傳播數學知識．促進數學教育

Interview with Prof. Kenji Fukaya

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

**Interviewers:** Tai-Ping Liu(TPL), Jih-Hsin Cheng(JHC), River Chiang(RC)

**Interviewee:** Kenji Fukaya(KF)

**Date:** May 10th, 2009

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

Prof. Kenji Fukaya was born in 1959 at Kanagawa Perfecture. He received his B.S. in 1982 and Ph.D. in 1986 from University of Tokyo. He has been a faculty at the University of Tokyo and Kyoto University, and in 2013 he joined the faculty of the Simons Center for Geometry and Physics at Stony Brook University. His research interests include symplectic geometry, Riemannian geometry and the introduction of Fukaya category. He was awarded Japan Academy Prize in 2003, the Asahi Prize in 2009, and the Fujihara Award in 2012. He is a member of The Japan Academy.

TPL: So we have a few people joining us.

JHC:They are Professor Kukaya’s joint author, Hiroshi Ohta, and his student, Manabu Akaho.

TPL: Okay. Let’s start. So thank you for taking time off for this interview. We would like to hear from your thought and your friends from Kyoto. Let me start from this canonical question, because Professor Jih-Hsin Cheng did prepare canonical question. So how did you get interested in mathematics ?

KF: First ... somehow most mathematicians are not very good with talking to other people, just keep to themsevles, that’s my case also. What’s good about mathematics is somehow you don’t need to do any political things. In mathematics, something correct is correct. That’s somehow a nice thing. You don’t need to negotiate.

JHC: When did you start to love mathematics ? from high school ?

KF: Yeh... high school... I read some books in university, but that’s really independent.

JHC: You decided to work in mathematics as a serious career in high school ?

KF: Yeh, I was serious about it, maybe at the end of high school years just before starting undergraduate courses. In my last year of high school, I was ready to enter the field of mathematics.

JHC: So you entered the Tokyo University as a student majoring in mathematics ?

KF: Yes. In Tokyo University, it’s decided at the third year. In the first two years, you do not need to decide. However, I was certain about doing mathematics in the first year.

TPL: At that time, in Japan, already you start to learn some rather serious mathematics in high school, for example, calculus ?

KF: Yes, there were some calculus taught in high school time, but I read some books by myself.

TPL: I see. What kind of books ?

KF: These books I don’t recommend so much. I read some Lang’s books about analysis. In the last year of high school, I read Frigyes Riesz and Bela SZ. Nagy book on functional analysis. That’s what I like.

TPL: Riesz’s Functional Analysis. You read that while you were in high school ?

KF: Yes, in the third year. In my second year, I read some books written in soviet union. It’s a kind of books on Calculus, translated into Japanese of course. I don’t remember the names of those. That’s what I like to do. It’s a bit more applied-oriented books.

TPL: There is Takagi’s book. That’s also very well-known and widely used in Japan.

KF: Yeh, that is right. But I didn’t read any Takagi’s book. I wonder why I did not do so. There is this entry exam to university in Japan. It’s a rather hard thing. Before the entrance exam, you were not suppose to study as you like. You were supposed to prepare for the exam. So the learning was much frustrating. After I passed the entrance exam, I want to read some mathematic books. I read two. One was van der Waerden and the other was a general topology book. I don’t know why I chose general topology John I. Kelley’s book. At that time, the general topology book was very interesting for me. Maybe because I was so young, after several years, I was tired and lost my interest.

JHC: So were you interested in other subjects ? other than mathematics ? after entering college ?

KF: I was interested in astronomy when I was younger. But I soon realized that I was good at those kind of things. I jointed some club and went to the mountains for star-gazing, but I was very bad about it. So I decided that it was not my type of things. People knew how to move telescope and prepare cameras, but I was very bad.

TPL: While you were reading those books, did you have friends reading with you , or were you by yourself ? How did you know about those books ?

KF: Actually, when I was in high school, my favorite thing was to go to bookstores. There was a section of mathematics books. Just to walk around there and browse around was my favorite behavior.

TPL: That reminds me about Mark Twain. He said ‘I never let a schooling to interfere with my education.’ So, therefore, you were not so much hindered by the university entrance exam, which was easy to you.

KF: Not easy, but somehow, I passed it. That’s enough.

TPL: But you spent a lot of times doing things which were in no relevance to the preparation of the exam.

KF: I spent some time, which was not very good. My grade in English was pretty bad.

JHC: But still you entered Tokyo University.

KF: Yeh, I got into it.

TPL: Then what did you do after you entered Tokyo University ? Having learnt Riesz and Nagy, Kelley and van der Waerden. So what do you do in Tokyo University ?

KF: At the beginning, I tried several things. The time when I read most books in mathematics was at the second year of university. That time I start to read something in English. So I found out that there were many books. After that, I don’t have that much time to read. Nowadays, I can only read one book at most. So those days were ...

TPL: But I mean that you have no courses. You have no mathematics courses to take as an undergraduate, because you learnt all those subjects already.

KF: It’s always interesting to hear somethings. Some courses were useful at third and fourth years.

TPL: I see. You were the first one I heard to have read those books, van der Waerden, Kelley in high schools.

KF: But having said I read those books in high school, it’s actually between graduating from high school and entering university. A couple of free months.

RC: So in two months time, you read two books.

KF: That’s very correct.

JHC: So how many courses you have to take as a math major student in college ?

KF: You mean in first year, just calculus and algebra.

JHC: But you said that you starting taking majors in third year ?

KF: In third year at my time, at Tokyo University, there were three hours of lecture with ninety minutes of exercise everyday from Monday to Thursday. Two analysis, one algebra and one geometry for one year. That I think is a standard course of Tokyo University. Kyoto University has more flexibility, but, they still have this kind of intense course, which is good.

TPL: Well, here, in Taiwan University, we also have a very dense course, at least for me. But I found out that it’s good for someone exceptional like you, but it may not necessary be good for everyone, don’t you think ?.

KF: I mean, to have a very dense course is always very good. I believe. Otherwise, unless you have much of smartness, you may just use your free time, and it’d be difficult. I like this of University of Tokyo, because most of people know something, which is different from speciality. Most of mathematicians who graduated from University of Tokyo know something about topology, something about functional analysis, something about Galois theory. To know three, usually is not so easy. To have a course supposedly to take all of them, you know at least something. I don’t know how much a student can understand.

TPL: But they offered an opportunity.

KF: I think at least those who became mathematicians understood it. Most of the mathematicians. I mean, after you became a mathematician, you could not find a time to learn those kind of things, and most of the things, I would never use in those days. I could use them later on. So I consider that’s quite lucky. General topology is something of use right. At the beginning, I was very far away from algebra , but nowadays I am getting closer to algebra, so I am happy that I can use it.

TPL: We were talking about Charles Conley before the interview, I know him since 1972. One time, he said that he is studying algebra, I said that ‘do you need it ?’. His answer was ‘If I don’t know it, I would never need it.’ (Everyone laughed.) Could I ask you a sort of technical question ? You work on Arnold’s Conjecture, and you’ve done important work on that, I was also told that Conley has done something on that, so what was his role with that ?

KF: I think that he proves somehow very big purpose. First, enough big part, then as Conley and Zehnder proved case Tori. I think that’s the first step. That’s always very important. He showed I can prove something about it. Probably after that, people get more serious and try to solve it. Before that, I am not sure how much seriousness Arnold was put in it. I mean, in mathematics, you put some difficult questions. At the beginning, the questions look too much and seem so far away, nobody even tries to solve it seriously. Then time comes. People become more serious, and there comes correlated work, Floer ‘s work. That happens within ten years. Those days is changing this problem from something you believe you could never do to something you suppose to work on. That’s kind critical moment.

TPL: So you are not so sure how much Arnold’s ....

KF: I don’t know. Because when Arnold’s time, I never met Arnold. I don’t know what kind person he is. Maybe he is an adventurer and believed that he can do it. But at that time, there was no way to do those kind of things. Those kind of ideas. There were nothing really proper in simplectic geometry.

TPL: I see. I met Arnold a couple of times. He has quite a unique personality.

KF: Ah. Please do not tell him what I have just said.

TPL: No. the usual Arnold, he usually doesn’t listen. He only talks. But he would probably appreciates your comments really. He is an interesting person.

KF: But of course he is a very important person. Arnold mentions several questions. That, somehow, he is the first to propose a problem that suggests in global symplectic geometry, otherwise, people cannot use that.

JHC: How about A-infinity algebra ? When did you get the idea of this structure?

KF: Something I was interested to do in Floer homology at the beginning of 90s. So A-infinity algebra category kind of things come from 92 or 93. Also, there was this talk by Donaldson who did something about gauge theory of Floer homology. He suggested to cook up some kind of category. Then the problem was, in some case, chain complex, in others, we have Lagrangian submanifolds. We need to find something which was a mixture of chain complex and Lagrangian submanifolds. That’s the answer, the functor from this category. Sorry that’s too much technical. But try to cook this category with this problem. So in the nonlinear things, people introduce nonlinear analysis to geometry and topology. Donaldson and those people. Then, they cook up numbers and distinguish something by numbers, then Floer gets more to cook up groups, I feel, somehow more algebraic things, because Floer said in some lectures, algebraic topology is to replace algebra by topology. Geometry is interesting and you have a lot of intuitions. But it’s difficult to handle. Algebra you can calculate and it is more attackable. But if you want to understand the geometry, to replace geometric thing by algebraic thing is a natural idea. But topology goes in that direction so much. Then somehow with these nonlinear things. That’s an important issue. That’s the kind of thing I want to do. Then this kind of thing come together. After we start working on this nonlinear Floer theory, then we founds that to correctly organize the work, the category that we did in 92 fits in naturally.

TPL: I shall go back to this educational aspect. Now, you are in Kyoto University, knowing that you learn things so fast, you read books by yourself and you like the fact that University of Tokyo provides this high level basic course in different directions. So how do you feel the students in Kyoto University ?

KF: Kyoto University... when I first came to Kyoto University, the system was very flexible. Students are free to take anything and free to take nothing. But Professor Mariyama, he passed away, he introduced that every student should take some kind of seminars. I told him that this is the kind of way Tokyo University to make students work hard, to train them ; Kyoto University is the opposite, to let students free, why do you want to change it ? Mariyama told me if you let the students free all the days, only a few, one or two students really use this free time to do great things. Nowadays, if you let every student free all the time, then no body come. (Everyone laughed.) So you need to put some duties. But still, there are some differences. For example, if you want to take some courses, you are free to take it at second, third or fourth year. I was surprised when I was teaching introduction to manifolds. One of my fourth year student came to take the exam, so that kind of freedom exists in Kyoto University. In University of Tokyo, math major students who know enough about kinetic dynamics are much less than students in Kyoto University. Because physics and mathematics are more separated in University of Tokyo. I learnt symplectic geometry but I never took a course in kinetic dynamics.

TPL: That’s very nice, very insightful information.

KF: Kyoto people always say that Kyoto University is a university of freedom.

TPL: So you did not suffer so much from the university entrance exam ? I heard people complain that we have entrance exams for university and so on. But for you, there was no problem. You made minimum effort.

KF: I didn’t like it at all.

TPL: But you did not really suffer from that, right ?

KF: I don’t know how much I suffered. But in the sense I passed it, I think it’s okay. I think that mathematics still suffers. Many Japanese do not like mathematics, because they had bad experience with the entrance exam in mathematics when they were high school students. So mathematics is always a bad memory for many Japanese people.

TPL: Well, except for you. Mathematics is difficult. So I think that is universal.

KF: As you know, it is a difficult thing, but if you can learn it for your own purpose and as an entertainment for fun, it can be a joy and fun. But the difficult mathematics is just something you have to pass at the entrance exam, it usually is just a pain. So that I think it is the reason why people don’t like it.

TPL: I once had this impression that Japanese mathematic, if there is such a thing, as it is very international now, but is there certain characteristics of Japanese mathematics ? Can you identify some characteristics that ‘yes, this is the kind of thing we Japanese do’ ? Is there such a thing ?

KF: I think that I talked with you about this kind of thing. You have some kind of object. Japanese people see this is one part of million things. They just concentrate on it and do not bother with anything else. Then they do it and are satisfied. When other people came and they like general things. They will compare with other examples. For example, he will compare Kiyoshi Oka with Henri Cartan. Oka found some most basic things about coherent sheaf. Oka did not like the names of coherent sheaf. Of course, Cartan is a great mathematician. He developed it enough to make it global to be used in many other things, like algebraic geometry and algebraic topology. Oka somehow just addressed the problem, and proved some basic problems. It’s a kind of strong contrast, but both are important.

TPL: So do I understand you right ? Japanese like to do more explicitly.

KF: Not explicitly. If you have some problems, and you think that this part is the most decisional part of this problem, then the interest only lies on the particular part of the problem.

TPL: I see. The key question.

KF: Yes, the key issue. I mean there are some other kind of Japanese mathematicians as well, but, rare.

TPL: I see. Of course, French sometimes are criticized for being general. Criticize the Bourbaki, abstract thing also. But clearly, Oka’s contribution is fundamental and widely appreciated, don’t you think so ?

KF: Yeh, I believe so. In Japan, Oka is of course very famous. But I don’t know how famous he is in other countries. If you ask American Mathematicians, maybe they know of him.

TPL: But you have to ask the leading mathematicians, right ? Not the general publics. The general publics only follow the paper before you, they do not go to the source. They do not have the capacity. You have no particular idea saying that ‘I wish I did not go to this particular field of research, I would rather have done something else.’ It seems to me that you are very much definite about what you want to do.

KF: I am not so sure. I am always thinking about moving to something else.

TPL: Like what ?

KF: I mean, something I really want to do through homology algebra to do field theory, quantum field theory. I mean, what I am doing now is something related to that kind of things, but more directly to physics. To put the rigorous mathematical foundation of quantum field theory using homology algebra, that’s another kind of dream. Actually I don’t know when I can do that. I am getting old for it.

TPL: What aspect about quantum field theory that you think about ?

KF: That’s a very strange thing. Most of the fundamental side of physics, after a while requires rigorous mathematics. Like, general relativity at the beginning, quantum mechanics at later, Von Neumann and other people. Of course, other things like Delta Function just came. But quantum field theory, exists almost 50 years, exists to be a fundamental side of physics, but that’s out of reach of mathematicians. That’s quite frustrating situation. So...

TPL: There is Glimm-Jaffe theory, which they could not complete. We actually interviewed Glimm for this series of MathMedia. By the way, this series is called ‘Friend from far away’, which is a Confucius saying. Glimm also made a try on the quantum field theory. He said that he did this C* algebra thing, and he was told that this was maybe related, so he made a serious attempt. But when he entered it, it turned out he was using more functional analytic, path integrals those kind of things. But in any case, it was not finished, so everybody should give it a try.

KF: I mean, after this kind 80s, 90s, many geometry insights were related to it. As Glimm did his attempts. It was a time when people thought about these things in more purely analytic point of views, like functional analysis and C* algebra. We know somehow, some kind of field theory has some geometric background. For example, it’s very likely some global geometry are related. That kind of things people never tried. Maybe some big geometers already tried. Maybe Alan Connes already tried. I don’t know. Somehow when we do this kind of geometry, we just do it using traces of path integrals. We just do very different things. Then something happens, we just do it directly by physics. Some geometry shall be involved in the story.

JHC: So you have some ideas to deal with path integral using homology theory ?

KF: You know, homology is some kind of cancellation and boundary operators. In the case of path integrals to make sense with cancellation, I think homological integrals is more of a systematic way to see this cancellation. The physicist they usually do it by hand, somehow. To do this kind of thing systematics, shall be a mathematician’s work. For many things, physicist draw on diagrams, they look very much similar to graphs drawn by topologists. I don’t know if it’s possible, but...

JHC: You skip the analytic part?

KF: I think that one has to join them.

KF: I mean, analysis shall be always there. For example, in some case, topologist provides the thinking. Usually you have the operator, which is discrete spectrum, but for real physicists, you have the continuous spectrums. In that situation, we have to join this algebraic thing with functional analysis. I mean, in some part of this C* algebra or functional analysis, they use very elementary part of algebra. They never use this highly developed algebra. I don’t know if there is any hope to this all.

TPL: Sounds very exciting.

KF: If you succeed.

TPL: But first, you have to get yourself excited.

KF: I dream of this kind of things all the time, but it’s very far away.

TPL: There is always the struggle with infinity. It is quantum field theory.

JHC: So how do you balance your life between family life and work? You said that you work in the evening.

KF: Actually, I usually work after my wife and children are asleep, like 11 o’clock to morning, then I go to bed.

TPL: But your wife knows that you work at evenings, right?

KF: Yeh yeh, at midnight. She knows.

TPL: I see. But, then, how about your day time duty?

KF: The day time work, somehow… The trouble is there are so many paperworks. You should have the same problem. Nowadays in Japan, this kind of things are happening more and more.

TPL: Yeh, I was once told about that. Why is that? There are more of these bureaucratic things.

KF: You know we need to somehow write many things very frequently. To get any money, we need to write up something like this pad of papers, then after that, we have to report it. So we take more time to explain what we are doing than doing it. To justify the work. To report what is the research of this year, the number of students had and those kind of things.

TPL: Why come to this? Why is the society becoming like that?

KF: I believe if somebody decides and that person understands the thing that person decides, then he does not need to read so many report because he can understand the content. But in Japan, the government officials who can never understand mathematics, has the power of final funding decisions, so they need some reports to understand, otherwise they can never decide. That’s the reason I think.

TPL: But Kyoto University is such an established institution. I suppose you have the least of those things. Because people of course trust Kyoto University, don’t you think?

KF: I would rather the need to write these kinds of reports and proposals to be declined. It’s kind of a competition. They force us to compete then they decide.

TPL: So does this mean that the prestige of a university, like Kyoto, is not sufficient to deal with the political situation that we need to spread the funding around more to Kyushu, Hokkaido and so on. I imagine in the old times, they would say, ‘Well, Kyoto is so good. Let’s not ask them questions and just give them the funding’. Does it happen that way?

KF: I mean, it is the government officials who decide it. They don’t want to do it, otherwise, they have to take some responsibilities. I mean, they understand that, if they leave some universities to their own ways to succeed. But for them, they need to understand the applicants. Maybe for Shigefumi Mori it is okay, because they know of his name by Fields Medal. But for others, you have to provide them with information. For Taiwan, don’t you have this kind of problem?

TPL: Yeh, very much so. I think it’s universal. But I think Japan has less problem.

KF: No, Japan is bad.

TPL: But I can see that Kyoto University becomes more and more beautiful. Around this bell tower, it’s much more beautiful than before.

KF: Tower? Ah, they rebuilt it. There is a conference block. But mathematicians use it very rarely, used once or twice only. This conference block is somehow too good for mathematicians. We don’t need this kind of beautiful things, because chalks and dusts will dirt the place and they don’t like it somehow. We just use ordinary rooms with blackboards. That kind of beauty is a waste of money in my opinion.

TPL: But the math department building is very beautiful.

KF: Math department building is okay. RIMS have trouble, because they do not have enough buildings.

TPL: The office is very big.

KF: Yeh, it is.

RC: So I want to know this canonical question. What do you do when you get stuck?

JHC: What’s your attitude when you get stuck?

KF: Maybe shochu, why?

TPL: He never got stuck.

KF: No, no, I do. (laughs) But somehow, the biggest thing to stop from doing mathematics is these paperworks. So it is very easy when I get stuck, I can just do some paperworks. But that’s very good from recovering being stuck. I mean, if I stuck in mathematics, I will do paperworks, then I will never come back. That’s the something I am afraid of. So…

TPL: But, really. When you got stuck, for your own record, what’s the longest period of time you got stuck but you still push on?

KF: You know, there are some experiences that I tried to do but never succeed. That maybe I should not count. But there are some things which I try to do for more than 10 years, but I never succeed. That kind of things I cannot concentrate for too long.

TPL: Right, on and off.

KF: Something I start and come back for couple of months later.

TPL: But there are some questions you actually keep thinking about them on and off for ten years.

KF: Yes, for example, we wrote this book with four authors, something we claim in general. I found some troubles. Then I tried several things to get rid of these troubles, it took two months, then I gave up and decided that it did not go in that way. I stopped and put some conditions. That’s two months. I mean, during those two months, I did not do any mathematics other than that. So…Actually, my wife has the children at that time. I was at the hospital frequently, but …

TPL: Still thinking of those problems.

KF: Yeh… (laughs)

TPL: So you are already doing something about your dream, doing quantum field theory now? You are already taking your time off and to think about quantum theory now?

KF: Thinking is very difficult now. Maybe I need to learn something also. I feel that I need to read these papers by Jaffe and other people.

TPL: But you are thinking about doing that?

KF: Yeh.. sometimes. Maybe I need to get rid of some paperworks for this.

TPL: But you should have no particular feeling that you need to perform something and therefore, you cannot afford to think about such long range difficult problem. You don’t have that problem right? You can do whatever you want now. Don’t you think?

KF: Now?

TPL: Yeh.

KF: Actually, somehow, we just almost finished this ten year project. That’s in Field Theory. Then we need to do some calculations to synchronize that. We announced 5 or 6 papers which are not yet finished writing. So to finish that one is still …. It take a couple of years at least I think. I hope to find something on that direction during that time. So I don’t know when I can finish this project. So… I also feel that this project is of enough interest to continue so for the moment I don’t have so many degrees of freedom to move. But maybe it’s better to move during the time when it is still interesting. Because if you move and feel that there is no interest left in that, then probably at that time, the situation in that is not so good. Probably you don’t have enough power to do such a thing.

TPL: I was told by Professor Sone who is visiting us here, he said that in the old time, I don’t know how old time was, maybe when he was young, he is now 73 years old, he said that in Kyoto University people over age say 50 or something, they don’t write papers, even a good professor. They just supervise and try to get knowledge and be a good scholar, know things and can direct students. I guess that you are the wrong person to ask, because that’s before your time. Then he said that, people now write papers even after age 60, 70 and so on. This has a plus and a minus according to him, with obvious plus and minus.

KF: What’s the minus?

TPL: The minus is that then people do not become what an American would say ‘Professorial’ to have a broad knowledge and then be able to bridge very different disciplines and nurture the more broad culture for the next generation.

KF: I don’t think so. Really, to understand something, the only way for me to understand some mathematics, is to write a paper about it. So if you just read something, usually is not enough. So if you learn something, when you seriously want to use this, then you can learn something. If you somehow educate students, if you just observe the areas and you give a question, usually it’s a bad question. Maybe you need to be ready yourself. I mean in the old days, maybe professors are very big, and they look things down from up. But that kind of thing is not so good nowadays. I think now, professors and students are the same level to talk, just 50 to 50 at least. That I think is much better. So in that sense everybody shall continue to hold the same passion as young people. Of course, it gets harder and harder.. But, also, on the other hand, recently, it takes more time to learn something than the old days. So in the old days, maybe when people get to 30s, they have known most of the things to do their research. Nowadays, if you want to have the correct or enough knowledge for your own research, sometimes it gets more than 40 or 50 to do that. So still, you have something that you need to learn, but that’s a happy thing.

TPL: Yeh. We saw that prolong this process of maturing and always feel like a freshman. I am happy that I asked that question, and, I am happier you answered this way. So, after you have this 5-days conference, you must be exhausted. But I can see that you have so much energy.

KF: Talk in mathematics is always fun.

TPL: Yeh. Maybe we could have stopped here. You have very insightful comments. That’s very refreshing. So you still feel as good as you were in high school.

KF: hahaha

TPL: I can feel that your spirit is very high. So thank you very much. Maybe some years later, we get together again.

- Tai-Ping Liu and Jih-Hsin Cheng are faculty members at the Institute of Mathematics, Academia Sinica.
- River Chiang is a faculty member at the National Cheng Kung University