Mariano Giaquinta is a mathematician trained in the deep-rooted tradition of the calculus of variations in Italy. Following traditions of Dini, Fubini, Tonelli, Caccioppoli, and De Giorgi, he has made important contributions to the modern theories of calculus of variations, and the existence and relularity of elliptic partial differential equations. He had held professorship at University of Florence, University of Pisa before he was appointed as a professor at Scuola Normale Superiore di Pisa and as the director of Ennio De Giorgi Mathematical Research Center. Scuola Normale Superiore di Pisa was founded by Napoleon, and has been since one of the most important training centers of scientists in Italy. The famous physicist Enrico Fermi was trained here.

FCL: We are interviewing you for a section of the journal " Math Media" which is published by our Institute for the benefit of students who are interested in mathematics. The section is called " Friends from afar". Certainly, Italy is far enough and, besides, you are a well-known mathematician. Could you please first give us a brief background of your studies in mathematics or your education in general, and, if possible, tell us what brought you into mathematics?

MG: This is a good question. (All laugh!) When I was a student in high school, I had no particular difficulties with mathematics. Then I heard that in Pisa one could learn good mathematics and I went there to study. Well, I must say that I was not a particularly good student. My main goal at the time was to finish as soon as possible my studies and become economically independent. After the first university degree I was offered to stay at the University and to start doing research in Pisa. At that time Pisa was really a great place for mathematics, people like Bombieri, De Giorgi, Stampacchia, Andreotti, Campanato, Prodi and Vesentini were all there. They are all good mathematicians. It was a good period of time for me.

FCL: When was that time?

MG: I was student at the University of Pisa from sixty-six to sixty-nine. In the seventies, immediately after my degree, I first spent a year in Paris, then returned back to Pisa. I had remained in Pisa until seventy-six when I got a position in Modena as a professor. Then I first moved to Ferrara for two years, then to the University of Florence. Six years ago I moved to the University of Pisa, and in the last three years I have been Professor at Scuola Normale Superiore. I believe that one important experience for me was to spend quite some time at the University of Bonn. I have spent about two or three years in Germany. This has been very fruitful for me, since I always found myself in a very stimulating atmosphere in the Mathematical Institute in Bonn.

FCL: You mentioned a good period in Pisa, could you say more about it?

MG: It was a really special period of time: De Giorgi had just developed the theory of minimal surfaces and proved the regularity theorem for elliptic equations; Bombieri, Ge Giorgi and Giusti were giving the counterexample for Bernstein problem, Stampacchia was developing the theory of variational inequalities. Many interesting techniques were being developed at that time , like the ones for the regularity and partial regularity theory that are sort of models for the kind of regularity theory developed in more recent years.

FCL: How did you contact with these good people when you were there?

MG: Talking and listening to them, attending seminars and lectures. I remember that at the time there were not so many books like nowadays. But, there was a kind of tradition that people communicated orally. This tradition of oral communication was very positive because people would get to know things quite fast. Besides there was less pressure in publishing. This was part of the culture of the period. In Pisa, there were not so many professors at the time. There were probably ten professors in mathematics, now we have about seventy. So, it becomes extremely difficult now to keep track of what is happening. It would be easy for ten people to talk to each other; for seventy people this would be hard. It is not hard to imagine that if there are too many seminars, then people do not even attend any. Suppose that there are only two seminars, one feels obliged to attend. But if you have too many seminars, then you start thinking that there is no time to attend all of them and decide to attend none. On the other hand, it is good for students to be offered many different topics so that they could acquire different points of view.

FCL: Many years ago, I read an article written by S. Hildebrandt, describing the activities at Oberwolfach. Could you say something about it?

MG: I had been participating every two year at the session on the Calculus of Variations at Oberwolfach from mid-seventies to mid-nineties. Nowdays Oberwolfach is even more active than in the past; the conference on the Calculus of Variations is still going on. At the beginning, there were only few participants mostly working on the same subject. We used to have lectures in the morning, while in the afternoon people just talked to each other, discussed problems or chatted. There was a lot of free time. It was very stimulating. Recently, more and more people become working in the Calculus of Variations. In the seventies there were about twenty-five or thirty people participating a conference in Oberwolfach. Nowadays, there are up to seventy people, the result being many lectures in the morning and in the afternoon. But, of course, there are positive and negative aspects in a big conference. Now let me talk about an aspect of life in mathematics. I think that life from the seventies to the eighties was a little bit relaxed than what is now; I never had any pressure in publishing papers, for instance. There were not many positions, but enough there were so that people could have chance to improve their positions and later it even became not so difficult to get an appropriate position. But now there is a very strong pressure on young people to publish, and I think it is not good. I have been in selection committees with young people presenting forty, fifty papers. Of course, there are always exceptional people, but published papers sound sometimes just like kind of refined exercises or works studying details that might interest only a few people. I don't think that this is positive for mathematics because having too much detail often doesn't really help. A consequence is that people have less time to talk to each other, everyone is interested in his own problem. This is probably why we have so many journals with thousands of papers published each year. One wonders how these papers are to be assimilated, as it sounds simply impossible. When I started my career, there were just a few important journals. People used to publish probably one paper per year, and attend one conference every two years. Now we have plenty of conferences and speakers are forced to repeat or make variations on the same topic. On the other hand, I understand that young people are under pressure and conferences provide chances for them. I don't see good solution for that. Maybe we mathematicians are too many, but I am not sure. I don't really understand the mechanism, but this situation is surely difficult for young mathematicians. This causes problems. For instance, there is less connection with other fields. Mathematicians used to work together with physicists, nature scientists, physiologists and so on; now we might talk to people from other fields, but somehow we end up with our own way, we turn away immediately from others' problems. It's my feeling but I might be completely wrong.

FCL: Yes! I also have the same feeling.

MG: All that seems to me not positive for mathematics.

FCL: Do you know that in Taiwan it is usually very difficult for young people to talk to senior people?

MG: Well, it is not that difficult in Pisa. Because Pisa is a very small town: there are about hundred thousand inhabitants and most of them live outside the center of the city. In the city center you meet mostly students and professors. People are just there. For instance, I live now in Florence and commute to Pisa. I spend three or four days a week in Pisa. When I am in Pisa I am there from eight in the morning to ten or eleven in the evening; either I am in my office or I meet somebody for work or for dinner, and most of the time he is a colleague or a student. Even for students, not only PhD students but also undergraduate students, it is not difficult to talk to professors. I must say that this could be special of Pisa. Pisa is somehow special in Italy, particularly for mathematics. This is a long tradition which might go back two centuries ago. One reason may be the presence of the Scuola Normale Superiore. The Scuola Normale Superiore recruits, if not the best, surely some of the best students in Italy. In fact it is the only place in Italy where students enter by passing a very severe competition and get supported on university fees, food, living expenses and also some pocket money; to be sure this is not through, say, extra courses and or direct contacts with professors. So it is particularly attractive to good students and in particular to good students in mathematics. Thus, most of good students try the competition to enter our school. Nowadays, however, students probably feel more attracted by big cities. Besides, I am not sure that all good students who want to study mathematics do really study mathematics. Often they choose different subjects. At the time I went to university, becoming a mathematician or becoming an engineer was not such a big difference in terms of salary, but the freedom you would have later and the possibility to do exactly what you would like to do played the difference. Now, it is not any more so with mathematics. Salaries in Italian universities are not so high compared with salaries in private companies, for instance. I believe that if you are good in mathematics, you might as well be good in engineering. Why? There is no particular reason for a person to be good only on one thing. Actually, I always advise students not to restrict their interests to mathematics alone, and try to acquire a broad spectrum of knowledge. My opinion is that if you have a very good training in mathematics, you should be able, for example, to be a very good manager.

FCL: When I said that it is difficult for young people to talk to others, I didn't mean that they don't have chance to talk. They do have chances, many chances. Usually the difficulties originate from themselves. They are afraid to talk.

MG: Oh, That's the same in Italy. Students are afraid to talk. They have chances, but they are afraid.

YCC: Are they afraid to talk about mathematics only, or in all situations?

MG: I think in all situations! A reason is probably the difference in age and position. Students usually prefer to talk to assistants than to professors, because they are more or less of the same age. I think another reason could be the system of exams. They know they have to pass exams, so they don't want to show themselves up. They are always afraid of being judged when talking to others. This might sometimes be even true.

YCC: Every time I went to see my supervisor, I was always well prepared. (All laugh!)

MG: Students might be afraid of being not well-prepared. Nevertheless, I cannot say I am always prepared. I like working with other people, not alone, because it is more fun; but I often end up with stupid mathematical statements or with an incorrect proof. This probably never happens to good mathematicians, but to me, this happens quite often.

YCC: Can you mention some open problems or new research directions in the Calculus of Variations?

MG: Well, Calculus of variations is a kind of collection of problems and methods. It is hard to say about good open directions, but I think one main direction is to understand variational problems involving nonscalar functions, for instance geometric variational problems like harmonic maps, Yang-Mill's functional, models in modelling microstructures, etc. Typical of these problems is the occurrence of singularities. These singularities are important. Sometimes singular solutions are even more important than smooth ones. Understanding these singularities, the way they arise, and their structure are very important.

YCC: How about some old but still important problems?

MG: There are also many open old problems. For instance, classifying the isolated singularities of minimal surfaces or studying minimal graphs or surfaces in higher codimension.

YCC: You mentioned the minimal configuration. Why do we always try to find minimal configuration rather than stationary one?

MG: Probably because it is easier. Also because minimizers have stronger properties. For instance, there are energy functionals of physical relevance with critical points which are everywhere singular, while minimizers have singularities only on some lower dimensional sets. Sometimes minimality can be replaced by some sort of stability.

YCC: Maybe you could offer me an answer on a question that I always have in my mind. Why there is usually a variational principle in the physical world. How do you think?

MG: In my opinion your question can be answered in many ways. We can say, like in the past , that God created everything in the universe in the best possible way, or we can say that mathematics works this way, or we can say that when we try to explain natural phenomena, we look for simple principles, and the minimum principle is a very simple principle.

YCC: Quite the same as that many mathematicians or physicists like simplicity.

MG: Yes! It is the same principle, probably. If we explain something on the basis of a very complicated principle, we are not explaining very much, just as we might actually be explaining something complicated with something which is even more complicated. (All laugh!)I think there are many, many answers, and probably the real answer lies in history: variational principle have worked for centuries. I think the history of science just gives the answer. Not only in mathematics; actually, you see more minimum principles in the history of physics. By the way, you asked me about possible development of calculus of variations . Think of Lagrangian and Hamiltonian theory. Both pictures are equivalent under suitable conditions; without those conditions can we relate them in some week way?

FCL: In Italy, do the students go to school at six?

MG: Yes, at six. But, there are proposals to allow children to enter school at five. Presently children enter the school at six, attend five years of elementary school, then three years of junior high and then five years of high school. Starting from this year we have a new system at university: three years to get a first degree, two more years to get the second degree, and then three more years for a PhD.

FCL: In the elementary school, do students just learn Italian, or they also learn other languages?

MG: Apart from Italian, now in the third year, they start to learn a foreign language in some informal way. I think they formally start the study of a foreign language at junior high school. Some schools allow student to choose two foreign languages. The curricula are quite strict in Italian schools; students do not have the right to choose their topics in high schools, everything is fixed. Usually what happens is that if a second language is offered, additional hours have to add to the ordinary schedule in the afternoon. In the past, students were used to learn French, but now they usually also want to learn English, but there are too many teachers in the schools who teach French and could not be fired, so some students have to learn French. (All laugh!) Usually students who learn French do learn English. The result is that they learn two foreign languages and this is good.

FCL: In Taiwan students start to learn English in junior high. Now there are proposals for students to start learning English in elementary school. What's your opinion about this?

MG: I think it is good. In my time, for instance, we had foreign language, but mainly we studied history and literature and we hardly learnt to speak the language. The mechanism was that students were required to learn basic grammatical rules and their exceptions, and so on. They never learnt to speak. But the situation is changed now. I still remember different rules and exceptions in French, but I learnt only a few words. Now whenever I speak French I realize immediately that I make many mistakes and have hard time in expressing myself. Though I never studied English and I make even more mistakes than in French, I feel much more confident with English (All laugh!).

FCL: It is a really interesting remark.

MG: I must say however that Americans are very tolerant as far as their language is concerned, while French are not that tolerant.

FCL: We don't have too much time left now. I would like to ask the last question. What is your advice to the young people who intend to study mathematics?

YCC: Perhaps some advices on the issue whether young people should learn as many topics as possible to broaden their view or to specialize as soon as possible.

MG: It is always difficult to give advices on this. If you specialize yourself, then the advantage is that you go deeper and deeper, but this could be very dangerous. On the other hand one can learn a lot when one is young, while it becomes more and more difficult to learn new things when one gets older, although experience helps. Now, due to strong competition, people have to specialize very early since they have to publish quickly. Learning might take time away from publishing. I think that young students should combine the two things. Of course there are people who can learn quickly, and very good students who can learn a lot and quickly, while there are students who need time, like me.

YCC: So, according to one's experience, one decides which is the best way for learning. But, it takes time.

MG: Yeah, unfortunately. There are no rules that you could apply to provide you with sure results. All ends up with one's personal experiences, that is, with his own life.

FCL: Probably we have to stop here now. Thank you very much. This is really a very interesting interview.

• Fon-Che Liu and Yi-Chiuan Chen are faculty members at the Institute of Mathematics, Academia Sinica.