傳播數學知識．促進數學教育

Interview with Prof. Kalyan B. Sinha

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

**Interviewers: **Tzuu-Shuh Chiang(TSC), Chii-Ruey Hwang(CRH), Hui-Hsiung Kuo(HHK)

**Interviewee:** Kalyan Bidhan Sinha(KBS)

**Date: **August 11th, 2010

**Venue: **Restaurant, Allain

Prof. Kalyan Bidhan Sinha was born on June 3, 1944. He obtained his B.S. in 1963 from Hindu School, Calcutta, M.S. degree in 1965 from Delhi University, and Ph.D. in 1969 from the University of Rochester. For his fundamental contributions to mathematical physics, especially to quantum probability, he was awarded the Srinivasa Ramanujan Medal of the Indian National Science Academy in 2019 and was elected as Fellow of TWAS in 2002. He served for Indian Statistical Institute as the Director between 2000 and 2005 where he is now professor emeritus.

CRH: Usually we start the interview with questions like, how come you got into mathematics. Why? I mean, especially you know, nowadays you are working in quantum probability? You started when you were very young, is it?

KBS: Well, in school I had always been interested in mathematics, but, the mathematics one learns in schools is not always exciting except geometry. But, I was also quite interested in other sciences, like physics, particularly physics. So that’s why after school, when I went into college, actually I had difficulty choosing; whether I would go for physics or mathematics. My father wanted me to be an engineer, which I didn’t want to be. So I made some compromise and became a physicist. I went to college to earn a degree in physics.

CRH: Where did you go?

KBS: In Kolkata. An old college called Presidency College. I think it was established in 1818 by the British. And I went to that college and read physics. Looking back, I think it was a good choice because the mathematics department was not so great in the college. But the physics was indeed very good. So I did my master and undergraduate in physics. But even during that time, I was fascinated by the mathematical issues behind various physical problems and physical theories. I must admit that even today, to some extent, physical intuition and physical requirements do guide me to ask, what I would think at that point in time, the right question. As I have often said to my students, asking the right question is half the problem solved. It’s often not easy to ask the right question. In that sense, sometimes, physics is indeed a guide to me. After college, I went to the US to do my doctor’s studies and that was in Rochester New York. That’s where I gradually switched. Mathematical physics was just becoming fashionable at that time. Mathematical physics is, sort of, running into bad times.

CRH: So that was in the 60’s?

KBS: Yes, in the late 60’s was when I got interested in mathematics much more directly. Actually I learned all the serious mathematics since then. I mean rigorous, serious mathematics. Well, let me present, you this book of pictures for Indian statistical institute (I.S.I.) . First, this has something to do with Indian statistical institute (I.S.I.). You may remember this building from the old times. First part contains the past, reminiscences from the past. You have here the founder of the Institute, Prasanta Chandra Mahalanobis with his wife Rani, and the prime minister of India Nehru. This is the early days of I.S.I. and there are many personalities who passed through the I.S.I. Fisher’s picture is here and this is Madam Curie.

HHK: When was I.S.I. founded?

KBS: You see, I.S.I. has a curious history.

TSC: in 1931.

KBS: Yes, it was started in 1932. Mahalanobis was a physicist by training. He went to Cambridge to do his Ph. D. and he did not do his Ph. D. . He met Fisher, and he changed to Statistics.

CRH: Changed to statistics?

KBS: Changed to statistics. He was fascinated by Fisher’s way of approaching problems and he became a statistician. Then he did not complete his Ph. D. . He came back and was teaching in the 30’s in Presidency College where I went for undergraduate studies. Then I think at that time he started a statistical institute, as a sort of a private enterprise. Seven people, seven learned people, working together as a group, decided to learn statistics and teach each other. So, they would have other professions by which they earn a living. But, in the evenings, they would meet and they will tell each other what they have learnt during the day. That’s how the Indian Statistical Institute started in the 30s. It was entirely a private enterprise. Mahalanobis then became more confident in statistical knowledge and started to do projects for the British Indian government.

HHK: British government?

KBS: British Indian government. British government was still there. One of the major projects that he did in the 40’s, was to study what is known as The Great Bangal Famine. It was early 40’s, just before the British left. It was a very big famine. Many people died. British government wanted him to find out what is the cause. Was there a less production? Was there a problem with distribution? or what? And he did the concluded statistical study that it was a mostly man-made famine. It was not because there was any sharply falling production. It was because the distribution was so poor and the government was not well organized to see to the distribution properly. It was one of his early projects. He earned some money from the projects and with that money, he supported youngsters, pay them a small amount of salary, but they would still be interested in learning this new subject of statistics and working with him. And then, he got to know Nehru, the first prime minister of India. And that helped him to get government support, but that was already about 1960’s. That’s when the present campus in Kolkata was got where the institute is located now.

HHK: The name? ISI?

KBS: it went back to 1931. Then, they had their own place, own lands and own buildings and then that was how it started at about 60s. But still there was no structured training or teaching. I think in the mid 60’s, the British biologist, JBS Haldane, came to India because he was a communist and was harassed in Britain by the British government. He left Britain and came to India and was taken into I.S.I. by Mahalanobis. It is he who suggested to Mahalanobis that there should be structured training. And he went onto draw up a syllabus for bachelor of statistics and that’s how bachelor of statistics started in the late 60’s. At that time, in Indian universities, statistics was not yet a major subject of study. Mathematics was there but statistics would be taught as one paper in mathematics. But more structured statistics course did not exist. Mahalanobis with the help of Haldane created that in the 60’s. And then in the70’s, master of statistics came. Then C.R. Rao joined Mahalanobis around late 60’s, early 70’s. They made a good team. C.R.

TSC: That’s still in the 60’s?

KBS: Late 60’s and 70’s. So, here you see the pictures of various personalities. Ho Chi Minh came and there is Fisher. This is S. N. Bose of Bose-Einstein statistics. The name Boson came from him.

CRH: Did he get the Nobel Prize?

KBS: No, no, he did not.

CRH: But who in the 40s?

KBS: In the 30s, there a Nobel Prize was awarded to C. Venkata Raman , experimental physicists. I think that he is the only physics Nobel Laureate from India. This is John Galbraith, who visited also, and Haldane, and many statisticians, Neils Bohr, the famous Nobel laureate physicist, Wiener. This is Wiener with C.R. Rao and Kallianpur, who was very young then. You would almost not recognize him. I think Gopinath Kallianpur did his post-Doc with Wiener, did he?

HHK: Yes, but his Ph. D. was with Herbert Ellis Robbins.

KBS: Then he did work with Nobert Wiener for a little while.

HHK: That’s right. There was also Paul Edward.

CRH: So he was a student of Robbins, you mean?

KBS: So Wiener is visiting.

HHK: How about Andrei Nikolayevich Kolmogorov (Andrey Nikolaevich Kolmogorov , Russian: Андре́й Никола́евич Колмого́ров)?

KBS: Kolmogorov should also be there. Kolmogorov visited India in the 60s. Here is Kolmogorov being given the honorary doctorate of the Institute. Here is C.R. Rao. Mahalanobis had also brought quite a few young people with him at that time. He was a very autocratic person. So sometimes, people could not stand that. I have heard of people leaving because of that. R. C. Bose left because of that, and went to UNC-Chapel Hill. So that’s glimpses from the past, now here are the glimpses from the present. Here you would see occasionally my pictures. This is at the time when I was in Kolkata.

TSC: When was that? When were you the Director of I.S.I.?

KBS: I was the Director from 2000 to 2005. I decided that we should give honorary doctorate to distinguished scholars who passed through I.S.I., Kallianpur and S.R.S. Varadhan. They were given the honorary doctorate of Indian Statistical Institute during my time as Director.

CRH: Also S. R. Srinivasa Varadhan.

KBS: Kallianpur and Varadhan. Only two were given and they were the two. So these are some of the events during my time. We had a small conference also. Here Kallianpur is seen giving a talk there. People who participated in the conference, I think that Parthasarathy and Balram were there.

HHK: How about Masani?

KBS: I don’t think Masani was there. I think Masani was already not well at that time. Masani had come earlier but not during the conference. In Delhi, he was there. He was visiting. So this is sort of a present for you.

TSC: So in the 30s and early 40s, India was a communist country?

CRH: It was under British ruling. India was a colony of Britain.

TSC: But it has a very close tie with Russia, right?

KBS: Well, not in 40s, 50s and 60s yes.

TSC: Alexei Nikolayevich Kosygin was there, the Russian primer.

KBS: Yes, Kosygin was also visited.

TSC: I.S.I. now has not only statistics, but many fields of applied statistics, like social sciences, economics, or physical sciences.

KBS: Yes, you see, Mahalanobis’s idea was that statistics was an applied science. So it should get inputs from other sciences. Other sciences would give the data, all social sciences, it may not be natural sciences, and all social sciences provide the data for the statisticians to analysis and find new methodologies, new theories and new ideas to analyze those data. That was what he was thinking. But, I mean, I would guess that it is something not quite realized. The reason is very clear. If you have a physicist and ask him that his only role is to produce data for the statisticians, I don’t think he would like to hear that.

HHK: I did not know Ho Chi Minh had a doctorate degree.

KBS: He came to I.S.I. .

CRH: Maybe he got his Ph.D. in France, maybe.

TSC: How did Mahalanobis think of mathematics? The relation between mathematics and statistics?

KBS: It is very curious. I think that the most famous mathematicians that visited I.S.I. at that time were Kolmogorov and Wiener. But their interests of course were closely related to the subject of Probability, Wiener is more of an analyst, a very high order of analyst. Kolmogorov also, I would say that Kolmogorov had an enormous range of interests.

CRH: A very wide range.

KBS: So let’s call him a mathematician. So they were the two eminent mathematicians who visited I.S.I.. Probably the reason was at that time there were four young mathematicians at I.S.I., who were doing doctorates at various stages. Well, maybe three, because the fourth one had already had a doctorate from elsewhere. He was sort of a post-doctorate fellow. These four made their mark in later years; one is Varadarajan, who has since been in U.S.A. for many years. He was the senior amongst the four. Youngest of the four was Varadhan and in between were Parthasarathy and Rauga Rao. They started learning things in different ways. Varadarajan gave regular courses on many subjects. So they learnt things and taught each other. They were very bright and could do that at a very rapid pace. Then for example, Varadarajan got interested in quantum mechanics. He wrote two books, called geometry of quantum mechanics. Actually Von Neumann and Birkhoff wrote two massive papers on what they called logic of quantum mechanics, but, strictly speaking, I would say both were not quite correct with their choice of titles. It should be the probabilities in quantum mechanics. For that, Varadarajan interest sort of is a wrapped off to the youngsters, namely Varadhan. So, you may also notice, when they talk, they talk more like a physicist than like a mathematician. He would try to motivate why he is doing what he is doing, rather than giving details on the boards and things like that. That is there anyway, in their paper or papers. So they had an attitude of trying to understand the problems so that that there is a possibility of reading a lot more. So these four had a great influence on others in I.S.I. to have an orientation towards mathematics. That’s how I think mathematics made its beginning in I.S.I., as distinct from probability theory, which is naturally connected to statistics.

TSC: So in I.S.I. now, how do you balance between probability and statistics?

KBS: That’s a difficult thing.

CRH: or even mathematics.

TSC: Mathematics is not a major concern of I.S.I., right? I mean to say that I.S.I. concerns more on probability and statistical applications and so on?

KBS: Well, the law which created I.S.I. said that I.S.I. is empowered to give degrees in studies in statistical science and any related sciences. So that wording allowed a lot of freedom, and has been used by I.S.I. quite deliberately. I mean, arguably, one of the example is, M.I.T., one of the finest institutions of the world. Many people do not know that it has possibly the best economics department of the world. But Massachusetts Institute of Technology would not be trained that they are supposed to have it there. But it is there and extremely good. So some of us has taken that attitude in I.S.I., but not everyone agrees, and there are detractors.

TSC: For example, I believe that I.S.I. has an agreement with the Institute of Statistics, Academia Sinica, and every two years there shall be a joint conference. But it seems that it is taken over by statisticians. I don’t like this but it is true. It seems that the probabilists are excluded and not involved.

KBS: Yes, when I visited Academia Sinica in 2002, this was great.

CRH: We signed an agreement. The most recent joint conference was held. There was no message to the Institute of Mathematics, Academia Sinica.

TSC: I think that the agreement was signed by three institutes, right?

CRH: Four institutes to be correct, which are Institute of Statistical Science, Institute of Economics, Institute of Physics and Institute of Mathematics of Academia Sinica.

TSC: Four? It’s taken over by the Institute of Statistics and the other two are out of the game.

KBS: I don’t know really.

CRH: Alok Goswami was very mad about this.

KBS: He was visiting?

CRH: He was visiting here and he found out that we were not informed. It was very upsetting.

KBS: There is reason to be upset and I can understand that.

TSC: That’s why I ask the question. How do you balance between probability and statistics?

KBS: You see, there is a very simple answer. Arguably it depends on the people at the helm of affairs. After my time is over, the new director came in.

TSC: Is that Bimal Kumar Roy?

KBS: No, Roy is just about to become the Director. But I am talking about the past five years. He was not sympathetic to either probability theory or mathematics; let me put it that way. Statistics for had to be catered to, at least in the Statistical Institute, he cannot really deviate too much. So that created some problems for Mathematics and probability theory. That could be the reason. I am not in the know of exactly what happened, but if some individuals took some initiatives, that could have been avoided. But you see, I.S.I. is structured in such a way that the Director is extremely powerful. So that has plus and minus sides. The plus side is you could have things done rapidly. The negative side is which you were talking about. But this conference which I would have hoped to keep on happening every two years, here and there, is not happening, which is a real pity. In mathematics and statistics, maybe, I take some initiatives, he takes some initiatives and some younger people take some initiatives to start the process. But Physics (I consider the physics in I.S.I. is not one of the best) someone has to take the initiative. The original idea was to have physics as a source of generation of data for statisticians to analyze, finding new methodologies and new ideas. But that is not the way it really moves. If you want a good physicist, he or she would do exactly the thing he or she likes to do.

TSC: But I remembered that you have said in yesterday or day before yesterday, under the new director, there should be improvements.

KBS: I would expect some improvements.

HHK: The director, is that appointed by the government?

KBS: No, there is a governing council of the Institute, which is the one that appoints the director. So when the end of the term of the director approaches, the council forms a committee and searches for a new director. They ask various people’s view and so forth, like I was asked this time when the new director was appointed. Similarly, many others were asked.

HHK: But it’s often an Institute is like an university. You have many students.

KBS: Yes, it does and it also advices the government.

HHK: That’s right. It serves the government very much.

KBS: That is where I had the maximum problem. I wanted I.S.I. to be basically an academic institution. My perception of an academic institution (which I have seen in the West, particularly in the United States) is very different. I.S.I. simply is too closely related to the government. I did not like that. So I had clashes with some bureaucrats of the government quite a few times, saying that you cannot consider us as an extension of your structure. I would offer mean in our meetings that you are funding us and as, you can cut our funds, but after with whatever you have given us, you keep your mouth shut.

HHK: That’s like Academia Sinica, receiving the fund directly from the central government.

KBS: There are problems and so forth.

HHK: How are the students admitted to I.S.I. ?

KBS: There is an entrance test which is held nationally. Through that they are selected. The entrance test is followed by an interview for all of them.

TSC: What’s the distribution of the faculties? Probability, Statistics?

KBS: Well, Statistics is certainly larger than all others.

TSC: Much larger or slightly?

KBS: Much larger. The number of probabilists has decreased considerably. The emphasis of I.S.I. has been on measured studies. In 50s and 60s there were a lot of people working on just pure method theory, pure method theory, selection theory, .

HHK: Are there many international, foreign students in I.S.I?

KBS: Not many, we have some, yes. Mostly are from Africa. Nowadays there are a few students coming from Iran.

HHK: I.S.I. has a very long tradition on Indian tradition of good probabilities, statistics, I guess you would have more international role.

KBS: We do have an international exchange program which was started by Mahalanobis and is called International Statistics Education Center, which the Japanese are deeply involved. But every year about twenty or twenty-five of them would come from various countries, but, mostly in very applied statistics. The official statistics and things like that. Many of them are government officers who are in statistical departments.

CRH: One thing, in the 60s, in I.S.I. in Mathematics and Statistics, you have a lot of very good students, Varadhan, Parthasarathy, Rauga Rao, Varadarajan and etc. How do you compare them with the students in the 90s and later?

KBS: They are very good, but you have to consider one that that many of them after graduation have gone to the U.S. Earlier people used to come back, but, now they don’t come back as many. That has changed the situation.

CRH: So do you still get very good students?

KBS: Yes. Also, you see the model changes; particularly in the US and the rest of the world follow that. Emphasis on pure academics is decreasing. You know that very well. To learn something for the sake of learning and just understanding something, that emphasis is decreasing. People are looking into more applied things, and so the old things have changed somewhat. For example, one of the systems which was developed in my time, which we spent mostly in Delhi rather than Kolkata, I spent only 5 years in Kolkata as a director. What the students of masters used to do, particular the bright students, they would do the first year in Kolkata then come to Delhi in second year, because in Delhi, some of us would look at things in a more modern perspective, and they wanted that. The younger students they want to have a more modern views and so forth. So we had about ten years of this program running this program which was very successful. We produced some really bright students. But then, all of these have been scrapped. That course and that kind of training have more heavy weight on mathematics, of mathematical ways of looking at probabilities and more mathematical. That has been scrapped because it takes too much of mathematics. You have to learn much more. Question is it worthwhile. Of course this question brings a broader question, what do you mean by something is worthwhile. Anyway, that has been scrapped in favor for a more applied side of course so the students can get a job quicker and that is the idea. Particularly with the rise of Information Technology (I.T.) industry in India, it has become a rather tough competition. Some of the bright students they want to quickly finish and get in the high paid jobs, instead of struggling for five or six years for a Ph.D. They want to have a good jobs and settling down. That has also been elsewhere which is negative as the academics is concerned.

HHK: So what have you been doing to attract the best students to Statistics?

KBS: Well, we cannot compete with them moneywise. It is clear.

HHK: In U.S., the National Science Foundation (N.S.F.) put a lot of money. It’s very easy for the young people to get money.

KBS: I mean moneywise is has improved, but I don’t think there could be a competition.

CRH: The salary afterwards.

KBS: Afterwards also. Moneywise it is not a competition. So you have to attack something else. You have to challenge to the intellectual content to the person otherwise there is no way you can attract them. Recently, there is a special scholarship which is higher than the normal type which the graduate students get. So I have been involved with that for many years, but this time I was surprised to find that five in mathematics, but only two in physics, two in chemistry and nine in biology. Biology would be large as it is emphasized. So nine in biology, five in mathematics, two in chemistry and two in physics and that sort of gives you a little idea of the relativity. One of the kid I interviewed, he had two years gap after master. So I asked him what he had been doing. He said that he had been working for Barclays Bank. Then I said, why are you here? You are in Barclays Bank. He replied that they promised him to work in modeling and analyst section, but ended up as an accountant. He did not like that and want to come back to the academics, losing a considerable amount of money, for he did not find it challenging enough.

HHK: It is similar in Rajeeva L. Karandikar’s case.

KBS: But Karandikar is much readily. Karandikar’s case is a little different. He was paid a considerably high amount of money.

KBS: About three or four months back (interviewing time August 2010) he has come back to the academics. Part of the reason is that you see with the downslide of American economy and European economy, there has been many impacts in IT industry as well in India.

TSC: What if a student asks for your advice about what he/she should do? Like going into I.T., finance or stayed in academics?

HHK: Well, I am biased. I must say right from the beginning that I am clearly biased. Having been in the academics for so many decades, I cannot but be biased. I have seen a few bright people who have this experience of great disappointment. After going into the I.T. or private sector, let say, of course there is a good salary, but they squeeze the last bit out of you.

TSC: Yes, of course, they squeeze the lemon.

HHK: Absolutely. I mean, you don’t have time to yourself to think about the bigger things and both the big picture.

TSC: That’s right. They become a subject.

HHK: You become a part of a huge mosaic. So I try to appeal to that. Do you want to be the canvas where you are a small dot or do you want to be the canvas itself? Some people do get challenged. Not everyone is interested in being intellectually challenged. I feel that in academics it is one of the best things you can do. That’s why the role of good teachers is tremendous. If you have good teachers, it makes of all the difference.

CRH: Am I right that your wife is with Indian Institutes of Technology (I.I.T.)?

KBS: My wife was. She was with I.I.T. but she was retired. I have retired from I.S.I. and she has retired from I.I.T. .

CRH: So, when you took this attitude. How about your wife?

KBS: You have a good question and a difficult question to answer.

KBS: She was a teacher in I.I.T. . She also studied as a physicist but become a computer scientist later on.

CRH: So what is her opinion about getting into I.I.T. and going into pure academics?

KBS: She doesn’t agree with me. I.I.T.’s undergraduate program is extremely successful. They get very very good students, but the negative side is, many students, I would say about 99.9% of students they go to what is called a cram school. That is terrible.

TSC: Your name, Sinha, that’s a popular Indian name.

KBS: Yes, that’s a common last name.

TSC: Does it have any meaning?

KBS: Yes, when you turn that into the main Indian language, it came from Sanskrit word Simha and is actually Singh. The British could not pronounce it, so they made it a little simpler. That means lion.

HHK: I know many Indian mathematicians who went to U.S. and got a Ph.D. and stayed in U.S., but some returned to India. What is the percentage of returning in the recent years?

KBS: you see, it’s a very low percentage.

HHK: Oh really?

KBS: very low.

TSC: You get the good four, Varadhan and others all stayed in the States.

KBS: Only K.R. Parthasarathy came back. He went to US, somehow, did not like US.

CRH: worked with Kolmogorov.

KBS: He spent one year in the US then he went to England.

HHK: I think that his reason of dislike US was for medical system. I know.

CRH: Is B.V. Rao still there?

HHK: B.V. Rao was there. He did his Ph.D. in India.

TSC: I notice that you have a co-author called Goswami. He’s not Alok Goswami?

CRH: No, not Alok Goswami.

TSC: If you could do this all over again, what would you choose? Mathematics, Probability, Physics, Statistics?

KBS: Mathematics, definitely. But, which particular area of mathematics, I could not say it. I believe that there is a very liberal and wide outlook in mathematics. Though which area of mathematics you may use sometimes you could never predict. Of course you have to learn a lot, but that’s the price one has to pay. Probably.

CRH: Why you got into Quantum Probability?

KBS: I mean, I started in quantum mechanics which means “essentially mathematically speaking”, theory of operators on Hilbert spaces, in particular the Schrödinger operators and in the spectral theory. So that is what I used to do earlier, in the 70s and 80s, when I came back to India. I have written a book on Schrödinger operators, published by Benjamin, which does not exist anymore.

HHK: Oh, the book by Parthasarathy, which pointed out the three Indian gods, creation, annihilation and preservation. That’s what the book is, preservation.

KBS: He had the three gods of the Indians deities, one for creation, one for annihilation and one for preservation.

HHK: That’s right. It is written in his book, on the first page or second page.

KBS: But you see, Indian gods are rather unpredictable. Same god may show up in a different form, the creator may become the destructor in certain situations.

CRH: So it means that they are commutative. But in your case,

KBS: highly non-commutative.

CRH: Exactly, so how come?

HHK: So that’s right. So I called three operators, P.A.C. operators.

CRH: So you also have worked with Parthasarathy? A lot of cooperation with him.

KBS: Yes, a lot.

CRH: Can you say a few words about Parthasarathy ?

KBS: Actually I learnt probability theory from Parthasarathy. I mean serious probability. We had a rather interesting complimentary expertise. He was expert in probability theory and I knew operator theory quite well. So we joint our hands and I think that was a rather successful collaboration for many years.

HHK: I have heard a story about Parthasarathy. Later on you may find it useful. That was in 1982, in a Bangalore conference, first, participants stayed in ISI overnight, and the next morning, we went to a local airport to Bangalore. While we waited at the airport, there was Parthasarathy. I didn’t know him then. He was there in the front greeting the participants. I was with Ito in the back. Ito said to my ears, I am surprised he is very famous and he comes to the airport to welcome everyone. After that, Ito thinks very high of him.

KBS: He has a book, Probability Measures on Metric Spaces, in particular, this book is very highly regarded in many places. (K.R. Parthasarathy, 1967, New York: Academic Press). (53:54)

CRH: So the reason I asked about quantum probability is that we don’t have anyone in Taiwan really working on quantum probability, which I have mentioned to you before. I think that’s a fair statement. What’s your suggestion?

KBS: On quantum probability? That’s a very difficult question. Indeed, you are right. The number of young people doing quantum probability is not very large. That’s true.

CRH: You see, we have not many people working on probability in Taiwan, but we could still survive in some sense as we have our own seminars and things like that.

HHK: No, in Taiwan, you have many people working in probability.

CRH: But if you try to do something. That’s hard. In my opinion, to get one that’s dangerous. Unless there are good probabilists, then, we should get more than one.

KBS: You see, there are many other areas that also have very few representations. Even in the US, it’s an enormously large country, not just in the size of population but the number of universities, but still, when you think of an area which is relatively new, like non-commutative geometry, you find very few people doing that, very few indeed. Now, why is that? Why does Europe have some number? Although it’s not a very large number but it is a significant number. Of course it has the impacts of one great mathematician, Alan Cohn. But that influence does not affect the entire Europe. Europe means Western Europe, which has not moved to U.S. so much, very limited. So I don’t know what to conclude, except it’s possible to say that young people do not like to go into risky areas. So I guess it’s quite natural for young people not wanting to take the risk. I would be understanding of that. But I don’t know really why. The same thing happens by the way. It is known in many areas of Mathematics,. It’s like Eastern Europe, before it was taken by other countries, like particularly US, a very large academic country. For example, even matrices, do you know that US imported the British guy, James Joseph Sylvester, at the end of 19th century, beginning of 20th century. He was taken to the US to teach matrix theory. He was invited and he gave lectures. He was in South somewhere.

HHK: That was in University of Virginia.

KBS: Similarly, when quantum mechanics flourished in Europe, J. Robert Oppenheimer and many others who studied this new science called Quantum mechanics in the late 20th. Oppenheimer later on was famous because of the Manhattan Project, the Atom Bomb Project in other words. So it takes time.

HHK: I think that in the early days of Jimmy McShane, after obtaining the Ph.D. degree, they were paid to go to Europe to have further education.

CRH: Even in Probability Theory, Joseph Leo Doob was not that well regarded.

KBS: Who?

CRH: Doob.

KBS: Doob was a European descendant. He is a third generation or second generation Czech.

HHK: I think that he is a second generation Czech.

CRH: Actually, in 1985 or 1986, when we were in Minnesota, we were surprised. When Doob showed up, apparently the American probabilists nobody greeted him in respect. He was hanging in the back by himself. When you look at Ito in Japan, everybody would stand up when he entered.

KBS: That of course the culture is very different. I remember that when I went to US as a graduate student, the first thing that shocked me was in the class the students had their legs up to your eye level. That was a whole new experience.

CRH: Everybody in Taiwan or other country, are curious about mathematics in India, why do you have so many mathematicians, and everyone thinks that the Mathematics in India is very very strong? How come?

KBS: I wish that I could agree with that.

CRH: Even IT and software industry is in the same situation, because it takes very strong mathematical background.

KBS: For that is a different level. You are talking at a different level. I am a prejudist. Again, I must say that I am heavily prejudiced for many reasons. For software industry, it does not really need real mathematics. They want you to be able to do the programming with just enough background information Writing packages for example, if you are writing statistical package, of course you need some basic information and if you are a little more intelligent you can make things go a little faster, but, I don’t think that needs a very deep knowledge of either mathematics or statistics.

CRH: What can you say about the mathematics education, high level mathematics education?

KBS: Well, as I said, a good teacher is at least, a great teacher is at best. If you could have a great teacher that is a fantastic thing, particularly at school and graduate school level. It does make a difference to young people. I would say I was fortunate to have an almost a great teacher in school. As I have said, mathematics in school was so trivial that I could not get excited about it except geometry. That was the only thing which was interesting for me in school, and this teacher who was a very good geometrist, he really taught me how to look at geometry problems at a proper way. That I still remember even till now.

TSC: But you are not in geometry.

KBS: I do use geometry but of course in a modern language. That brings me back to this. Occasionally we do strange things. This is written by R.L. Karandikar, C. Musili, S. Pattanayak, D. Singh, A. Dey and myself, we joined together to write a book for school kids. We thought that we are doing a great job but maybe we are wrong.

CRH: So this book is for high school?

KBS: Yes, the last two years of high school. Essentially it is set at a little higher standard which hopefully can challenge the students.

CRH: So for mathematics, what do you think about it?

KBS: Well, that is not so high education. That is at school level essentially. There is one chapter in Statistics.

CRH: But including the undergraduate, what do you think?

HHK: I have a small correction. This definition assumed is to be continuous, is not necessary. You define uniformly continuous.

KBS: Correct, but when one emphasize uniform and not uniform, it creates troubles for the students.

HHK: You are right. Students maybe easily confused.

KBS: You see, in fact, in school, I have noticed that continuity and uniform continuity cause the maximum confusion. They are unable to differentiate continuity and uniform continuity.

CRH: But including the undergraduate, what do you think?

KBS: You see, we have a system which is essentially obtained from the British. So you go to school for twelve years. Then you go to what you call college where you spend three years as undergraduate, then two more years for masters as preparation to do Ph.D., then Ph.D. if one continues. I think that the most ending place for an average Indian student is at college.

CRH: In college? Not in high school?

KBS: In college, not in high school. In high school, they are very young and still full of enthusiasm. But in college, they are badly taught and they lost it all.

CRH: So they burnt out in some sense?

KBS: Yes, they are burnt out and finished off. So that’s what I.I.T. does. It teaches mathematics from a point of use. For example, I taught in I.I.T. for a year a long time back. I was in Switzerland for that time and I went back to India for a year and I was teaching in Chennai I.I.T. and I had to struggle. You see, they need exponential functions, they want to solve exponential problem as quickly as possible but they have not done any limits. How to introduce exponential functions when you have not done any limits, that’s a challenging idea for a teacher. So that’s what I mean, the kind of teaching is done, in a haste way. You are taught to not to ask but just take it and go to the next level. So that is the danger, particularly if you want to learn a subject well.

HHK: I have a small correction. This definition assumed it is to be continuous, is not necessary. You define uniformly continuous.

KBS: Correct, but when one emphasize uniform and not uniform, it creates troubles for the students.

HHK: You are right. Students maybe easily confused.

KBS: You see, in fact, in school, I have noticed that continuity and uniform continuity cause the maximum confusion. They are unable to differentiate continuity and uniform continuity. So it is not true that Indians are good mathematicians. It is true that many Indians are doing well in mathematics. Of course there are few have done well in US, for example, there is Manjul Bhargava, a number theorist, who is very well and he is in Princeton. There are also quite a few others. He has researched the Gaussian method in number theory. But he is not a first generation Indian. He is a second generation Indian. Similarly there are others second generations who are doing quite well. You see, India have some very good mathematicians, of course. But it not the heights which matters. It is the average that matters. It is the critical mass that matters. That’s why US is doing so well, because they have a mass that is way above average.

CRH: So you think that in India though you have quite a few good mathematicians but not enough.

KBS: Not enough with average good level. So that you can depend on them to teach well and produce good level of students and there will be a generation of good mathematicians coming up and etc. That is not enough. That is the unfortunate thing. For example, Tata Institute of Fundamental Research is doing very well as well as their area of research is concerned, Algebraic Number Theory is their main interest of concern, and they also concentrate on Representation Theory, these are their strong points. They have some very good researchers. But unfortunately, they do not produce enough students.

CRH: Not enough students, I see. So in comparison to Japan, population wise, Japan is not that big. But what do you think of the level of their critical mass? For example, even in probability, they have a huge number of probabilists in Japan.

KBS: And first class ones. Also when you look at analysis, they have Yoshida along and he has many students. You see, that is the tradition of having students missing from India.

CRH: Yes, in Japan, from Ito, he has a lot of students. Yoshida, he also has a lot of students. They are also very good in algebraic geometry, number theory. So you mean they consecutively have good students in Japan, but you don’t think that is happening in India?

KBS: That is not happening. It is like a branching process, you know. It has to be happening with a critical branching ration so it can grow.

HHK: This is a very nice book. For the book, how long does it take you to do it?

KBS: Well, you need to take time off. Each of us writes a few chapters and put it together. To put it together, all of us took some time off and stayed in a village under the foot of Himalayas. It took us about two weeks to put it together. In a quiet place.

TSC: You know Kolmogorov was a champion of high school education. He wrote many high school textbooks.

KBS: Russian did very well. They wrote a lot of very good textbooks for high school students. In general, I think Russians write good books.

CRH: I think even at elementary schools, they were involved in writing the syllabus of mathematics. I mean Kolmogorov, Lev Pontryagin, those famous Russian mathematicians, they were serious about their mathematics education. How about India? How about the syllabus? Do you have any influence on the syllabus design?

KBS: For syllabus, we have influence, yes. I give you an example, in University of Delhi, which is a premier university of the country, I use to be in their syllabus committee once in a while, I saw once in their master program, so this is after bachelors. They have these called optional papers. You know you have some compulsory papers and some optional papers. I was shocked to find two papers on basis in Banach Spaces. I said that do you know how many people in the world study basis in Banach Spaces? You can almost count them. It is such a super-specialized area of mathematics, to teach it to a master student; I think it is too much. If you want to research, I think he would learn at some point, there is no problem, but to teach as a regular course, I think that is too much. Too much as it is super specialization. I think that super specialization is necessary at some point, but, to learn it is overdone at an early age.

HHK: That can discourage students from studying in mathematics easily.

KBS: Absolutely.

TSC: The teachers teach the things they want to learn, but not for the students.

KBS: I mean it has both pluses and minuses. I do agree that it sometimes does help to learn a subject well if you have to teach and explain it to a class. You sometimes discover new viewpoints which you have not thought earlier, it can happen. But then, there are limits to what you can do, I mean. You cannot go beyond certain limits.

TSC: Yes, dual purposes. You have to serve to both the benefits of teachers and the students.

HHK: Do you think that Indian mathematicians have common sense?

KBS: Well, I don’t really know what you mean.

HHK: Well, the intersection of mathematician and human common sense is very small.

KBS: That is true.

HHK: You see, when I was a university student. Many of my classmates, well, we were just starting mathematics, and they think that mathematicians should have more common sense.

KBS: I understand what you mean.

HHK: They tried very hard to behave differently from other people.

TSC: That’s nonsense. Mathematicians shall try their best to behave like usual people. Not the other way around.

HHK: So one of my classmates had said, in this class, only two persons are not crazy. “I am one and the other one is Kuo.” So he is the first one and I am the second one.

KBS: On that whole thing, mathematics being a little exclusive to common publics, some may find it to their advantage to behave like a crazy person.

HHK: That’s right. I have many Indian friends and I find that they all have common sense and they are all very civilized.

KBS: Also, I feel that in mathematics, one of the greatest powers of mathematicians is the freedom to define things. But unfortunately, I would say that bad mathematicians are one who misuses the power. Like all powers, it has to be used responsibly. So the power of definition also, shall be used with certain sense of responsibility. I give you an example, when we learn basic topology, these various axioms, T1, T2, T3 and T4, spaces, Hausdorff, normal, and etc. Someone can twist the definition with a little bit of new property and a new topology would be defined. But if you ask give me an example of a function which is continuous in this one but not continuous in the other one, it completely falls through. They have not thought by just twisting the definition a little bit, it does not get you a responsible mathematics.

HHK: That’s exactly what I have said. Many mathematicians do not have common sense.

KBS: So this is what you meant. Okay, I agree with you.

HHK: For example, in writing a book, in Chapter 0 and Chapter 1, many definitions you have created which are never used later on.

KBS: That is a terrible waste of time and effort not only of you but the readers.

TSC: One of the Nobel Laureates, a physicist, Chen-Ning Franklin Yang (楊振寧), once said, there are two kinds of mathematical books. The first one is you can only read the first line then you want to quit; the second one is you can only read one page, then you quit.

HHK: I remember when I was in Courant Institute in NYU, one time, Peter Lax was giving a colloquium talk from 3:30pm to 4:30 pm. When it was 4:28pm and he was in a middle of a proof, and then, it was impossible to finish the proof in time, so he turned around and he said something which influence me ever since. He said, most of the colloquium speakers would say another thing at the end of the talk if it was overtime, to his own satisfaction, not audiences’ satisfaction, therefore, I stop. Everyone applauded. One time in a conference where I was the chairman, and this speaker he kept asking me how much time he has left. Can you believe that, someone who gives a lecture without knowing how much time? There is no common sense at all. Eventually, I gave up.

TSC: United States is better than here. Here a seminar talk easily goes over one hour and a half. But the thing is, after one and half hours, everyone gets tired.

KBS: Also in colloquiums, usually in colloquiums there are not just mathematicians but sometimes also non-mathematicians and mathematicians of other fields may also be there, so you have to keep all that in mind before you give a talk.

TSC: The psychological, the concentration spam of most people is about 45 minutes.

KBS: That is already a lot.

TSC: Yes, if you think really hard, 45 minutes is a lot, especially to my age.

HHK: Actually, mathematicians are very good human being. You don’t see mathematicians who are mean, they are all very nice.

KBS: Well …

TSC: For that, I take preservation operator.

KBS: I don’t think mathematicians per se are like that. I mean some scientists or mathematicians can be little difficult human being. That’s possible.

HHK: Yeh, that’s difficult but not mean.

KBS: Some great mathematicians or great scientists can be poor teachers. For example. William Paul Thurston comes to my mind immediately. Thurston is a great mathematician. William Thurston from Cornell University, Fields Medalist, but he is a poor communicator.

CRH: He is in Topology.

TSC: Any idea who is going to get the Fields medal this time?

KBS: Well, I have heard some rumors. That it is possible to be a Vietnamese mathematician. He is making a big wave. I don’t know him and I don’t know his work, but he is currently in France. He is working in Langlands conjecture, Langlands program.

CRH: Are you in the committee?

KBS: No, no. The committee is IMU. The committee decides. IMU has a Fields medal committee and that decides. It has already been decided. What is happening is now, previously, there will be lectures. Fields medalists would not give the lecture but some other mathematicians will present the work of Fields medalists. But this time, apparently, the Fields medalists themselves will be asked to lecture. So the people who are compiling the lectures, they probably have already known.

CRH: You mean one hour lecture?

KBS: Special lectures. Varadhan is giving one lecture. It’s called Abel lecture. So one of them is giving one lecture and I think Parthasarathy is chairing the session.

CRH: There is a Chinese who is going to give a one hour lecture, right? Peng.

KBS: Who?

CRH: Peng Shige (彭實戈). He did work on backward SDEs (Stochastic Differential Equations) and FBSDE (Forward Backward Stochastic Differential Equations) But I think he is well over 40 years old, I think.

KBS: No, that has nothing to do with it. Only Fields medal has an age limit.

CRH: So Varadhan got Abel Prize. It was in 2007. Did that cause any impact in India?

KBS: Not in the media, but, academics, of course he made a big wave.

CRH: Did that have any impacts on young students going into mathematics?

KBS: I won’t say that there has been much impacts on that. Well, I mean.

CRH: Well, I mean one million US dollars for the prize.

KBS: That, of course, is a big thing. But these guys don’t even know the prize is one million dollars. They know Abel Prize, but they did not know the prize money would be so much.

TSC: So what do you regard these things, for example, Varadhan got Abel prize, but actually he is an American. Americans would think that he represents America in this prize.

KBS: Well, actually he has been in the States since 60s something and he was more of less in one place.

HHK: But he was educated in India.

KBS: He got his Ph.D. in India in I.S.I. but immediately after that he left.

CRH: Yeh, he was with Courant Institute for a long long time.

KBS: Actually, he had an Indian passport until recently. Only recently he changed his passport to US passport.

TSC: Kallianpur is also like that. He didn’t have an American passport.

KBS: Right, recently he also changed.

TSC: Would you still regard him as one of Indian Mathematicians?

KBS: Officially, maybe we cannot, strictly speaking, because he spent most of his working life in US. But he comes very often. He does come to India quite often. In fact, every year he does come at least once if not twice. So in that sense, he has kept in touch quite a bit. But on the other hand, things like taking a student somewhat that has not happened much. To have an Indian student or associating with a student, that does not happen much. But he does come quite often for conferences and otherwise.

TSC: If you take the world as a global village, everyone is part of this universe. Then it’s okay. Everyone is part of this universe. However, if you consider this world as countries, then that will become a problem.

KBS: Well, as you said, if you consider the world as a global village, then it has no issues. But unfortunately or unavoidably to some extent human nature being what it is, people do look at it. Otherwise, why shall I talk about first generation Indians or second generation Czechs. It does not make no difference. People are people. We are all human beings and we do make those alienations. Even though maybe otherwise we do not need to make those alienations.

TSC: In Taiwan, even this person is remotely related to Taiwan, we take him as one of us. If he got prize or won some competition we would be very happy.

KBS: National pride, in other words. But it’s part of being human being to have a national pride.

TSC: But in China, this is not the case.

KBS: Is it?

TSC: China is very nationalism in this sense. We take anyone who is related to Taiwan as one of us. China makes is clear, if you want to have certain position in China, you have to stay in China, do your job in China. If you get your honor or overseas then it’s very hard for you.

KBS: But I’ve heard that mainland Chinese in US are asked to go back and pay them with American salaries.

TSC: Yes, but they do not have any power.

KBS: Okay, that’s different.

CRH: But the salary is ridiculously high.

TSC: Salary is high. So they pay you to work for them in some sense, but you are not part of it.

KBS: What do you think of this? I have been noticing a little bit that the rate of plagiarism has gone up tremendously, and in mainland China it is happening a lot, particularly in sciences like chemistry and biology. Now it is showing in India also, which is, of course, a very dangerous trend, plagiarism in sciences or fabricating data, particularly for important experiments. It happened in US. I remembered that when I went to US as a graduate student, I remembered reading an article on newspaper talking about even in a famous institute like Sloan-Kettering Institute one young post-graduate or Post-Doctoral fellow, he was under so much pressure, it was at the time when heart transplant was just beginning to happen a lot, they were under so much pressure, they were studying and wanted to invent some kind of handy injection drug, because one of the major problems for heart transplanting at the time was the body rejection reaction. So they wanted to discover a new drug to reduce this rejection rate. So this guy did some experiment on mice and wanted to color the skin and it is different color so they wanted to see if this new skin is being rejected or accepted. He was caught and he admitted that he painted the mice.

HHK: I think this happened about 30 years ago.

KBS: Exactly, that’s what I mean. It was a big story. There is pressure on young people which you can tell.

CRH: Even for famous people I mean, they do fudge the data.

KBS: Yes, it is happening.

CRH: The recent ones, like a couple of years ago in Korea, the cloning experiment. I shall say that in biological science, pressure is the highest. In 1987, when Joel E. Cohen (Rockefeller University) visited us. He was a probabilist doing random matrices. He was quite good in several areas, not only in mathematics, he was good in public health, in population science and things like that. I remember that he visited us and we were walking by the Institute of Biomedical Science and he said that in US the problem of that particular science was that the pressure was so high, between different laboratories, most of them did not collaborate scientifically and even tried to hide their results from each other, and a lab would even try to steal results from other labs. So he said that was not science. That was not good for the development of science. He was a little sad about things like that.

KBS: Well, it is happening more and more. I remember when I was a graduate student there was a famous experiment done in US in physics, and Boson, who knew he would get Nobel Prize if it comes true. He announced his result to Time magazine before he even actually wrote up his paper for publication in appropriate journal. So the pressure of this has also become a little bit negative. There is too much pressure and that tells on the peer compliments of the areas of learning.

CRH: Let me switch to another topic. If you remember in this afternoon there was a talk by Professor Hsieh of National Taiwan University. In the beginning he was talking about his Ph. D. student, his current Ph.D. student, co-authors of the paper, and he was complaining about SCI. I was surprised that even in National Taiwan University the students have to publish in SCI journals before getting Ph.D.s . I was surprised that in Mathematics. What do you think about this?

KBS: Well, I think in US there is no real pressure about publication before obtaining Ph.D.. In US, by in large, they give you a stamp. You are ready. You are being trained. You are ready to go and be a professional mathematician. On your own and do whatever you can do.

HHK: There is some discussion between the Japan and US system. In Japanese system, before Ph.D., the Japanese standard is much higher than US standard. But after Ph.D. the US standard is much higher.

KBS: This is the miracle of US. We cannot understand what is the real reason of this happening, the reversal of what you have just talked about.

HHK: In Japan, before Ph.D. they perform very well, but after it, they get tired.

CRH: Burnt out.

KBS: Same thing is happening in Norway or Norwegian system. Average Ph.D. age would be between 35 and 38, but they have published quite a few papers and make a book out of these papers and submit that as their Ph. D. thesis. People are already 35 plus or 38.

CRH: That’s too old.

KBS: So after Ph.D. they have lost all their incentives. That’s why I think America has something different and that has something to do with it. I never understood really. Before Ph.D. there was no pressure on publication, but after Ph.D. there is pressure.

CRH: Even that, for example, in India, even in this country, applying for grants, they look at SCI.

KBS: That is a different thing.

CRH: I don’t know in India if that’s the case. What do you think about SCI?

KBS: Let me go back to the previous discussion. Namely, in I.S.I., I can say without a doubt that the average level of I.S.I. Ph.D. thesis in mathematics and statistics is higher than that of American universities. But Ph.D. out of American average universities has higher probability of doing better in later life than a similar person from I.S.I. I have observed many cases. This is a mystery to me and I want to implement this in India. But people won’t accept that kind of pressure after Ph.D. They just won’t accept. I was not successful in introducing that kind of system. I had a theory later on. The salary difference is considerable. What you earn as a post-grad. Unlike in Europe, you are essentially almost a government employee of the government. So every year you have a little increment in your salary until you got your Ph.D. and after you become a post doc., you get another increment. Nothing much.

HHK: Here’s my theory, the reason is back in Japan or in India, in the early days, they are forced to do it, but in the US they do it because they like it. In the recent years, in US, undergraduate students they are very good students, truly good undergraduate students in the last ten years and I send them to universities like Princeton to study. They want to do mathematics because they like it.

KBS: That is the proper way of doing mathematics. Mathematics is not like other sciences. I believe there is a fundamental difference. I would say one shall not do it unless you have the love for it. Not like, I would use the word love.

HHK: That’s right. Oh by the way, what’s your perception for mathematics community in Taiwan?

KBS: I don’t know many of them, I must admit. I only know a few. Thanks to Prof. Hwang, I came to know a bit. In Academia Sinica, whom I have met, I don’t even remember many names, in statistics, in physics. I visited all the departments one by one. I visited the Institute of Statistics, Institute of Physics and Institute of Economics, and also the President’s office. Then I also visited your National Science Council and talking to someone there.

CRH: You also visited National Taiwan University.

KBS: At that time you told me already that the Institute of Mathematics may move and now it does happen. I think I don’t know most of them, but I have good feelings about people I have met and I have enjoyed their company. Though both trips my times here were very short but it has been a very happy and positive experience.

HHK: Shall there be more exchanges?

KBS: I work toward it, but between two institutions it’s not as easy. He is a design theorist and then he has moved over the years . He has a group doing . But he is not a mathematician in the strict sense of words. When I was there, there was a project with to design for Defense. I remember initiating that project. There is also IndoCrypt in 2010.

TSC: So when was this?

KBS: The first , but you see what happens, the leading countries, but this is a sensitive area of research. I was not involved.

KBS: I like being adventurous, to move into slightly different areas and explore, because I sometimes feel that there are beautiful tracks connecting different areas. For example, astronautics, and geometry, many people do astronautics and geometry. I still remember Laurent Schwartz, looked at Brownian motion on manifolds. His way of looking at it was rather spectacular. So I find it very intriguing and amusing. Personally I often do these excursions into other areas. Sometimes I just bang my head and don’t get very far, but sometimes interesting things do come up. So I like this sense of adventure in mathematics as well.

CRH: You are right in that sense. You know my teacher, Ulf Grenander, so actually I am atypical. I am not a typical mathematician in Taiwan. Because my mathematical problem came from applications, so usually I invented my own mathematics problems from square one. For example, the last twenty years I looked at Monte Carlo methods, but the problem came from a paper in 1988 in Image analysis. There were some discussions about which algorithm was better and things like that. So that’s the influence from Grenander. So I look at Stock price, but looking at real data. I ask my assistant to gather information from real data and store it in a certain way so I can look at the behavior of the real data and try to formulate problems from that point of view without assuming any mathematical models first. From there, I try to find some invariance. But things like that will take a lot of time, even to formulate the problem.

KBS: That depends on the risk. Because once you complete the thing and successful, you gain satisfactions which is much higher than the normal route.

CRH: I am fortunate in that sense as I am not that young and I can afford to do things like this, but for current young people, they need to have publications under time pressure.

KBS: The fine line is there. The depth of adventure one wants to do and the amount of time to spend. It depends on to some extent, how bright the chap is. If the chap is extremely bright, he/she can take some risks. For example, I can tell you that I have an experience. I have started in India introducing new areas of mathematics interrelated areas. The so-called non-commutative geometry was introduced by me in India, which exists in Europe ten years before that. I had two students at that time. I was lucky to have two extraordinary bright students. They with me did some rather positive work. So much so, Alain Connes, the great man was sufficiently impressed and came to visit us and stayed with us for nearly three weeks. Now I think these two students are on their own and doing independent research. They have grown up. My aim and still remains an unfulfilled aim namely to marry non-commutative geometry with non-commutative probability. I have looked at some specific models only, there is no general theory. For the matter there is no general theory for manifolds in non-communitive language. This is again a kind of experimental mathematics, which is happening and very exciting, because you really don’t know as the definition is not given to you. You are in the process of inventing that appropriate definition and that is very exciting, I think.

CRH: Thank you. Thank you very much.

- Tzuu-Shuh Chiang and Chii-Ruey Hwang are faculty members at the Institute of Mathematics, Academia Sinica.
- Hui-Hsiung Kuo is a faculty member at the Louisiana State University.