Interview Editorial Consultant: Tai-Ping Liu
Interviewers: Tai-Ping Liu(TPL), Fon-Che Liu(FCL)
Interviewee: Luigi Ambrosio(LA)
Date: December 3rd, 2009
Venue: Institute of Mathematics, Academia Sinica
Prof. Luigi Ambrosio was born on January 27, 1963 at Alba, Peidmont. He received his B.S. in 1985 from University of Pisa and his Ph.D. in 1988 from Scuola Normale under the supervision of Ennio de Giorgi. After the passing of De Giorgi, he took over the teaching position of De Giorgi at Scuola Normale Superiore di Pisa. Due to his important contributions to the calculus of variations and geometric measure theory, he was awarded the Caccioppoli Prize of the Italian Mathematical Union, Fermat Prize, and Balzan Prize. He is a member of Academia Nazionale dei Lincei.
TPL: let’s get started. So we have worked you hard this time, but that was a good crowd. They asked questions.
LA: When you teach, usually you feel the atmosphere. I think that the atmosphere was good. Someone was really interested and into it.
TPL: As I have told you that I was listening and I was very much awake. Your answer was very impressive. Your presentation is one that we do not need to talk about it. So you said that you come from the south of Italy. Where about in south?
LA: Well, it’s actually my family was originated in Naples, they have been living still in the south of Italy, but on the other coast, the city is in a region called Puglia just on the Adriatic Sea. Actually Ennio de Giorgi comes from the same region, not the same city, but the same region.
TPL: I have been told that most people in academics come from two situations, one is children of someone of academics; or come from usually , how shall I put it, from not the high income family. Do you come from either of these two categories?
LA: My father was a judge, a lawyer, but he is now retired. My mother was a school teacher. Actually you know, my family was pushing, and putting a lot of pressure on me to be an engineer, because it was not so clear of how rewarding could be becoming a mathematician, in economical terms. So we had an agreement and I had to go to Pisa to try to enter the Scuola Normale (Scuola Normale Superiore di Pisa). If I succeeded then I could be a mathematician. If I failed, I should have gone to engineering. That was the agreement. Then I succeeded.
FCL: Either of your parents was not in mathematics, why did you choose mathematics as your subject of interest?
LA: It could be that there were some early experiences that when I was a child about 6 or 7 years old, actually, my grandfather, he was not an educated person, he was only educated in elementary school for 5 years of school, but, he had a real passion for reading, for chess, for games, maybe, it could be that, he started to play those with me. I don’t know. It’s very difficult to trace the origin of this. I mean I have always been very … let say … first very successful in school in mathematics when I was 12 or 13 and really interested in knowing more. So there is always more or less this kind of drive.
TPL: But your father wanted you to go to engineering. Today, is your father happy with your situation?
LA: Yes, he is. As soon as I entered the Scuola Normale he didn’t question my choice.
FCL: I see. So you are from a great school, the great Italian School of Calculus of Variations. Is that part of the reason that you choose subjects related to Calculus of Variations?
LA: No, that is not the case. I have to say that, well, first I did mathematics logic. But then my first ``research’’ experience in that area was not really good. I don’t know what happened, but eventually I got dissatisfied. Then I went to de Giorgi. He was teaching to very young students and I already attended his classes when I was a freshman. His classes were very nice and he was a very quiet person, not portraying himself as a great mathematician. In fact, basically, I had no idea of the level of De Giorgi. I realized that De Giorgi was a really great mathematician only during the PhD studies. Actually, I can tell you a story which is really nice, how well. Of course I had the feeling that De Giorgi was a good mathematician. But, you know, good and outstanding, there is a difference, right! Then I was in the library and I think I was in my second year of PhD. I was reading the introduction of the book by Ladyzenskaya and Ural’tseva about linear elliptic equations. I don’t know if you have ever read this introduction. It is very impressive. Because this book appeared in, 1960 or so, just a few years after De Giorgi’s celebrated paper on regularity of solutions to elliptic equations in divergence form with measurable coefficients. In the introduction, they say, well, this topic has been treated by many authors, but, basically, the first part of introductory was saying, well, there is a lot of literature, but we really do our things by our own means, we do not rely on all those papers. But then, in the last few lines there was this sentence: it says, well, we don’t use most of the things, but we use some ideas contained in this paper by De Giorgi. So, actually, when you read the book and you see that it is in large part based on De Giorgi’s ideas. Then I started to realize the greatness of this person. This has also to do with the fact that he was very modest. For instance, he never told me about his previous work about minimal surfaces. Since we were working on different subjects, so more or less previous stories I did not know. Of course I came to know all these things growing as a mathematician. You gain more and a broader respects. So my choice was not motivated by particular reasons. I have been lucky in this respect.
TPL: So you like all kinds of mathematics. Talking about Calculus of Variations, next week, we have this Australia-Italy-Taiwan meeting. I think originally that Calculus of Variation was very much what we have in minds, right?
FCL: That’s right. That was in 1994. The first Taiwan and Italian joint conference here. The subject was Calculus of Variations. So the first one was Taiwan and Italy.
TPL: Australia then join?
FCL: Australia joined at a much latter time. In 2002, I think.
TPL: So we went to Australia, then to Rome and now back here, every 3 years. So Calculus of Variations, we know it went all the way back to Leonhard Euler and Joseph Louise Lagrange. But in Italy, serious things happened around what time?
LA: Okay, let me see. Well, I should say that Leonida Tonelli has really been a leader and a pioneer for Calculus of Variations in Italy. Of course, some aspects could be traced back to Levi, to Vitali, I guess. But I would say that the one who really gave crucial inputs was Tonelli, in his work about the regularity of minimizers of Lagrange problems in Calculus of Variations. Indeed, the class of Lagrangian to which his result applies is known as the class of Tonelli Lagrangians. Now in perspective, many years later, you can see that already in this one dimensional problem you could see the origins of much harder problems appearing when more than one space dimension is present, namely the appearance of singularities, their characterization and the estimate of their size. So I mean how sensitive could this problem be in the function of space. So I really think that Tonelli in this respect has been a pioneer. Then of course there have been Cesari, Picone, I mean all those names.
TPL: There is somewhere, a name of an institute or a building named after Tonelli. Where is it? Somewhere in Italy, the name of an institute or a building is named after Tonelli.
LA: That Institute is in Pisa. The Mathematics Institute of Pisa is named after Tonelli, yes.
TPL: You mean, in the University of Pisa?
FCL: I know that the first time I visited the Scuola Normale, there was this lecture room called ``Aula Tonelli’’.
LA: Yes, I have been teaching there. That was before my departure for Taipei. I think that it’s nice to give this name to the room, because it brings some responsibilities.
FCL: I would tell you a story of mine. You know, the first mathematical paper I ever read was the paper by De Giorgi on perimeters. That was a beautiful paper. That’s why I learnt a little bit of Italian, because the paper was in Italian. Although I did not know much Italian, but still, I could read it. It was interesting. Tell me if I am right, it seems to me that in the beginning of the last century and the end of 19th century, Italy became to have calculus of variations, some schools of calculus of variations starting with Tonelli as you have mentioned. You have all these big names, De Giorgi, Picone, Caccioppoli, Stampacchia. Can you tell me the reasons at that time why calculus of variations flourished in Italy?
LA: I wouldn’t be able to give a specific reason. My feeling is sometimes mathematical research is sensitive to very small numbers, can also be the effect of some particularly gifted person, also particularly gifted students, I mean I cannot imagine a deeper explanation. Of course, for instance, talking about Caccioppoli, De Giorgi interacted with him and shared with him his vision about the theory of perimeters and theory of functions of bounded variation. Because of this interaction, De Giorgi did this project and you really see that the successful interaction of this person, De Giorgi, potentially outstanding in every field I would say, together with the vision of Caccioppoli, made the miracle. Good. If they weren’t met, De Giorgi for sure would have become an outstanding mathematician, but maybe in another field. Maybe, still in analysis, but perhaps complex analysis or so…
TPL: That’s nice to know that there is a trend, but, individuals do matter.
LA: Even if you look at my personal story, you see that the interaction with De Giorgi, and later on other mathematicians, has been fundamental. it could have been completely different. I think this is true for everybody.
TPL: I have heard a very vague story about how De Giorgi discussed mathematics with younger people. So can you describe it a little bit. You have the first hand experience.
LA: As students, we had a joke. We measured the way the appreciation by De Giorgi on this student by the time you were able to receive on the blackboard telling something. Typically, there were two options. The first option was De Giorgi becoming completely absent doing things with his own problems, or reading a journal; the second option was he asked for chalk and started to say to himself the answer to the problem. So you could receive the one minute or two minutes, or more, this was a sign of how much respect you had gained. The interaction was very informal and I would say except maybe in my first year of PhD when it was still possible to enter the details of proofs, you have a problem and you started saying this is what you have tried, but then, the interaction was mostly at the level of problems, so to speak. So De Giorgi at that time was about 55 years old, he was not interested in proofs. It was not possible to discuss with him at this level. On the other hand, it was very fruitful to interact with him at the level of problems. He had a clear vision of what is an interesting problem, what is a nice direction, what is a dead end and so on. I could imagine that it would be difficult for some students to interact with him. For junior students, you really need to be trained at the level of details, but for senior students, it was very good. You know, maybe when De Giorgi was younger, things were slightly different, but I don’t know. That was my interaction.
TPL: So De Giorgi, you said that when you entered, he was around 55 years of age. So that was the time he did not feel compelled to write more papers or to be still on top, that kind of psychological problems.
LA: Let’s say, when I started to collaborate with him it was in 1985. In the 1970s there was this boom of Gamma convergence. For De Giorgi’s school that was really the climax, so at that time, there could be 10 or 15 people sitting around him thinking in parallel about different problems, because Gamma convergence was really the source of many different applications. This was the generation of Dal Maso, Buttazzo, Acerbi or so. So when I came in, basically, this boom was over and almost none was around him. In fact, I have been one of the last students. He got me interested in this problem, about the bounded variation functions (BV functions), on which I have been working for 8 or 10 years. Already he was not really interested in writing any more proofs. He wanted people to follow his visions. From a personal side, I have to say, of course, I am not comparable to De Giorgi, but I see a similar trend also in my life, as years go by. This is why I think that interactions between a senior person and younger students are fruitful, because senior ones have a vision. But, what was really impressive about him was that, while it was very easy to fall in a kind of wishful thinking, his visions were correct. What was really impressive about him was that he was very sharp and he was giving the right definitions of and right proposals. There is this example which I would like to make and really struck me. So in the end of his life, it was in 1995, he made a proposal about how to develop the theory of currents, the Federer–Fleming theory, in general metric spaces instead of Euclidean spaces and so on. Then he wrote a very short paper, again, just with definitions. There was not even one conjecture. He usually made all sorts of conjectures which are still to be proved, in many cases. But in this paper, he just stated the problem. So it was just a bunch of very nice definitions. Then, in 1997, I took some articles, I was looking for a new problems to think about. Then I started this project with a colleague from Germany, Bernd Kirchheim. To be really honest, we were really skeptical about it. When we first started, what we can do in general metric spaces, presumably, the results could be very limited, because we are aware of the fact that many proofs of the Federer–Fleming theory used heavily the Euclidean or Riemannian structure. Well, the final result was a surprise even for us, because we came to a new theory with these definitions. It is valid in general spaces and it even simplifies the Federer–Fleming paper, since we use more essential tools. This was a complete surprise for us. Of course, proofs were given by ours, De Giorgi only gave the definitions, but he gave the right ones and this was absolutely not easy. You see what I mean. It was not just a wishful thinking. This is really impressive. This really struck me because I did not believe in this project, at least at the time when I started working. I had some believes, but for sure, I did not believe that in the end the result could be so nice and so general.
TPL: So in 1995, how old was De Giorgi?
LA: I think that De Giorgi was born in 1928, so he should have been almost 70 years old by now.
TPL: I see. Was he a good teacher?
LA: Yes. I mean he has a peculiar style, so how can I say. First of all, he had a very particular mathematical taste. For instance, his presentation of Measure Theory or Calculus of Variations was very original and you could not recover it in any textbook. Sometimes he got slightly confused. But I have to say that his point of view was so original, maybe, at the end you were was disguised because you could not recognize these things. But eventually of course you would appreciate it. Also he had the habit of never giving heuristic ideas, which is something I don’t do actually. So he was very Bourbaki style. This is the theorem and this is the proof. No comment on the proof. Once, I asked him about this, when I was older and more confident. He told me that giving heuristic ideas to students, already, gives the students a privileged viewpoint, why students, this was his opinion, students just from the dry proof have to make their own mind, which of course was difficult. His way was putting students in difficulty because you could not form an understanding of the proofs without understanding what the idea was behind. He would say well, this is the students’ task, it’s not my task. So I have an idea doing this, but I would not give the ideas to students. It’s better for them to try to get it. Well, at least, my personal style is to give some heuristic ideas. I understand the reasons behind De Giorgi’s choice but, I still believe that at least some hints are useful. Although it’s true that by giving some hints you are putting all students on the same track in some way, which is not good for developing new points in some way, so to speak. They all start to believe in the same way and maybe a new perspective does not develop.
TPL: Whatever the truth is, he was definitely a compassionate person in his own way.
LA: He was a very nice person.
TPL: I meant his thoughts for students.
FCL: To figure things out themselves. I have heard a story about De Giorgi a long time ago. A few people like to play bridge, De Giorgi and the other famous guy who have been in Princeton.
FCL: Yes, Bombieri. He, Bombieri and some other guy liked to play bridge. De Giorgi never cared about win or lose, but Bombieri cared very much about winning. Is that right?
TPL: That’s a sensitive question.
LA: Well, this I do not know. I came many years later. So … Before, I was almost a professional player of chess. De Giorgi also liked to stay with the young students very much, even having lunch with them. I remembered that once I played chess with him. We did the first game, and I completely destroyed him in a few moves. So I got the impression that he was a poor player. So for the second game, I relaxed completely and he defeated me in the second game.
FCL: That’s a very interesting story.
LA: You know with chess you can never know.
TPL: You have been interested in this optimal transportation theory for a number of years.
LA: More or less around year 2000 when I started.
TPL: I see. This series of lecture you have just given here is really wonderful. So what made you get into it?
LA: It really was a random start as usual. Actually it was because Bouchittè and Buttazzo started writing a paper on Shape Optimization, where they found a this connection with Optimal Transport. Apparently their problem had nothing to do with the optimal transportation. But it turned out that the optimality conditions for their problem are closely related to Optimal Transport. Since this paper was submitted to the Journal of European Mathematical Society, and to me as an Editor of the journal, I started to have a look at it. Then I came to the paper by Craig Evans I mentioned in my lectures, and I was really fascinated by the subject. Actually, I also got an important suggestion from Craig Evans. He told me, since I know well of measure theory and probability, why don’t you look at this paper by Sudakov? He told me, well we are not able to understand the paper, and eventually, I found the paper had a mistake. So..
FCL: Which paper are you talking about?
LA: There is this paper by Sudakov, a famous Russian probabilist, dealing with the optimal transportation problem in the case of cost equal to distance, which contains a subtle mistake. The mistake has been fixed, with the contribution of many people, in the last few years. So this also is something interesting.
TPL: But the paper contains interesting ideas.
LA: Yes. The Sudakov errors shows how our intuition of Lebesgue measure can be deceptive in some cases.
TPL: Can you elaborate on this point a little bit further?
LA: Yeh. Basically the Susakov mistake is the following. Susakov idea is the following. Take the Lebesgue measure in the n-dimensional Euclidean space, at least the first meaningful example, take a family of disjoint lines which corresponds to the transport rays in an optimal transport problem. Well, his idea was that if you disintegrate the Lebesgue measure along this family of rays, i.e. you write the Lebesgue measure as a superposition of measures, each concentrated on one ray, then almost all these 1-dimensional measures are absolutely continuous with respect to the 1-dimensional Lebegue measure on the ray (a kind of generalized Fubini theorem). Also, he sketched a proof of this fact which sounded quite plausible. But whenever you try to make the argument rigorous, you fail. Actually, the property is false and we were able to build a decomposition where the absolute continuity property claimed by Sudakov fails. The next question is to find an intermediate regularity property satisfied by ray decompositions in optimal transportation and sufficient to have the absolute continuity. After many contribution by many authors, the key property has eventually been identified by Bianchini and his school.
FCL: That’s a difficult problem. That question is difficult. I remembered that last time when Bianchini was here. He asked me something about it. But at that time, I really did not know it. Really, the problem was mine. Now I understand it better.
TPL: Yeh, Bianchini was quite excited about this, resolution about this problem. He began to educate me why measure was so important.
FCL: I see.
TPL: I can see that the original source comes from you.
LA: The possibility of this is very high.
FCL: So is this also connected with what you have just talked, some kind of concepts of calculus of variations in general metric spaces?
LA: A Little bit. I would say more of Calculus of Variations in gradient flows in metric spaces. The leading idea behind many applications is that you can start to consider a space of probability measure as a metric space. Then you would like to have a description of some evolution problems, for example of gradient type. It turns out that, even in situations when you do not have a differential structures, you might use the only structure you have, namely the metric structure. Then you are forced to describe a gradient flow using a very fundamental idea, the energy dissipation rate. Just using this as a fundamental tool, you get a clearer understanding of the problem even in situations where the metric viewpoint could be avoided and you might use the differentiable structure. You see what I mean, in some sense, it is to look for in higher generality, because then, by looking in this perspective you are forced to use more essential tools. Sometimes, studying a problem, you use too many tools and assume so many things. We use the minimal and only the essential.
FCL: So you mean, if you look at more general situations, it’s easier to identify the key points of the arguments.
LA: Exactly! That’s exactly my point. It could be technically more difficult, because to derive precisely only that thing, you have no choice. But if it happens, then you have understood something …
TPL: You mentioned the definitions given by De Giorgi. It is difficult in general to identify how much generality you should have?
LA: Exactly! As I have said, it is a very difficult task, presumably only a very few mathematicians are allowed to have this vision, e.g. Caccioppoli, De Giorgi. Also some pages by Caccioppoli were very impressive. You know Caccioppoli was very criticized because he was often giving only formal arguments. So when studying this kind of very fundamental problems, not being able to turn this into a rigorous proof by modern standards. But on the other hand, when you read them in a later perspective you realize many things. When you read his papers, it is very impressive, because it was clear that he had in mind which were the right directions, which were the dead ends and so on. So he was like De Giorgi in the second part of his life: he had the vision, but maybe he did not care about the proofs. But if you have the right vision, that is not so important. Also, to make a more modern analogy, another mathematician with the same style is Gromov. I mean, Gromov, has absolutely the right idea with the right definitions. Sometimes, maybe, he claims that he has proofs, but they are not really complete…
FCL: Who is this you are talking about?
LA: Mikhail Gromov, a Russian Mathematician.
TPL: One time, Arnold was in Stanford and he said that, “I don’t know how Gromov arrived at that. He did not give an indication of how to prove it, but he was always right.”
LA: That’s exactly the point. The thing is also the work of those mathematicians who tried transforming these ideas into rigorous proofs should be appreciated in some sense.
TPL: I was told this great school of Italian algebraic geometers are also like that. They were very intuitive.
LA: Yes, it’s not my field, but I have been told that many later generations of algebraic geometers have been spending their time in trying to make rigorous proofs out of these informal and intuitive arguments.
TPL: Could I just go on a little bit, we have talked about Italian mathematicians now. You have given two examples already, in algebraic geometry and in analysis. There must be something specials about the Italians. You want to speculate a little bit? Of course you guys solved the first the 3rd degree polynomials.
LA: I don’t know. It is difficult to say why mathematics blooms in a country sometimes. I had to say the traditions have always been very strong, even at the end of 18th century, the 19th century. I mean you have people like Dini, Vitali, Levi, and so on. It is weird also, because during the fascist period Italian mathematics suffered a little bit from isolation. For instance I noticed, when studying the papers by De Giorgi on elliptic PDE’s, that the theory of distributions, arrived much later in Italy. For some time, we have been isolated. It’s difficult to say. I would say that tradition in education is important. I mean, in Italy, many universities are very old. They established in very old times. Of course this does not explain why mathematics bloomed.
TPL: Somewhere I read, some cities pride themselves as the city of some certain mathematics discovery was made, for example, the third degree polynomials were solved by us.
LA: Yes. There were mathematical disputes and mathematical gains. There typical one was:, I know the solution of this equation, and you have 6 months of time to give me your solution. Then I would give you the solution.
FCL: So in the 20s and 30s of 20th century, mathematics was really blooming in Italy right? You have a very good school of Calculus of Variations and a very good school in Algebraic Geometry at that time. Also in Differential Geometry, you have Levi Civita. But you say it suffered from the fascist?
LA: I think that we suffered a little bit. For instance, all the developments in differential geometry, in the work of Riemann, and the communications were not good in general. In countries like France, it was not so.
FCL: So talking about the dispute in mathematics you have just mentioned, remember that Satsuma interview, when we interviewed him, he said that in certain places of Japan before the temple, they have posted problems on the wall for people to solve.
TPL: That is in the Shinto Shrines, and the problems are about plane geometry.
FCL: So that's a certain kind of cultural bias.
TPL: Although the Japanese mathematics, as I have been told, eventually inherited the European one, not that tradition; but the soil, the ground, was prepared to be able to absorb the European tradition of Mathematics. Without that, they would not be able to do so. So that proves scientific research is a slow business. I remember that Fon-Che always says that we shall not rush as it takes a long while for the general culture to change. One cannot push it. So you are still very much interested in this optimal transportation theory for the moment?
FCL: Are you writing a book on that?
LA: I already wrote it. It was published in 2005, mostly devoted to the theory of gradient flows. In 2008, we had the 2nd edition.
FCL: I see. Published by which?
LA: Birkhäuser. It’s a series of books which are related to the NachDiplom course (an advanced PdD course) in Zurich. I mean, teachers of this PhD in Zurich are expected to write some notes. So I was in invited in 2003, I don’t remember the exact year, to give a course there and then I had to write the notes. That was also the occasion for writing something more extensive, so it became a book.
TPL: How about the Italian analysis in general? This is such a vague question. This is one which very much De Giorgi would not be happy to hear such a question.
LA: I don’t know. We are now suffering a little bit because of the brain drain. I mean of course we have some very good mathematicians, even after my generation, we have in Analysis people like Alberti, Bianchini, Malchiodi, Mingione and others. But I would say that after them, I start to see some difficulties. Some of our best talents are going abroad. So… you know, it’s not a very exciting moment now. Maybe, I think more helpful to start to think mathematical research in a kind of European perspective, which makes things easier. But on the nation, purely from a nation’s perspective, it’s not so nice at this moment, because our best talents are moving away.
TPL: Just a specific example of what you have mentioned, when Aberto Bressan moved to Penn State University. I was telling him that why don’t you spend a few months each year in SISSA, but he did not. On the other hand, there are maybe positive things happening. Bianchini began to develop his own ways and he told me that now he has excellent students. So…
LA: No, in fact, it is more to the younger talented scientists in places like Scuola Normale, after this generation of those who are now 35, or between 35 and 40, I don’t see really big names coming in. I mean there are big names who have been trained in Italy, but they are all abroad. There is Camillo De Lellis in Zurich, who is a former student of mine; Alessio Figalli who is now in Texas, also a former student of mine. So I see a little difficulty. It’s not just looking down in my own field. I mean whenever I look around in other areas of mathematical analysis, there is this moment of difficulty.
TPL: Well, the underlying culture and tradition is so long and so deep, I think that this would be a temporary phenomenon.
LA: Yes, I hope so. I hope this is only temporary because it’s related to the difficulties in the recruitment and so on, few positions and also kind of rules for recruitments need to be improved. There will be a second generation. So there is now this young generation of mathematicians who did their PhD abroad, maybe, there is a possibility to recover them in the future.
TPL: Well, people talked about the Russian schools, of course they are in terrible shapes now. On the other hand, still, they produce someone like Grigori Perelman.
LA: A singularity.
TPL: But as you have said before, it’s all because of certain individuals.
FCL: Is it still true that best students are coming to Scuola Normale, or are they spreading out?
LA: No, I still believe that the best students are coming in.
TPL: So maybe I would like to ask some questions on your institute, Scuola Normale Superiore. You have no undergraduate students?
LA: We have.
TPL: You have undergraduate students! Oh! Okay.
LA: Actually, this is the main difference of us with SISSA in Trieste. We have undergraduate students. The thing really matters. Actually, in my opinion, it’s the undergraduate program of Scuola Normale Superiore the one which works best, not only the PhD program. Of course we have very good PhD students, but in the sense that it works well because many of them are former undergraduates of our school. Because of that, this makes it possible to start working with them when they are extremely young. Figalli, the latest example I can make, is now 25 years old and he is professor in Austin. He came to me when he was in the second year of his undergraduate study. I realized that he was really exceptional and I started to really spend time with him and in his third year of university, I sent him abroad. So actually everything happened faster, he did his PhD in one year and so on. Figalli is really an exception, but I could also make other examples with very good students I had. All of them have been working very well because I exposed them very early to advanced mathematical research. Of course, without the undergraduate program, this is not possible. So this is the real reason why they are doing the undergraduate program. Already at young age, they are exposed to meetings, to visit abroad and so on. Of course, this would work on really few students only. I am not saying that you should do this to other students because it can be destructive. For many students, they need their time and then they can become good mathematicians. Anyway, I say that this is very efficient when you spot a very good person, it is a very efficient policy.
TPL: We have nice good conversation. I really appreciate that. You have many good points and very nice stories. Maybe we could quit for now and thank you very much.
LA: You are welcome. I appreciate you take your time for this conversation.