傳播數學知識．促進數學教育

Interview with Prof. Yakov Eliashberg

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

**Interviewers:** Tai-Ping Liu (TPL), Jih-Hsin Cheng (JHC), Yng-Ing Lee (YIL), Mei-Lin Yau (MLY)

**Interviewee:** Yakov Eliashberg (YE)

**Date:** November 3, 2011

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

Prof. Yakov Eliashberg was born on December 11, 1946 in Leningrad (St. Petersburg). He received his PhD from Leningrad University in 1972 and held a teaching position at the Syktyvkar State University of Komi Republic of Russia from 1972 to 1979. Between 1980 and 1987, he served at a computer software company as the head. In 1988 he moved to the United States, and he has been a professor of mathematics at Stanford University since then. His main research interests are in symplectic topology. He is a member of National Academy of Science and was awarded the Crafoord Prize from the Swedish Academy of Sciences in 2016, and Wolf Prize in 2020.

TPL: Thank you for coming. I would like to say first that this interview will be published in our Chinese journal MathMedia, though we plan to publish the English version someday.

YE: So from now on I should speak Chinese? (All laugh.) I speak some French. When I first came to the West, I was invited to give a series of talks in Lyon. I just wanted to speak French, and when I started I explained in English that I will speak French, but please apologize my French. Then I said “from now on I will change to French,” I thought a few seconds and said “Okay, ..” Everybody started laughing because my first word in French was Okay.

TPL: Did you speak French in the talk?

YE: Yes.

TPL: St. Petersburg has long tradition of French speaking.

YE: Well, that’s long time ago before my time, I just studied French in my school. That is unfortunate thing for all of people speaking non-native languages, that at some point we say what we can say and not what we want to say. When I want to say something and can’t immediately find the right word, I find something suitable but not exactly what would be the best, in that sense when I speak English I feel myself more stupid, than when I speak Russian.

TPL: That has probably some advantage, because there is a book written by Andre Weil, which told a story in Brazil: He met two guys, one is very good in English, all sentences are crafted perfectly, another one is such a person just like you have described, he could not understand the one who speaks perfect English, the other one he could understand perfectly.

YE: You read about Andre Weil… First time, I saw Andre Weil when he came to Moscow and was giving some lectures. Actually I lived in Leningrad (St. Petersburg), I just happened to be in Moscow and somebody told me that today Andre Weil was giving a lecture, let’s go there, so we went to the Steklov Institute where he was giving the talk. Yuri Manin was translating, and Andre Weil started speaking, he spoke English with very heavy French accent, even I could understand precisely because of this reason that he spoke so bad English that it was easy to understand. He said just a few words and stopped to let Manin to translate, and Manin started translating. He translated and translated for about maybe 15 minutes and then stopped, then Weil started to talk. He talked and talked and Manin was not interrupting him. Then at some moment Weil stopped, waiting for Manin to translate, and Manin said: “I already translated this”. (All laugh.)

TPL: We have worked you hard for this series of lectures you just finished, they are great lectures. I think everyone is happy. Now we are fresh from your lectures, you mentioned most of the Russian geometry school, Arnold, Gromov, yourself…

YE: Of course Russian geometry school goes much earlier than that.

TPL: I know, I know. I read an article somewhere that education on geometry in Russia is taken as a central role of undergraduate education. Is that so?

YE: That depends on which year you talk about. Certainly when I started in school the textbook we used was Kiselev textbook. Actually this book went back many years before. In our home I’ve seen its edition of 1898, and this was not the first edition, it was a bit earlier than that, some late 19 century. The book changed a little bit since, but not much. It essentially treated plane geometry all from scratch. Of course, it was indeed a very important part of mathematical education, because it was used as the way of teaching logic and the idea of a proof. This was till essentially my time and already at the end of this period (I finished my school in 1964), at this moment, a new wave of Math reform came. I think it was probably all around the world. In the Soviet Union it was led by very good mathematicians like Kolmogorov. They had some good ideas, but unfortunately this was pretty much a failure, in my view, because they tried to over formalize all geometric stuff. The results were not great, mainly because teachers could not understand it. I knew this because I was in a university and was giving some lectures for teachers. It was clear that teachers could not handle this. For instance in classical Euclidean geometry, you talk about the equality of triangles, say when the lengths of sides are the same. But in Kolmogorov’s book this was rejected, because, of course, they are not equal, the word “equal” was replaced by the word “congruent”, because of course they are not equal, they are not exactly the same. They tried to carry systematically this distinction, but students and teachers couldn’t really understand. For instance, I was giving once some lectures for school teachers about geometric transformations. When I finished, one teacher came to me and said that I did all this wrong. Why? Because I used the letter “R” to denote rotations but in the textbook it was denoted by some other letter, hence I was using a wrong letter to denote it.

TPL: Could you understand the psychology of these great mathematicians, like Kolmogorov, why people like him would cook out things like that?

YE: He was extremely interested in mathematical education, in fact I was told that Kolmogorov, when he was still pretty young, decided that after 60 he would stop doing mathematics, and after that he just started to be full time involved in the mathematical education. I do not know if it is exactly true that he completely stopped doing mathematics, but at least really he was just teaching in a school and was really very seriously involved in this. There was a very special school in which he taught, extremely special with selected students, for these students maybe he could teach them this kind of book and he then extrapolated this to all schools, which was pretty much a disaster. I don’t know what the situation is now, but after many changes back and forth, somehow the system was destabilized.

TPL: It didn’t destabilize, it becomes quasi-periodic, well, we have that in California, right?

YE: I don’t know why, but we’ve seen all this wave of people with completely different intentions coming to education. First they were really professional people, good mathematicians, maybe with some good ideas, then maybe people with wrong ideas came into this play.

TPL: It seems to me that education is about how to get some knowledge into your brain, right? It’s a plain truth that we don’t know how the brain functions very well, and your brain is different from mine. Yours is greater but different.

YE: Our brain has the following fantastic property. In a computer you write a great program, the computer does a fantastic thing, but unfortunately if you just make a small mistake in this program, then this great thing is not working anymore. And the way how we, human, are programmed, we are programmed by all our interactions with the outside world. Most of what we get is some kind of garbage, a very small amount of the information we get is useful. But somehow our brain is able to resolve this extremely bad software which we are getting and able to function well, produce good things. But at a certain point, when the amount of garbage that we are getting exceeded a certain level, then, of course, our brain stops functioning properly. I think this is the same when teaching children, there is an extremely high level of toleration of how much nonsense we can teach them and they still can reasonably function.

TPL: Just like a filter…

YE: Yes, and after all, maybe teaching them badly might be beneficial: those of them who are able to resist the garbage we are teaching, may develop into and some great minds who could succeed despite of all our education.

TPL: They built up some immune system.

YE: But when this is a little bit above a certain threshold, then essentially nobody survives, that’s bad.

TPL: Education is a difficult thing. I would like to follow up: Now that we have named Kolmogorov, his educational idea is not to everybody’s liking but as a mathematician, of course he is a great, great mathematician, can you say something about him?

YE: Unfortunately, I cannot say much about him. In Soviet Union he was in Moscow and I was in Leningrad, which is now St. Petersburg, links were fragmented, there was not much interaction. Of course I knew about Kolmogorov and I heard a lot of stories from people who personally interacted with him. My only personal interaction with him was only once. When I was a school child l participating in some Olympiad. Kolmogorov was personally giving diplomas, and he shook my hand. This was only once, I did not have any other interactions with him personally. I knew quite well Arnold, he was Kolmogorov’s student.

TPL: He was quite a character.

YE: I heard stories from Arnold about Kolmogorov, and all the stories I heard from Arnold were at some point becoming a little bit nasty. The point of Arnold’s story about Kolmogorov which I am about to tell is to say something a bit nasty about Gelfand. So here is this story about Kolmogorov and Gelfand. Gelfand published his collected work, and according to Arnold, I did not check personally, in this collected work of Gelfand there is only one paper signed by Gelfand without coauthors, and one paper, which is not signed by Gelfand at all, but signed by Arnold. The question is that how come that in the collected work of Gelfand there is only one paper by Gelfand alone, and why there is a paper of Arnold in Gelfand’s collected work. Gelfand was a student of Kolmogorov, and Arnold was also his student. According to Arnold, Kolmogorov had a point of view that a student cannot write his first paper himself, because he just does not know how to do that. So the role of the advisor is to write the first paper for the student, and if the student is good then he will understand how to write papers, and after that he can continue successfully himself. So, that’s why, according to Arnold, in the collected work of Gelfand there is only one paper signed by Gelfand himself (because that was the paper written by Kolmogorov). Then the story about why Arnold’s paper is there was as follows: when Arnold was a student, Gelfand asked him a mathematical question related to the mathematics of the heart (Gelfand was very seriously interested in mathematical biology). Arnold thought about it, invented and proved something, and came to tell about it to Gelfand. Gelfand was happy, ”Great, let’s write a joint paper,” but Arnold replied, “No, I will write it alone”. Gelfand did not object, and Arnold published the paper signed only by his name. However, when Gelfand was preparing his collected work, he asked Arnold to give this paper for his collected work.

TPL: I heard another story about Gelfand, the story goes as follows, in the US-Russia meeting, there was a banquet, in the party Gelfand began to say something in front of Petrovski, of course Petrovski was more senior and was president of University and so on. Gelfand tried to say the right thing “It is not important for the young guy to prove the hard theorems, much more important things are for the senior people to see the landscape and point out to the young people the new directions.” Of course this was not too subtle a reference to the importance of the senior people like Petrovski, and Petrovski was no fool and responded, “Mr. Gelfand does not always think that way.” I heard this story from Peter Lax. From what I gathered, all these exchanges are very productive, because they talk to each other so the mathematical and scientific communications continue, is that so?

YE: Yes, so in Moscow, of course this was a great mathematical school with a lot of great mathematicians and this especially was really a golden time of the generation of Arnold: Sergei Novikov, Yakov Sinai, Yuri Manin are in this generation, many extremely bright stars. They were very excited bringing and developing new Mathematics. In general Soviet Mathematics did not have much interaction with the West. Of course, we were getting journals but this was not the same as to be able to talk to people. Moscow was in some sense an exception, because in Moscow quite a lot of foreign mathematicians were coming, very very few of them get to Leningrad, only trickled down effect. These young people in Moscow were able to interact with best Western people like Steve Smale, bringing new ideas. In Leningrad the situation unfortunately was not so great. There were several extremely good mathematicians. For instance, there was a very good school in algebra, main person there was Dmitry Faddeev. Maybe you know Ludvig Faddeev, the mathematical physicist, he is his son. Ludwig’s father was a very good algebraist and the school was around him. There was very strong school in mathematical physics with Ladyzhenskaya, young Uralceva, Birman. L. Faddeev came from this school. And there were very good analyst. For instance Victor Khavin , do you know him?

TPL: The complex analyst?

YE: Ya, Ya.

TPL: Yes, I remember the time when Louis de Branges, the mathematician in Purdue, solved the Bieberbach conjecture. Because he has over-claimed something else before, after a few wrong claims, nobody believed him any more. A Chinese mathematician Ky Fan, who was once the director of this Institute, knew de Branges well and said to him, “The only way to save you is for you to go to Leningrad, and if they say yes, then people will believe you.” So then I knew there was a very strong group in analysis.

YE: Khavin had several students. Nikolai Nikolvskii is one of them, he is quite famous now. Also geometry school left after A.D. Aleksandrov: Victor Zalgaller, Yuri Burago… Oh, also there was the probability school of Yuri Linnik. But overall there was really nothing comparable to Moscow. In fact, these people did not really have power. For instance, in the university people in power were pretty mediocre, I would say, and they very much wanted to preserve this mediocrity level. Strong people were a threat to them, so it was very important for these people not to have too strong people. For instance, they really disliked my advisor Rokhlin, who came to Leningrad in 1960. Rokhlin had a very complicated life. He was Jewish. During the Second World War he volunteered to the army the first day to fight. You know that the beginning of the Second World War was a complete disaster for Russia. Rokhlin was captured by the Germans and sent to a prisoners’ camp, but he managed to conceal that he was Jewish. Otherwise he would be killed immediately. Rokhlin was born in Baku, Azerbaijan, and he spoke Azerbaijanian. Somehow he convinced Germans that he was Azerbaijanian, and he was sent not to the concentration camp for extermination, but to a military prisoners’ camp, which was much better. He managed to escape, and in Europe, joining all kinds of resistance groups, and by the end of the war he managed to get back to Russia. But in Russia everybody who was in a German camp was immediately sent to a Russian camp: if you were in a German camp, you could not be trusted. He was 21 when the war started, but he was a brilliant student and he already had some important work done. Some mathematicians knew him, for instance Pontryagin. He wrote some letters to authorities, and after one year, they managed to get him out of the Russian camp. But then he could not find any job, and Pontryagin hired him in Steklov Institute at the position of his helper. Pontryagin was blind and he had an official secretarial position of a helper. He hired Rokhlin to this position of helper. Then Rokhlin being at this position got his Ph.D and the second doctoral degree (in Russia there is the second doctor degree). But when he got his second doctoral degree, they fired him because now he was overqualified for his position as helper. Then he really had hard time; he could not find any job in Moscow. For many years he was working in some small places far away from Moscow, but still was coming to seminars in Moscow University. He was at that time working in ergodic theory and running a seminar in Moscow University. His work place was a few hours by train from Moscow. People like Arnold, Sinai were participants of this seminar. In 1960 something happened. Aleksandr D. Aleksandrov, the geometer, was at that time the rector of Leningrad University. He knew Rokhlin and his work, and he had power to bring him, avoiding all kinds of political committees. So he hired Rokhlin. Immediately, when Rokhlin came to Leningrad, there was an explosion of new mathematics. For instance, Gromov was among his first students, and he attracted several other bright students. Really something started to develop, but Rokhlin was always extremely inconvenient for the administration, because every strong person is a nuisance, right? Because a strong person has his own opinion and not just ready to do what he is told to do.

TPL: But Aleksandrov tolerated him.

YE: Aleksandrov moved to Novosibirsk in 1964. They actually got rid of Rokhlin as fast as they could. In Russia the official retirement age was 60 years, of course in universities nobody really retired at 60, they could continue forever, but in his case they forced him to retire, so he retired. And at the time Gromov graduated, again he was such a fantastically brilliant person that they could not just get rid of him, so he got some position in the university which was, I would say, a kind of lowest paid position possible. It was paid less than a janitor’s salary. But he was happy at this position, because he could do a lot of great mathematics. But when he got his second doctoral degree he became overqualified to be at this position. So they either needed to promote him, or somehow had to get rid of him, and of course they preferred to get rid of him.

TPL: So they managed to do that.

YE: Ya, they put some condition…You know in Soviet Union there was no unemployment. Everybody who was graduating from a university was assigned some job, it was called “distribution”, you were distributed to some places, you always got some kind of job, some people managed to get better jobs, and some not. In my case, when I finished my undergraduate study I was recommended to the graduate school, you needed to get such a recommendation in order to even apply. But they still did not want me. There were 2 entrance exams to get to the graduate school, in mathematics and also in a political subject, the party history, (TPL: You didn’t do too well.) No, still you studied the party history and related stuff during the whole five years in the university. This was of course an extremely political exam, and they could really manipulate whatever they wanted, because it was not the question of your knowledge but the question of interpretation. You could be accused that your point of view was wrong. They tried to fail me and I was saved only by Professor Nina Uralceva, she was representative of the mathematics department in this political committee, and she blocked them to fail me, so I got some low but passing grade. But when I passed, I was been punished by being assigned a special position, where I was not entitled to a free distribution but was required after I graduate to go to work in a place in Siberia, or somewhere sufficiently far away, not in Leningrad, not in Moscow. So when I finished I went to work in Syktyvkar, which is a city in the north of the European part of Russia, it is northeast from St. Petersburg. It’s the capital of Komi Republic, you may have never heard of it but its territory is bigger than France. There is an ethnic group, called Komi, who live there, whose language is in the Finno-Hungarian group of languages. At Soviet time and especially in Stalin’s time this was the standard place for prisoner camps and political exiles. All around Komi there were a lot of labor camps for people working and exiled there. In my time, there were still quite a lot, but already not to that degree. Syktyvkar was a normal city, and they just founded a new university there. The rector of this university was a woman, and Leningrad University was assigned by the party bosses the role of a senior university, supervising the new one, so they were obliged to send some people there. Of course, I was just finishing and getting my Ph.D at that time, so the Syktyvkar rector picked me to go there, and I was working there for 7 years. And then they also tried to send Gromov there, but Gromov did not want to go and put some conditions, but they refused and said, either go there or you will be fired, so he decided to leave the Leningrad University.

TPL: So where did he do?

YE: In Soviet Union there were a lot of somewhat strange organizations. Gromov worked, I think, for a while in a meteorological institute, and after that he worked in the institute of pulp and paper industry. And it was at that time, I think, when he left university he already decided that he wanted to leave Russia, this was in 1972 and in 1974 he left.

TPL: What happened to Rokhlin after he retired?

YE: After he retired, he continued for a while but he died of a heart attack three years after he retired.

TPL: Is this Rokhlin in Yale his…?

YE: His son.

TPL: He is also quite a unique personality, some genes survive.

YE: I remembered this young Rokhlin, a very small boy when I visited Rokhlin. And Rokhlin demonstrated the following ability of this boy, you could tell him any long sentence and he would immediately repeat everything backwards, without even looking at the sentence, just listen to you and repeat backwards.

JHC: In early 80’s you were working in industry, right?

YE: I was sent to work in Syktyvkar, and in fact I would say that I liked there. This was an absolutely new place. At that time there were a wave in Soviet Union of creating new universities everywhere, and that was one of the new universities. But in most places there were already some institutions, usually called pedagogical institutes, preparing school teachers. In most cases they would just rename it to a university, as an upgrade. But all the same people remained there. In Syktyvkar they decided to do something cleverer. I don’t know, but I think some good people were involved, and they decided to do it from scratch. So while there was a pedagogical institute, they tried to organize a new university separately. They just imported a lot of good people, so, for instance from Leningrad several young Ph.D came, and from Moscow. There was a very nice spirit there. Another thing, which was great there, was how they managed to attract some people to go there. In my case I had no choice but to go there, but some people did not have to go, but still came because they gave them a decent place to live. In Russia there was always an enormous problem with housing. For instance, in order to live in Leningrad or Moscow you needed to have a special right to live there. You could not just come and live there. Unless your family was living there for many generations, this was highly non-trivial. And even with this right the living conditions for most people were very bad. I was lucky, my parents’ family had a pretty good apartment, so we could continue to live there. But, for instance, in my wife’s family the parents and three children, five people, they lived in a so called communal apartment, they had just one 15 square meter room in an apartment where many other families lived. That was really not so great. Later they managed to get a better apartment, but it was always difficult, it was always a problem. But in Syktyvkar, the local government gave to the university a couple of newly built housing complexes for new faculty. People would come and they immediately get an apartment, this was almost unthinkable elsewhere, for many people to get to the university meant to get a decent place to live.

TPL: How is the university doing right now?

YE: Of course not so great, it was good, I would say, for five years. Then bureaucracy already started to take over. We had a pretty heavy teaching load there and this was general in Soviet Union. For instance, when I started I was teaching 21 hours per week and the second half of my stay there I was the chair and as the chair I had a reduced teaching load of 12 hours teaching per week. It was really heavy teaching, but from the point of view of the administration we were really not working enough. They could not understand how come, that they, administrators were working 40 hours per week, and we only 18-20 hours, so we were obliged to write a report, where you needed to write how you spend 40 working hours: this hour you were teaching, this grading a homework, etc., precisely what you were doing at any given moment. You see, this kind of standard bureaucracy started to take over.

TPL: People knew how to deal with such thing, there was standard way to deal with this bureaucracy, right?

YE: Yes, of course. In any case, back to my personal story. I have two brothers, one is a well-known physicist and he is still in Russia, the other one is an engineer, he was living with my mother and they decided to immigrate to the United States in 1975, about the same time as Gromov, one year later, and they left. Of course they tried to convince me to go with them, but it was an extremely hard decision, because at that time leaving looked like a completely dramatic act, because it was indeed an iron curtain, we never could go abroad or travel, and if you left, whoever your friends and whoever you left behind you’ll never see again, and this was very hard to think about. So when they left, then gradually it became known to the university administration that my brother’s family emigrated. I was then the chairman of the math department, and my position became extremely weak, because the role of the chair is to fight with the administration, but whenever I would start a fight, the Rector would say “why you are listening to him, he has relatives living in California and waiting for him”. In any case, then I with my wife decided that we will also go. I resigned from the university and returned to Leningrad and applied for emigration, but it was already the wrong time because the Russian war in Afganistan started soon after. After that all emigration has practically stopped. We were allowed to leave at the end of 1987, so from 79 to 87, for eight years, we were in this extremely bad situation. Because after you applied for emigration you were completely a persona non grata, politically unreliable person, unpatriotic. I could not work at any university after that, so I worked in school, teaching for a while, even in school I was teaching as a substitute teacher, because they did not hire me for a permanent position, and I had many other temporary jobs before I found a job in a company which was developing an accounting software. I worked at this company until I left.

TPL: But all the while you had been doing research.

YE: Ye, I was doing research of course, but it was quite hard. What I was doing in this software company was a pretty complicated thing and really advanced big software system. Because I could do it, I quickly became in charge of a big group, because there were not many people who could do this. And later I found myself most of my time, even when I went to sleep, I wanted to think about mathematics, instead I was thinking what to do with this or that file, how to make the software to work better…

TPL: But economically you were doing okay.

YE: Yes, I was doing okay economically, it was not great but it was livable. I could tell you precisely my salary, my salary was 250 rubles/month.

TPL: How much an egg cost?

YE: Apartment was pretty cheap, apartment was something maybe around 20 rubles, but if you go to movie theatre the ticket will be something like 2 rubles, bread was quite cheap, bread was something like 15 kopeks (=.15 ruble), or so. Anyway my salary was completely sufficient for basics, like food, basic clothes, apartment and utilities, but nothing will be left after that, you just spent all your money. To be able to take vacations, or anything extra, you needed to do extra work, called “haltura”. Literally the word “haltura” means “the work which is not properly done”, but it was also a common term for any kind of extra work. For instance, once, with a group of friends, as a “haltura” I contracted with sea port at the North of Russia to install there an accounting system. But we were very bad businessmen, so the money we received were ridiculously low for the time and effort we spent doing the job. So what I was saying, this was time consuming, I tried to do mathematics and I was going to mathematical seminars, but of course in no serious way I could be an actively working mathematician.

TPL: So these eight years you did not do much mathematics.

YE: Yes, I did something, they invited me to speak in the Congress in 87, I would say that in some sense I had great time mathematically the first two years after I was refused till I found this good job. When I was teaching in school it was really a temporary job, most of time I had no job, then I had a lot of time to do mathematics, but no money. I did lot of things during this time, but during the rest of the time I was just essentially developing this.

TPL: You have very good family to support you during the hard time.

YE: Ya…I wouldn’t survive otherwise.

TPL: Maybe like your lecture, we will now have a break and open up for questions.

YIL: You already mentioned something about your experience in mathematics, I have few questions {\color red}{on this. The first one is about your experience in mathematics from childhood, how you got into mathematics. The second one is that, despite all the difficulties you have encountered, what made you persist in doing mathematics? You mentioned your advisor and Gromov, probably many people surrounding you all had difficult time, did that give you some examples or courage to go through all these difficulties? And the third question as you said symplectic geometry was a fairly new field, at the beginning you did that without others, you are also kind of the key person to promote this field, how do you manage to achieve this?}

YE: Okay, let us start before I forgot all the questions.

YIL: I will remind you.

YE: So okay, first of all how did I get to mathematics. Till I was probably 13 years old, I was sure that what I would do in my life I would play violin. I didn’t think about any other career, I was playing violin and that was what I liked. At that time in Russia there was a well-developed system of Mathematical Olympiads, so first was the School Olympiad in Leningrad, then the Regional Olympiad, All-City Olympiad, then All- Union Olympiad. I went for the first time when I was in 6th grade. Actually I did pretty well although I didn’t really prepare for it. Then I was invited after that to a “mathematical circle”, there was in Leningrad and in Russia in general such a system, called mathematical circles, like small clubs for school children. The circle to which I was invited was run by a very good mathematician. Her name was Nina Mefant’evna Mitrofanova. She was the wife of Yuri Burago, a well-known geometer. Some members of this circle became extremely good mathematician, like Yuri Matiyasevich, in logic he solved the tenth Hilbert problem. So I liked that, I spent a lot of time going there and working on problems, and in the next Olympiad I did better because I was well prepared. You know, this Olympiad thing is something that requires a lot of training. By the time I was in 8th grade I already became a professional. I would come to an Olympiad, looked at the problems, and saw something that I immediately knew what to do about, because I had seen already something similar. So fresh people, even very bright, could not really compete. So I liked that. At that time in Russia there was a system of special high schools, where you could go after 8th grade. In particular they just opened special schools oriented towards mathematics and physics. For me the choice was either I go there, or to a special musical school. I was suffering a lot, but at the end I decided to do mathematics.

TPL: So that’s Eng-Ing’s first question.

YIL: Yes, the second question is how come you have the courage to go through all these difficulties…

YE: Ye, this would be very nice of you to say so, it probably is in some degree true, but whatever you know, you assume the life is hard, whatever comes you have to…

YIL: But you did not give up.

YE: But what is the choice, you give up? You need to continue, right? But I would say that if I were not able to immigrate in 87, and this kind of thing continued for few more years, I think I would be finished mathematically, because gradually it became impossible to think about mathematics.

YIL: There is some limit.

YE: Yes.

TPL: But before you suffered through that period, you have already tasted the nice thing of doing research.

YE: Of course, I already had mathematically pretty good time. Since I was in graduate school, I was extremely lucky to collaborate with Gromov. My collaboration with Gromov started in the following way: Gromov is four years older than me, so when I entered the university, he was 5th year in university, we have five years undergraduate education. He was very famous among students, someone would show you “this is Gromov”. Then I started to go to Rokhlin’s seminars, of course he was there, I saw him, I knew him, but I did not quite interact with him in any kind of serious way until the 4th year of my undergraduate, and he was already in the graduate school. Just this summer after my 4th year in university, I thought about some mathematical problem. Rokhlin asked me some question, I was able to pretty quickly answer his question, but I started thinking about a question in the opposite direction. I understood something, and I very much liked what I proved. The whole summer I did not see Rokhlin, and on September 1st when I came back to the department I told him what I proved. He listened and then said, ”You know, just five minutes ago Gromov told me something very very similar, just go and talk to him.” So I went to talk to Gromov. In fact, what he did was not very similar, he actually was doing a different problem, methods which we used were indeed pretty close. And then Gromov said: “Okay, let’s now try to see what the most general class of problems that we can solve with this method is.” That was the beginning of our collaboration. After that, I would say that probably mathematically I learned much more from Gromov than from Rokhlin, because mostly I interacted with him. I remember once I proved something and talked to Gromov. He thought about it and then said, that if this were true then something else should be true. He just didn’t believe me, because he didn’t believe that this second thing could be true, he just thought that this would be too much. But the next day I proved that this unbelievable claim was indeed true. I remember it was an extremely exciting period, every day I was telling him something new is true, and he would push me further and further.

YIL: How did you feel, you and Gromov are so brilliant with great performance, still with Gromov you were suppressed by the university.

YE: No, when you were there in this system, of course everything was perverted but you lived in the system, this is your life you just never think about this.

TPL: I don’t think this is unique to Russia. Here in this society, if you look it from a distance or from an angle, you see something ridiculous here too, you just get used to it.

YE: Yes, you get used to it, of course right now I looked at it, everything was extremely perverted, but if you were in there in this routine you did it everyday. So the last question, what is the whole story of symplectic topology? During this period of my very intensive collaboration with Gromov, that I mentioned, we were developing what we called Hprinciple. In mathematics you have many geometric problems described by a system of differential equations and/or differential inequalities. You can get out of it an algebraic problem by replacing all derivatives by independent functions. Normally you do not expect that solving this algebraic problems leads to solutions of the differential one. But amazingly there are cases when this is the case. For instance, there was a known phenomenon of this type, the $C^1$ isometric immersion theorem of Nash. If you take any Riemannian manifold and consider the problem of isometric embedding, this is a hard problem. For instance you take the 2-dimensional sphere and you want to deform it isometrically, if you try to do it $C^2$-smoothly, you can never do it, it is rigid, right? If you have a ping-pong ball, you cannot deform it preserving distances. But amazingly, if you allow to do it in the $C^1$ category, then preserving distance you can manage to pack it into an arbitrarily small ball. Fantastically counterintuitive theorem, that was a theorem of Nash, and this is one example of this type I am talking about. And these problems which I was solving along with Gromov and that I referred to before was problems of this type. So we together were looking for the most general class of problems, which satisfies this type of property: at first glance it looks extremely rigid but somehow it has fantastically many solutions with any kind of properties. We were absolutely sure at the beginning that symplectic topological questions are exactly of this type. For instance, when you try to construct symplectic structure, when obvious obstructions are satisfied, then you should be able to construct it. We found many results very close, but not precisely like this, for which this was indeed true. It always looked like that you should just improve a bit your construction and this construction would work. For instance, one of the key questions was the existence of certain Lagrangian embeddings. I was making some extremely clever constructions of Lagrangian embeddings, but it never quite worked, always some problem remained. Eventually we started gradually to believe that if this is so hard to prove then maybe this is indeed not possible. Gromov actually formulated an alternative. The soft resolution of the alternative would mean that symplectic topology belong to the domain of H-principle, and therefore not so interesting. The hard resolution would mean something much more interesting. It would imply that Hamiltonian systems have some special qualitative properties just because they are Hamiltonian. As I said, at the beginning we believed that the soft part of the alternative should be true. Then I gradually started to think that Gromov’s altenative should have a rigid resolution, and eventually I proved it. It was actually in the end of 1981, at that time nobody was doing this, at least I thought so, especially because I didn’t have any communication with any outside world. So I thought that I had all the time in the universe and nobody would ever do this. If I don’t do it today I can do it next day.

YIL: Sometimes people will feel lonely about this, but you feel even happier.

YE: No, no, this was 81, I was already without job, I had a lot of other distractions, so I thought okay if I don’t have time to do it today, I can do it next day, but suddenly something exploded. That was the end of 81, or the beginning of 82, Bennequin’s paper appeared. Actually some paper in this direction appeared even before but it was completely wrong. So when Bennequin’s paper appeared I was not so much upset by this because I thought it was also wrong. But it turned out to be correct. Then Conley-Zehnder’s paper appeared, I was extremely discouraged because I really didn’t publish my result and I didn’t have actually even a possibility at this time to publish anything, so the world around me exploded.

MLY: How the rest of the world got to know your work on this symplectic topology?

YE: I wrote a letter to Gromov, he was already in the States, was writing his paper about holomorphic curve, he knew my work. Then one unfortunate thing happened, in my first paper, where I proved the Arnold conjecture for torus and other surfaces, first I made a small mistake but then I corrected a couple months later. A. Katok in 79 got a permission to immigrate to the US and I asked him to take my paper. You could not just send an unpublished manuscript, it was actually illegal. I asked him to take my text to the West and gave it to him before I discovered the mistake. I then sent him the new correct version but somehow he messed it up and gave to some French mathematicians my first version. They had been running for a year a seminar. But I guess they didn’t even discover. You know this was fantastically stupid indeed, you cannot even make photocopies, it was illegal, all copiers were under control, some people managed to copy but it was illegal. The idea was that with the copier one can organize some mass production of antigovernment propaganda. And to send mathematics texts abroad was also illegal, some mathematicians were visiting, they would sometimes take papers. But some of them were extremely afraid. For instance once I wanted to send one of my papers, it was in 87, I came to LOMI (Leningrad branch of the Steklov Institute) asked one well known mathematician, whom I knew well, to pass something to Serre, who was at that time in Leningrad. I did not know Serre personally, so I asked this person that maybe he can ask Serre to take my paper. But later he told me that Serre said he could not take it because when somebody else before him was visiting and agreed to take some papers, the Russian border control shook him so much at the custom that when he finally came back to France, he was still shaking. So for foreign mathematicians this was also not quite innocent. With Perelman

JHC: Some article said that you have connection with Perelman?

YE: I should not exploit my connection with him. I have the following connection with Perelman. First thing to tell is that I did not know him when he was in Russia, he was too young. At the time I left Leningrad University he was maybe in school, so I met him when he came to the West in 91 or so, he was interacting with Gromov very much, Gromov was at the time part time in Maryland and I met Perelman there. We continue interacting a little bit after that, and then, when he came to Berkeley, he did not have many friends, and so we had some interactions. When he was leaving back to Russia, he just packed some box with some old tax returns, or something like these, and asked me to keep them in my garage, and also asked me to be his US address, because he had some bank accounts and needed the statements to be sent to somewhere. I still get his bank account statements. Up to very recently he contacted me regularly, telling where I should send all this stuff, so I mailed all this correspondence to him. But for the last couple years, I did not hear from him. At one moment he wrote to me asking to find one paper in the box which he left with me. I went to my garage and discovered the squirrels made home in his box and chewed all his tax returns and other papers.

JHC: You know what he is doing now?

YE: I don’t know. A Japanese company made a documentary about Perelman without Perelman, and they interviewed me, they quoted my sentences and now everybody is asking me about him. My last conversation with Perelman was already after he refused Fields medal but before he refused his million dollar prize. He called me again with this kind of instructions where to send his mails. We started to talk and I asked, “Grisha, why wouldn’t you come to Stanford, you don’t need to do anything, you don’t have to give talks, just come.” “Then what will I be doing?” “Oh, a lot of people would like to talk to you.” “Talk about what?” “Talk about Mathematics.” “I am not interested anymore in Mathematics.” I asked him, ”And what are you interested in?” “It’s too early to tell,” he replied. I just said this to Japanese filmmakers, and they quoted me, and now everybody is fascinated. I just read some comments saying that he probably is doing something fantastic now.

TPL: Where does he live now?

YE: He lives in an apartment with his mother in, I would say, a very unpleasant part in St. Petersburg. Recently they tried to elect him to the Russian Academy, of course this was a ridiculous idea. In the United States, they don’t ask you before they elect you, but in Russia in order to be elected, you need to apply, you need to write application that you want to be in the Academy. It’s an absolute ridiculous idea that Perelman would ever write such an application.

TPL: To what would you attribute the success of Perelman?

YE: First of all, certainly he is extremely clever, no question about this. My wife often says mathematics should be done in monasteries. Indeed, you can do some good mathematics being normal person, but probably to do something absolutely fantastic like this one, then you really need to forget everything else, just as if there is no other life, that is the only thing there. That was exactly what he did. The same thing, now I would say, Clifford Taubes is doing. He does nothing else, just completely concentrates on mathematics, like nothing else exists in the universe. Of course, most of us still have some other life, we enjoy something else and this is extremely bad thing for doing really great mathematics. In this sense, I can completely understand Perelman when he went back to Russia, right? Because he has an idea of how he wanted to do the geometrization conjecture. If he stayed here, accepted a position here he would have to teach. Even being in the Institute of Advanced Study, you really have some other duties, you have to write recommendation letters, review postdoc applications, a lot of duties which distract you every minute from concentration. So he went back to Russia, he is a kind of person who does not need much from the material life. Of course the salary is absolutely miserable compared to here, but from his life in the States, he was able to save a few thousand dollars, which were going long way for him. In Russia, he was working in the Steklov Institute. And Steklov Institute, again is a fantastic place, you don’t even need to come to work. In previous time you had to come once a month to get your check, nowadays, maybe they probably don’t need even that: checks could be deposited. So for Perelman it was an ideal arrangement. During this period he practically did not interact with anybody. For instance, Yuri Burago, Perelman’s former advisor once visited Stanford after Perelman left. We didn’t know yet that he had proved the Poincare conjecture and we didn’t even know what he was doing. I asked Burago, “What Grisha is doing?” He answered, “Well, he really became crazy, unfortunately he is not doing anything.” But, in fact, he was doing at that time some fantastic things, just did not talk to anybody and nobody knew what he was doing.

TPL: It would be a waste of time for him to talk to anybody, there is no point. But Gromov of course you know him…

YE: Yes, Gromov may be a counter-example to what I am saying, but I would say that at some extent Gromov is not an extreme counterexample. Gromov is also a workaholic, but he is interested in a lot of things, for instance he likes reading books. But Gromov also mostly was in a privilege situation working in some places where he did not have to teach. So Gromov is a little bit counter-example but not much, I could imagine if Gromov moved a little bit further toward the style of Grisha Perelman maybe he would prove more.

TPL: I have heard from other people basically saying the same thing. Gromov said this ought to be true, so there must be a trend of thought lead him to say that, but it is not obvious to most people.

YE: But in general, I found what is obvious, or not obvious depends on the person, because everybody has some kind of picture in his mind, what we say is just small thing what we can think of, right? And in this world in my head something seems obvious, but other person has a complete different picture of this, and my picture doesn’t fit there. Sometimes when in a paper it is written that something is “obvious” you cannot understand it. But sometimes just with one word it suddenly becomes obvious because we managed to see the picture from a different perspective. So our understanding works in many different ways. I remember Gromov actually was teaching me when he was a graduate student and I was undergraduate. He said if in a paper someone writes that something is “obvious” that most likely means that the author does not really know how to prove it. Because if he knew and it were indeed simple, then he would just write it, right? But because it seemed to be clear to him and he tried and tried but couldn’t really prove it, so he wrote it is “obvious”. Another Gromov’s teaching was that if you see some really tricky complicated argument then there are two possibilities, either the author does not really understand what he is talking about, or the argument is wrong. Because if the argument is really understood it should be simple.

TPL: This reminds me Arnold, I should ask you this question, Arnold said, “So and so they claimed they have proved my conjecture, but they don’t understand it, there is no understanding.” He actually made that statement at Stanford. So my question is that when you proved Arnold’s conjecture, how did Arnold respond, react to?

YE: My proof was kind of a very sad story. I first proved this Arnold’s conjecture for surfaces, actually this was precisely the year when we applied for immigration. I sent this paper to the journal called “Functional Analysis and its Applications”, it was really the best Russian math journal, Gelfand was the editor in chief, and Arnold was the deputy chief editor. Arnold sent my paper to a referee, and this referee was reading this, of course I didn’t know who the referee was. One day I happened to be in Moscow, I called this person who I did know very well, and who turned out to be the referee. He was extremely happy that I called him, “you know I am just refereeing your paper, I don’t understand something, can you explain it to me?” So I went to him and we talked the whole day. I explained to him the proof and he wrote a great report. Then, of course he confessed to Arnold that he had talked to me. Arnold was absolutely furious, he told me that I destroyed the referee, and now he had to start the process all over again. But by that time in this Journal they already learned that I applied for immigration. They were then extremely afraid, I didn’t know exactly what, but probably Gelfand was extremely afraid that some harm would come to the Journal if they published a paper of such a politically incorrect person, as me. One member of the editorial board called me and said that they cannot reject the paper because there is no mathematical ground for rejection. So they just asked me to withdraw the paper. I said no and did not withdraw it. The paper is still there, it was never rejected and never published.

TPL: But a lot people had read the paper.

YE: In Russia there was such a system, you could either publish a paper or you could send it to a special kind of depository of papers, then it was registered and after that it was refereed. It was not quite a publication, but anybody could write a letter to this place and request a copy, if you wanted to read it. So after this story with the journal, I just sent my paper to this place. It’s not quite published, but it’s in this place, and I know people who received it from there.

TPL: Now with the electronic copy…

YE: Yes, of course, it is ridiculous now.

TPL: Your life story is a little bit different from most of us, to say the least.

YE: I think everybody’s life story is different. If you listen to any person he has a unique life. You know, my story, the whole Soviet Union and our experience there, from today’s point of view look perverted. It was very strange what we did and how we acted. Looking back it looks ridiculous, but it was the way how life was.

TPL: In spite of all these not entirely positive happenings, mathematics at Soviet Union was then at the highest level.

YE: Yes, mathematics, but it was in some sense the same, I would say, let’s call it Perelman syndrome, because this is in some sense part of the same story, right? So first of all, if you have bright young people in the West, say, America, there are so many possibilities for them what they could do, in Soviet Union either they would go to some kind of hard science, necessarily, essentially the science like basic science, because humanities were also very much politicized, not everybody would like to do this, so what is left, physics, mathematics and chemistry. By the way biology was also discredited, because of people like Trofim Lysenko, so very much what essentially was left were mathematics, physics and chemistry. Or if you were very bright, you also could become a criminal, because many people wanted to somehow have good life and get some money, but there were no legal ways to get money, so to get money you have to do something illegal. Therefore some people who were very good at making money, they were either slightly under what is legal or slightly above, most of them actually did not end well. So it’s just sheer mass. In general I would say mathematics was not in a good shape, because what you see here, you see here just some really great mathematics in Moscow, not as great but still some good mathematics in St. Petersburg, maybe some excellent somewhere else, there was some kind of huge mathematical swamp, because in many places there were so called mathematical schools which kept reproducing themselves, Say, at some moment not so bad mathematician would get to a certain place, and everybody else would be his students, students of his students, etc. Maybe at first they did some reasonable mathematics, but then they were self-reproducing. This was extremely popular this type of thing, I would say in general the state of mathematics was not great, what we see are just a few bright spots.

TPL: But this bright spot had this big volume.

YE: I don’t know if it is that big. If I start to list great mathematicians so we can list maybe a few dozens of Russian mathematicians through the history.

TPL: You are now in the US, so how you compare the two systems? Of course US is an entirely different society, as you said most mathematically talented people do not continue mathematical career. What do you feel, these are two big countries and you have been in both places, in each for a substantial portion of your career.

YE: You can’t compare them at all, what existed in Russia only could exist because of all this extremely perverted system, everything was closed, not allowed to freely move, otherwise we now see that as soon as the Soviet Union collapsed, now the whole thing dissipated, right? What left in Russia is now not so impressive, most of well-known Russian mathematicians are now abroad.

TPL: Could it be possible that as soon as they put the economics in order the Russian school would come back? Because it is not just Soviet Union era, before that in the 19th century Russia had great intellectual gathering, I mean you have great writers…

YE: I very much wish them to become great but I should say that I am not so optimistic with all the recent political development in Russia. I don’t think that will happen in the near future, unfortunately.

TPL: But intellectual tradition in Russia did not start in 20 century, right? It went all way back, in 19 century you had great writers, musicians, so something deep traditionally there.

YE: Yes, I very much hope this will continue. Something strange about America is that of course we have great universities there, great mathematicians working there, despite of terrible school education. It is extremely scary to see what is going in the school education, in mathematics in particular, right? It is great that we have fantastic open country, I am personally extremely happy that I was able to come, to be at Stanford, but you look at the faculty of Stanford. Out of all our newly hired, how many American educated people, probably one, Brian Conrad, so you see, it is great that we are here, but it is a little strange.

MLY: I do have one more question. Back when I was in my undergraduate years, I studied pure mathematics, of course I liked mathematics, but all I learned was pure mathematics, I don’t think I knew much about physics, chemistry, other branches of sciences, was it the same type of situation when you studied mathematics in geometry, in topology back in college which was isolated from other disciplines of science, do you think this still is a good idea?

YE: First of all, there was always the Russian tradition that almost nowhere there was a mathematics department, they were always called department of mathematics and mechanics, so in Moscow it was mechanics and mathematics, and in Leningrad was mathematics and mechanics. Indeed they were deeply integrated, we learned quite a lot of mechanics, for instance. So this was good. Although looking back I did not quite like mechanics, which we learned, but it was always this way. We also had a little bit of physics education, for instance Ludwig Faddeev was teaching quantum mechanics. We also had standard classes of physics, some kind of general physics for a couple years. I would say there were some other sciences, but of course there could be more.

MLY: I don’t know the history of mathematics quite well, somehow I feel maybe in 19 century, mathematicians were physicist as well, they learned lot of ….

YE: This was actually more true in Moscow, very much coming from Gelfand and his best students, they were really promoting an idea of unified mathematics, there is one mathematics, and also in terms of applications. In fact, I think all mathematicians despite of whatever they claim that they care about pure mathematics and do not care about anything else, they all dream to have some applications, right? Of course, we all dream that what we do suddenly will be useful for something else. Of course, mathematics should not necessarily looking for applications, it is not necessarily from my point of view that all mathematics should be kind of applied mathematics. But in the long run I think it is good to have some kind of perspective that maybe what you are doing at some moment may become useful for something.

MLY: At some point, problems from other discipline may stimulate mathematicians to think deep and solving this problem from the mathematical point of view may create new tools and new ideas.

YE: Unfortunately I don’t understand this myself. You see, in recent years there were quite a lot of interactions between mathematics and physics. The classical interaction of physics and mathematics was that physics people were using mathematical tools, right? But now something quite opposite, new mathematics is coming from physics. There are, of course, many many mathematicians who are trying to learn physics. But frankly, I don’t know essentially anybody who succeeded in really understanding physics in the way how physicist understand it. So there is something wrong, like in the original mathematical education there is some problem that prevent us. When you were educated in mathematical sense there is a kind of a blockage not allowing us start thinking from the physics point of view, I don’t know why. Some physicists, the best of them well managed to do some good mathematics, maybe they didn’t learn it appropriately (from our point of view), but somehow they understand it. I have a very telling story. There is a physicist in string theory, Robbert Dijkgraaf, maybe you know the WDVV (Witten-Dijkgraaf- Verlinde-Verlinde) equation. He is a very respectable physicist, but of course he knows a lot of mathematics. I interacted with him quite a lot, and one year I was in Princeton, I spent a year in the Institute for Advanced Study, and was organizing a program about holomorphic curves. He was at the same time visiting the physics school at Princeton and I asked him to give a few lectures for mathematicians for our program. He agreed, we discussed what he will be talking about and then I just said, “I want to ask you about just one thing. In my experience when physicists talk to mathematicians they assume that mathematicians know all the mathematics.” He just looked at me very surprised, “Then what do you know?” Somehow it should be normal that mathematicians should know mathematics right? The physicists know all the physics, maybe some are more specialized, but in principle they are able to discuss all kinds of physics, so for them to hear that mathematicians don’t know all mathematics was strange. Then indeed, what do they know?

TPL: This is a troubling situation, the way we educated the students is something troubling.

YE: Ye, Ye, I agree.

TPL: At the worst we will become some logical machine, that’s very bad, we cannot even compete with those people who write software in that regard.

YE: So there is some blockage. When a physicist studies something, and say there is in something they know an extremely far fetch analogy, and they say now let us assume that the same thing may indeed work for our problem. In 90% of the cases, of course this does work, but sometimes indeed it works. But for us to do such thing is a kind of taboo.

TPL: We have lost this power of imagination of analogy, through our training we have lost some of the innocence.

YE: We try to do such thing but sometimes we put ourselves some limit not to that far.

MLY: Of course historically there is a lot of interactions between physics and mathematics, but up to now I noticed there are very few interactions between physics and now very popular biology. I noticed at some point maybe Gromov is doing biology.

YE: Gromov is always a mathematician. I don’t know what his biological achievements are?

TPL: How genuine is his understanding of biology?

YE: He certainly, extremely interested in it, and very much would like to make some contributions to biology.

TPL: How genuine is his contribution so far?

YE: I don’t think he achieved much so far.

TPL: In the 19th century, it is not that Laplace was a physicist, Fourier was a mathematician and they interacted. Instead, Fourier was a mathematician and also a physicist, Laplace was the same, and so they can interact within themselves. Now it is the interactions between mathematicians and physicist, I think that would be harder.

YE: But, in recent years with all these string theory development, of course physicists also don’t quite accept string theory in the community of physicists, they have some friction among themselves, but for us, nonphysicists they look like physics, and I think there are quite a lot of attempts of communications between mathematicians and string theoretic community. I think there are some good things happening.

TPL: But this is next to the best thing, the best thing really should be interaction between individuals.

YE: But not so many individuals within whom such interaction is possible.

TPL: That’s true. But now through our education, we almost try to minimize this kind of individuals, that’s the trouble. Well, perhaps we should spare you from that, it is really great, this is completely different from my previous interviewed people.

YE: I was very happy to talk to you, but I hope you don’t delete your part, because otherwise it would look very stupid, my monologue.

TPL: Could I end by one more question, it is really true that everybody can feel the tremendous enthusiasm in you. Maybe you were born with a lot of enthusiasm about life in general, or but could we also attribute this to this very winding road you have been in.

YE: Of course I heard people saying this about me, but I never thought in these terms about myself. Actually I don’t feel I have changed. Mei-Lin knew me, how many years ago? How many years ago you met me?

MLY: 1994, 17 years ago.

YE: So how much did I change?

MLY: You never changed, you look the same.

YE: So that means I was born with a lot of enthusiasm. You know this change or not change is an unfair question, because I remember, I saw Gromov last time before his leaving was 1974, and then I met him in 88 when I immigrated, so I didn’t see him for 14 years, then first time we met, for 30 seconds I was shocked, “This is Gromov?” It completely contradicted my interior image of him, but 2 minutes later, ye, this is Gromov, and I started to think that he always looked exactly like this. I think this is an unfair question because if you saw me, first of all you saw me few times in between, probably the first moment, the second when you saw me you did recognize that I was different, but then after 2 minutes I became the same person again.

TPL: This enthusiasm propagates to other people, you can see that in your lecture people just feel good.

YE: I am sorry, I never learn unfortunately, in all my lectures I always try to say 5 times more than I should. I know this, I tried, but somehow I never succeeded.

TPL: The way you say it is important, it is not just what you say. Anyway, thank you very much.

- Tai-Ping Liu and Jih-Hsin Cheng are faculty members at the Institute of Mathematics, Academia Sinica.
- Yng-Ing Lee is a faculty member at the National Taiwan University.
- Mei-Lin Yau is a faculty member at the National Central University.