Prof. Mourad E. H. Ismail was born in 1944 and grew up in Cairo, Egypt. He received his bachelor's degree from Cairo University in 1964 and earned his PhD at the University of Alberta in 1974. Since 2003, he has worked for the University of Central Florida, and his research fields include mathematical physics, orthogonal polynomials and special functions. The book, "Classical and Quantum Orthogonal Polynomials in One Variable", published by Cambridge University Press in 2005, is one of the most important publications of him.

TPL: First, thank you for coming. You have visited Taiwan for a number of times, right?

MI: Yes, this is my fourth visit so far. It’s a very interesting place here.

TPL: Perhaps I should not bring this up, but you are the first one with origins from the Arabian world that we interview. But of course you are really an American mathematician professionally. But because of that, I would like to ask this question first. How was your growing up in Egypt?

MI: Cairo was built around the year 1000 (founded in A.D. 969). I grew up in a very old part of Cairo which goes back at least five or six hundred years. So we have all these historical places which constantly remind you of the past. You always wish that the present or the future is as glorious as the past was. So there was this mixture of pride based on long history but at the same time not feeling very good because the present is not very pleasant. When I was in high school the Russian satellite Sputnik 1 came out. Suddenly there was all this excitement about science. So I went to Cairo University. At the time, there were several interesting mathematicians there. The economic situation was bad. So one of my professors went to Stony Brook of the U.S.A.. He is Egyptian for several generations. He was Coptic1, not Muslim2, like the majority. He was one of the best people I have ever met. He was a very kind and serious mathematician.

TPL: What’s his name?

MI: His name was Raouf Doss3. His son Hani Doss4 is a statistician. He was a student of Persi Diaconis5 . He went to Stanford. When I told Diaconis that my teacher was such and such, he said ‘Oh yeh, I had his son.’

TPL: Small world.

MI: It’s a small world. Among mathematicians, there was another person working for the United Nations. He left Cairo University for United Nations. At the time in 1960s it was a very interesting place. Then the economic situation was mostly destroyed by the military adventurism. In the last ten years, things are coming back. The libraries are improving. The education is improving at universities. So I hope you would see more.

TPL: So I take it that you come from a good family.

MI: Sort of. One nice thing about the area I grew up in was we were 300 meters from the National Library. When I was ten or twelve, I would go to the National Library in the summer. Not that I wanted to read anything but it was the only place that had the ceiling fans in the summer time. When you go there you have to do something. So I started actually reading a lot of stuff. It was a very interesting experience. The way it works is different from the way the libraries work here. The Egyptian library you walk in, you are at the reference section. If you want any book, you have to ask a librarian to bring the book to you from the storage. That usually takes half an hour to an hour. So you have to do something in this hour. So I read encyclopedias. You know these kinds of books that were in the reference section. That taught me a very good habit of concentrating on reading and so on. It was just basically trying to get away from the hot apartment in the summer time.

TPL: Of course there is a beach you can go to but you choose libraries.

MI: No, Cairo did not have beaches in those days. We are in the middle of a desert. The Nile was nice but you could not swim in it. The current was too fast. It’s huge.

TPL: What is your father’s profession?

MI: He works for some movie companies. He was not in the movie business. He was in the distribution of some of the movies and news reels. He made sure each movie went to the right theater and back by certain time.

TPL: You said that the library helped you with concentration, but it seems to me it’s the other way round. You can concentrate and that is the reason you end up in a library. In any case, what made you going into mathematics?

MI: That was Sputnik 1. At that time in the last 1950s and early 1960s both the U.S. and Soviet Union were having a lot of propaganda about space age, satellite launching, future in science. So I used to go to the American Cultural Center in Cairo and the Soviet Cultural Center. They had many educational movies and books about science, space, exploration, astronomy and so on. So this was really a very fascinating time.

TPL: At that time, Egypt had a good relation with both Soviet Union and the United States?

MI: With the Soviet Union mostly, with the U. S. it went down. At that time the Soviet Union was treating third world countries much better than United States did. Also when I was in college, there were some Russian books translated into English. The English version was extremely expensive for me. But the Soviet Union would translate them to French and sell them to the third world countries. So if you know French you can get these cheap books for a dollar or two, while the English translation would cost forty dollars. So I had all these French books as an undergraduate. These were among the best mathematics books. In the 1950s and 1960s Soviet Union was really something.

DCW: Mine are just routine questions for MathMedia. Please give a definition of “Special Functions”.

MI: It’s very hard to do that. It’s some specific functions that keep reappearing in applications. Also some of them were not actually discovered by mathematicians, like Bessel functions. Bessel6 was an astronomer. But at least his functions were studied by Euler beforehand. They became to be known as Bessel functions because he really developed a lot of their properties. The functions appear in a theoretical set up or an applied set up. If there is more need for more details then people study that. So there is really no definition because anyone can give you a function but nobody else would look at it. So there is really no such definition. It’s a collection of functions which appear again and again in many applications either in theory or in actual applied problems. Because many of these new special functions appear in groups of representations, for example the representations of quantum group and so on use the functions developed in 1980s. It’s things that has some common use. People study them.

DCW: Can you talk about their relations to difference equations, discrete Painlevé equations, q-difference equations, and discrete integrable equations? Because you introduce these functions through orthogonal polynomials.

MI: I claim that if you have something very natural it should touch upon many other objects. So the theory of orthogonal polynomials has been developed from many points of views. But the connection with the concept of integrability and discrete Painlevé equations and so on, was discovered recently since 1970s. This subject is about three hundred years old, thirty years is not much. It’s a growing area which I find very interesting. I personally keep an eye on some other areas where this subject is used. You will be surprised to see how many things touch on orthogonal polynomials and special functions. I regularly go through the science citation index. I look at the citations of people like Askey7 and Andrews8 , in addition to several people from my generation. I see where the work is cited. It’s cited in many areas of science which you would not think of. Physics may be close enough to mathematics, but there are some citations in electrical engineering of Askey’s work. So it is important for someone who is in the subject to know where the works is used and if possible to learn a little bit about where it’s used and maybe interact with people in that area.

TPL: Are certain special functions used in certain areas because this allows them to represent whatever they are doing in a simpler, easier to operate way?

MI: When I was a post-doctorate fellow, there was another guy, Thomas Nagylaki9 , at the Mathematics Research Centre of Wisconsin. He was in mathematical genetics. He had this certain boundary value problem where there is a Laplacian involved in n variables. His genetic problem reduces to knowing that the eigenvalues of the Laplacian are monotone in the dimension. When he tried to do this, he took one dimension, two dimensions, and three dimensions and tried to graph the eigenvalues. They looked quite the way he wanted, but proving it is actually very hard. The reason it’s hard is that dimension is a discrete variable. If you formulate this and write down the details, and treat the dimensions as a continuous variable because it comes as the order of the Bessel function, it became reasonably easy. So we did solve it jointly with Martin Muldoon10 . The reason was a match of talking to someone in a different area. It was also not taking his problem as he stated but change it a little bit, which was going from discrete to continuous. Then it was easy. Actually if you look at the proof, the proof does not work if you have a discrete variable. It is important that it is continuous because you are taking graphs and moving them a little bit. So this was one of the nice things that I did in retrospect and mathematically it was a very nice problem. It’s also important when you see an applied problem, to have your own input into it. You don’t have to take it to a high level as long as it is compatible with the problem. I actually spoke to Thomas Nagylaki recently. He has another problem.

JHL: Where was the place you did your post-doctorate?

MI: I was in Mathematics Research Centre of Wisconsin for one year and at the University of Toronto for one year.

JHL: I just repeat what you have said. So you have revised a problem that is compatible with an application.

MI: I treated the variable as a continuous variable. It is monotone and increasing.

JHL: Let me come back to a question when you are in Cairo. Where did you learn French?

MI: In high school, I took three years of French. I had six years of English and three years of French.

JHL: So after that you can read mathematics books in French.

MI: Actually when I was in high school I could almost speak French. It was a very good course. I lost it now. I cannot speak French anymore. But it was a very good high school course. That was a big advantage to have access to the French books. By the way, I found that from some friends in other countries that these same books were available, for example in Argentina. They had again the Soviet books were available and they were reading them.

TPL: The Soviet Union had systematically published those books.

MI: They were selling the English translations to American publishers or English publishers. But then they would publish translations in Spanish or French and some other languages. They had the lot in Arabic too. But these were mostly like Lenin11 ’s collective works were available in Arabic if you want them. I actually had one volume but that was about it.

DCW: Please talk about methods developed for special functions.

MI: Well, the beauty of it is that there is no systematic technique because it touches on other things. For example there is this classical analysis style which uses in complex variables and so on; but there is also another approach which uses group representations and more algebraic methods. It depends on what you want to do. If you are looking at Addition Theorem of course the group theory way is more convenient. Still you can find analytic proof of these but it’s not as natural as it comes in group theory. If you are doing combinatorics there are connections with orthogonal polynomials and so on. There are all kinds of counting techniques. There is actually a combinatorial theory of orthogonal polynomials developed by Viennot12 from France. He has lecture notes. It’s in French but it’s readable. It gives you interpretations for the coefficients in terms of what is called Moskin paths. So the techniques vary depending on what you want to do with it.

TPL: How do you view the role of special functions, it touches upon many other subjects as you have said, perhaps more so than a lot of other topics in mathematics.

MI: I can say something that is close to your question. In the 1970s, there was an attempt to start a journal on special functions. Askey who was at the time the young leader in the subject was against it completely. The reason he was against it is that if you publish in such a journal people in other areas would not see your papers. Also, you would not read as many as of the other journals as you would, to look for papers in your own subject and that is bad for the subject. Because it is the interconnection with other fields that you want people in other fields of mathematics to see your work and you also want to see their work. So it’s important not to have a journal in special functions. That was his philosophy. Obviously it was not shared by everybody. But he influenced my generation so much that we all sort to follow what he thought, did, and so on, including many personal opinions. I personally agree with his point.

TPL: I remember when Louise de Branges13 proved the Bieberbach conjecture he was using special functions. Of course special functions have existed for centuries, but at that time, a lot people were beginning to say that okay special functions. He is related to some people associated with our Institute. So when he found that I was told that people did not believe in him, because he claimed other things. After that he wanted to solve Riemann hypothesis. He had this credibility problem. So Ky Fan14 , the former Director here, told him that “now you need to go to St. Petersburg. Those are the people who are very much into the complex analysis and so on. If they say yes then we would believe you.”’ So he went there. I know this is a bad question because special functions are everywhere. Do you mind to say a few words about this particular incident Bieherbach conjecture solution?

MI: This was an inequality by Askey and Gasper15 . Askey has been asking Gasper to prove several inequalities involving certain hypergeometric functions. This inequality originally came from harmonic analysis. Many of the other inequalities also came from harmonic analysis. Askey thought this was the right thing to do. Why he thought this was the right thing to do? This was his gut feeling and has always been right. So he knew this work was good and would be useful. But he did not know which subject or who would use it eventually. So what happened was de Branges went to Gautschi16 to verify his conjecture numerically. Walter Gautschi verified it and numerically checked. So he called Askey and he said ‘do you know this?’ So the way it was originally formulated was not exactly the way Askey formulated. Anyway, Askey thought about it and called back ‘yeh, we did. This is known.’ Then Askey said ‘I don’t think that de Branges’s proof is right.’ Gautschi said ‘Why?’ Askey said ‘because this is a real variable inequality. How can you use it to prove a complex variable result? It does not make any sense.’ Of course eventually it turned out that de Branges was right and it was exactly the inequality that he wanted. Finally Askeyss prediction that this would be good for something came true in a big way.

TPL: Beautiful story.

MI: This credibility problem was that Askey thought that he was maybe asking the right question about this inequality, but you could not prove the Bieherbach conjecture by using it.

TPL: So Askey has played a role in both way very nicely.

MI: Many people from my generation worked on problems proposed by Askey for many years either jointly or separately. In his lectures he would formulate conjectures. In my own case, I remember he lectured on an electrostatic equilibrium problem which was solved by Stieltjes17 in 1884. He always said ‘this is a real gem. Somebody should really look at this and discriminants.’ I looked at it. Eventually it turned out to be very rich. So I figured out exactly what the general set up for it and how it relates to discriminants. There are some discrete and q-discriminants. All kinds of things that were related came out of this problem. So in many cases he would either tell you precisely to look at a specific problem or a general area. In both cases, his intuition was right.

TPL: He is still around, right?

MI: He is around eighty but he is doing education mostly now.

TPL: I heard him talking about that in Stanford.

MI: He claims that he has two proofs that he is getting senile. One of them is that he is doing education. The second reason is that his efforts will make a difference.

JHL: You mentioned that you had your undergraduate study in Cairo, how did you move to the States from Cairo?

MI: The Egyptian system is different. When you graduate you become like an instructor and you are expected to get a Ph. D. So I applied for Ph. D. studies abroad. The instructor is like a permanent job. I got an offer of research assistantship for one year by Professor Al-Salam and that’s how he became by advisor. So I went. He told me that it was fine if I wanted to work in special functions and if I wanted to work with somebody else he was fine too. But I liked the subject so he became my advisor. I taught for three and half years in Cairo after graduating. The reason it was delayed was because of the War between Egypt and Israel. They would not allow people to leave. It was a very big mess. It was in 1967. I graduated in 1964. People who graduated before me, the government sent them on fellowships to study abroad and so on. Then the economy became really bad and they had no money but they would not tell us. They kept on saying that we would send you out and so on. Then I applied on my own and got an assistantship. I could not leave because of the war. They would not grant exit visa.

JHL: At that time, was the military service compulsory in Egypt?

MI: It was compulsory but not absolutely compulsory. There was another war in 1956 when Israel together with Britain and France attacked Egypt. So after that war the Egyptian government decided to give high school student military training. So I had three years of military training in school. The assumption is if you graduate then you already have learnt most of the basic stuffs they do at military service, like using machine guns, taking the guns apart, cleaning the guns, load the guns, unload the guns and things like that. It was for civil defense. Of course I cannot drive a tank, but at civil defense you don’t need to drive tanks. At that time, the high schools had their own guns. There was a real threat of invasion. So the result of that was when you graduated from university, they only draft five hundred university graduates. In my year, there was something like 20,000 university graduates. They drafted only 500 by lottery. They don’t this anymore. Now it is compulsory for everyone.

JHL: Were you already working in special functions when you did the undergraduate course?

MI: I took a course in special functions in my fourth year and I found it very interesting. So I liked the subject. I was working on it on my own and got offered with this assistantship and I took it and met Al-Salam in Canada. I could not go to the States. At that time, the relation between Egypt and the States was really bad. At that time, if you want to study abroad, you can only go to either Canada or the Soviet Union. For Soviet Union, you have to get support from the government and not on your own. If you get an assistantship from the U.S. you will not get an exit visa. So I had to go to Canada before moving to the U.S. .

TPL: The special function is special. It seems to me that for people to work on special functions need to have special talents. Other fields are more systematic. The intuition you have in special functions seems to be a unique talent.

MI: It also comes with the experience.

JHL: You have a book “Classical and quantum orthogonal polynomials in one variable18 ” published in 2005 and revised in 2009. There is a book review published by the London Mathematical Society about it. Have you ever read it?

MI: I didn’t read this one, but I know the writer Erik Koelink19 .

JHL: He mentioned that you did not write all chapters.

MI: There are two chapters written by Walter Van Assche20 .

JHL: He also mentioned that other books are more eccentric. Some books are top down. He mentioned that your book is bottom up.

MI: This is actually an important issue. There is a book21 by Gasper and Rahman22 . If you read it, it starts right away with trying to develop the most general thing. So you have so many parameters and you get lost very quickly. It is difficult to do the most general thing and then take all kinds of the special cases. So the approach I used was to actually start from something with no parameter which is the q-Hermite polynomials. By doing a systematic approach we build the Askey-Wilson theory which has four parameters by putting in these parameters in a very natural way to generate the functions at the next level. So it is very straight forward when you think about it. So this approach I developed with Christian Berg23 . Then we also use some ideas by Andrews and Askey. You have to put in some details. Some of the details are borrowed ideas from others. I don’t think in terms of the most general case first. I try to understand the simplest case. Then if I understand that I should be able to pull it up. That’s a philosophical point of view. I do say this in the book but sometimes it gets lost.

DCW: Please talk about the universality of asymptotic for orthogonal polynomials, probability distribution of Random matrices and its impact in mathematics or physics.

MI: The random matrix series now is all over the place. The universality principle just says that there is a specific model which is in the random matrices, the Gaussian, or Gaussian Unitary Ensemble. In terms of orthogonal polynomials, this is the Hermite polynomial. What universality says is that if you take any other model for the study of the distribution of the eigenvalues of random Hermitian matrices, then after some scaling, you would get the same distribution that you would get from the Gaussian. This is easy to state, but proving it is another story. So these questions were around in the orthogonal polynomials, they were just not called this way. These are mainly the asymptotic of the polynomials, their zeros, and so on in terms of special functions. I think the theory of random matrices use a lot of orthogonal polynomial theories. It has also generated interesting questions for people in orthogonal polynomials. I personally worked on one problem which was Fredholm determinants, because the use of Fredholm determinants was in a very significant way. There is a certain Fredholm determinant which represent the probability that you have k eigenvalues of a random Hermitian matrix in an interval (A, B). So this quantity has to be non-negative. But if you write it down, it is not clearly non-negative. We extended this and generalized it and did it without probability. So this is a small thing I personally got out of the random matrices. But they bring in all these issues of different kernels, like Bessel kernels and so on. This is very interesting in terms of harmonic analysis and so on. So we are looking at questions that come out of matrix theory.

TPL: Before Derchyi’s serious questions, let me change the topic. As we know, the Arabs run supreme say over 1200 years ago. They were particularly strong in mathematics. You must have thought about this and learn about these things. Why was it, cultural background or whatever for that to happen? It was quite a spectacular contribution to the world of science.

MI: I would like to explain something about the contribution of Arabs to science. This is where I disagree with many of the religious groups. They claim this happens under Muslim rule. I claim it did not. If you look at which areas are the areas that were scientifically strong under the Muslim domination, you will see what are today Iraq, Iran, Egypt, and the area of Syria, Lebanon, Palestine, Israel, as well as in some of the Asian Turkish republics. So this is where there was scientific tradition before in the Arabs went in. The scientific tradition continued. Arabia never had serious science. This is where Islam came. So what happened was the political stability. When you have political stability and you have some tradition, very good things happen. It was political stability. It was tolerance and justice. It was liberal issues. It was not oppression. It was not strict enforcement of any dogma, or any religion. In fact when the translations from written Latin and so on started, this was during the Abbasid24 rule. The caliph at the time was Harun al-Rashid25. In his court, he was drinking every night. That’s sort of the peak in that time. The poet of the court was gay. He was known to be gay. He wasn’t bragging about it but it was known. During the time when there is very strict observation of the religion, a guy like that would be killed, even if he does not admit it. I am sure other Christian countries were like that too. I am not singling out any particular religion. It’s this attitude of let live, build, and don’t obstruct. It was the political stability, the encouragement of science, and so on. To someone who is not from Arabic background, you may not know that the literature of that era was also very rich, not just the science. Again, all these things were not in Arabia where the religion started. They were in these other places with scientific traditions.

TPL: Which period in history?

MI: It started around A.D. 800 and went on for about four hundred years. But after two hundred years it became very weak and other places started to take over. But they continued this tradition of having science and so on. Also in Spain, there was this tradition again. It was really the political stability, the tolerance, and the freedom of speech essentially. Also, one of the interesting issues was questions like the Earth flat or round. To them that was not a problem. They knew it was round but not flat. They also had many things like how old is the Earth, because according to the Biblical history it’s like six thousand or seven thousand years old. They knew that was not true. The Quran does not say how old the Earth is. So for the Muslim religion there was no conflict. But for Christian and Jewish religions the Bible says it’s so many years. If you disagree, you could not help it. For the Muslim religion it was not ever stated. So you can think about this freely.

TPL: What do you think the reason that religion can become very dogmatic?

MI: I don’t know. It’s very unfortunate because no religion really wants you to do bad things. Somehow when a particular religious group gets in power it causes disasters.

TPL: So it’s a political attitude perhaps. There are several instances in Chinese history, Korean history or other places where Buddhism became too much. There would be a time for the Emperor to begin destroying the Buddhism monastery, temples, and so on. I suspect that was also because the religion becomes too dogmatic or politically too powerful. So this is not unique to any particular religion. Every religion can become dogmatic. When Buddhism first came up, Hinduism was very dogmatic. There is a class structure and Buddhism means to break that. I shall not get into this. We are talking about special functions. This is not special. This is universal. Religion can become dogmatic.

DCW: Please talk about your most favorite work or the most challenged work you have achieved.

TPL: What is the most excited moment in your career?

MI: Actually the most exciting work was when I started working with Askey, although it is difficult to point out exactly which one. When I went to University of Wisconsin and started working with him. You know when you talk to a child; you can tell by the look at the child’s face whether he really likes your saying. Askey has this child-looking face where you could really see if what you are saying is something he really likes or he does not like. This is a very nice thing. You can tell whether you are doing something good or not. There were times when we sat to discuss something and something really interesting comes out. I remember once we were doing some combinatorial things. When we started the discussion neither of us knew how it would end. Then we eventually solved the problem. It was just really interesting. Some of my best work is joined with Dennis Stanton26 . He is at University of Minnesota. Some of the work I have with him is probably some of my best work. It has been very enjoyable. I also believe my work on the moment problem and continued fractions is important. The work on discriminants and electrostatic is also deep.

TPL: In Florida, a number of people, physicists, is Dirac in your university?

MI: No, that is in University of Florida. After Dirac27 , Ulam28 took the position. After S. Ulam, John G. Thompson29 went. They also had a part-time appointment for George Andrews. Thompson is there now as a professor emeritus.

TPL: Someone in information theory and control, Shannon, was he in Florida?

MI: No, there is someone called Kalman30 developed something called a Kalman filter. He is in Florida.

TPL: Florida has the same weather like Taiwan. Summer is humid and hot. You have been in Hong Kong City University for two years, how was it?

MI: It was extremely enjoyable. I really like Hong Kong. It’s a very interesting place. City University started as a poly-technique has the reputation of students being extremely weak and so on. The courses I taught were mostly advanced or graduated level, but I taught one differential equations course, freshman course. It was very good. The students were not uniformly good but the class had about maybe seventy students. There were ten of them who were very good, excellent students. They were from mainland China mostly. I think they wanted to go to a university where the language is English. Then they wanted to go to graduate school abroad. So this is putting a lot of pressure on the universities in Hong Kong because there are so many students from mainland China. So they have quotas. When you have quotas it becomes more desirable. Some of the locals were also good; don’t get me wrong. But the ones from mainland China were uniformly excellent. There was obvious a huge competition. These were just the top of the crops. So it was very enjoyable to work with these kids. There was also another guy who had a class for the incoming students from mainland to improve their English. It was like a summer camp. He was teaching them some of the things they have seen, and some of the things they have not seen but in English. He says that it was the most interesting course he has ever taught, because these kids were really good.

LCC: Actually, there is one thing I am interested in asking is that a conference which I have mentioned to you, to host a conference in three parts between Hong Kong, Mainland, and Taiwan in special functions.

MI: That will be very nice to do.

JHL: Are you talking about future things?

MI: China has quite a few people in special functions. One of the most visible people is Bill Chen31 and about 10 professors at Nankai University. He is in combinatorics. He dabbles in special functions. When I first met him, he was still a student working with G. C. Rota32 . They had a conference for Rota. There is a guy called Feng Qi33 . He works on inequalities for special functions. There are at least eight or nine very active mathematicians doing inequalities.

LCC: Actually the idea was mentioned to me by Xing-biao Hu34 of the Chinese Academy of Sciences.

JHL: The person she mentions is working at the Institute of Computational Mathematics, Chinese Academy of Sciences. Actually we already had a bilateral conference last year. But in Taiwan, we don’t have so many people in this field, even in integrable system.

MI: There is a guy called Liu in Shanghai. I don’t know if he is free to organize such things. He is very good and he is also calm. He can run things. He is not every excitable. Shanghai has several people working on special functions and their asymptotic.

JHL: In Taiwan, we already have some connection. For example, Li-Chen has connection with your other former student Ruiming Zhang35 . He is in Xian working in the Northwest A&F University. He almost visited us this year (2012) but failed because he was engaged by his teaching.

MI: He is excellent. He is very original. He is really exceptional. We had a conference a year and few months ago. This is when they told me about the integrable system conference in China but it conflicted with an activity in Copenhagen. It was the same time, same week. Who was running that integrable system conference?

LCC: Xing-biao Hu.

MI: I think Hu came to Hong Kong for the special function’s conference in May 2011. It was the end of May. We had about eight people from mainland China.

JHL: Was that in City University?

MI: Yes, in City University.

DCW: What is the most important future research direction for special function?

MI: I think for this subject it is really important to keep the connection with other fields. I really think that is where the gold is. Because there are problems that you would not think of which comes from other areas. It’s important for people to travel and have contacts with other areas. It’s mutual benefits of both, I think.

TPL: Perhaps we shall end here and chat the next time. Thank you.

MI: Thank you very much.

• Tai-Ping Liu, Der-chyi Wu and Jyh-Hao Lee are faculty members at the Institute of Mathematics, Academia Sinica.
• Li-Chen Chen is a faculty member at the Soochow University.