all bad. For the very top, it should not be necessary. For ordinary people or mediocre students, it is a very good reason to study. The more serious problem for Japan is that our system is so rigid; we cannot allow young people to study at universities. The minimum age is 18.

JKC: So students cannot enter university earlier. That’s not a problem if they can self-study EGA.

CLC: Thank you. It has been a pleasure. We hope there will be more interaction between Japan and Taiwan in the future.

YM: I believe we should be committed to making a closer contact.

- Ching-Li Chai was a faculty member at the Institute of Mathematics, Academia Sinica; now at the Department of Mathematics, University of Pennsylvania.
- Jungkai Chen is a faculty member at the National Taiwan University.