傳播數學知識．促進數學教育

Interview with Prof. Luis A. Caffarelli

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

**Interviewers:** Tai-Ping Liu (TPL), Chang-Shou Lin (CSL), Jong-Shenq Guo (JSG), C. K. Lin (CKL)

**Interviewee:** Luis A. Caffarelli (LAC)

**Date:** May 2nd, 2008

**Venue:** Department of Mathematics, National Taiwan University

Prof. Luis A. Caffarelli was born on December 8, 1948 in Buenos Aires, Argentina. He received his M.S. in 1968 and PhD in 1972 from the University of Buenos Aires. He wa a faculty at University of Minnesota, Courant Institute of Mathematical Sciences at New York University, University of Chicago, and Institute for Advanced Study in Princeton. Since 1997, he has been a faculty in University of Texas at Austin. Caffarelli is a world leader in free boundary problems and nonlinear partial differential equations including Monge-Ampere equation and Navier-Stokes equations. He is a member of National Academy of Sciences (1991), and was awarded Diamond Konex Award from Argentina (2003), Rolf Schock Prize from Royal Swedish Academy of Sciences (2005), Leroy Steele Prize (2009), Wolf Prize (2012), and Shaw Prize (2018).

TPL: We usually start with a very standard question. Your time in Argentina: how did it start? Your mathematical interest and so forth?

LAC: Well, I always liked sciences in general, and in high school we had a very inspirational teacher. In fact, from a group of 30 students, eight went to mathematics. After a while, I think only two or three of us ended up in math. I was undecided, I wanted to do either engineering, or physics or mathematics. So the last year of high school I went to some engineering classes and some mathematics classes, and I liked the mathematics much more. So engineering was out, and then I had to decide between math and physics. I liked them both, but just before I entered the University, in 1966, the Military took over and intervene violently the University. Many of the physicists and mathematicians left the Country, and it took Physics a while to recuperate. I did both Math and Physics for about two years, but then decided for Math…

TPL: Traditionally, Argentina has a very strong physics track, right?

LAC: They do have a very strong physics, yes.

TPL: How about mathematics?

LAC: Well, mathematics, too, but both in physics and mathematics we are all spread around the world now. When I was a student, we had some excellent professors, for instance, Luis Santalo, a Spanish mathematician, that was in the Republican army during the Spanish Civil war and had to come to Argentina. Also, Calderon was in Chicago, but he had some very good students that came back, and they all ad very good training. So I really had a very good training.

TPL: But how about Argentineans in the early part of the 20th century? There were some mathematicians…?

LAC: No, …I would say that the mathematics of Argentina… actually a lot of Latin America, grew out of the Second War and the Spanish Civil war, because many scientists came to Latin America at the time, for instance Andre Veil went to Brazil, and Beppo Levi to Argentina…Government supported science, it was from the 40s on.

TPL: In Chinese society, we value intellectual pursuit. But in Argentina, that’s so also, right?

LAC: Well, I think that now there is a conflict: They value sort of intellectual endeavor versus financial success… (laughs).

CSL: Ok, can you just talk about anything when you were in undergraduate study. Can you think of the inspiration from your professors in mathematics?

LAC: Well, we don’t have undergrad. In Argentina you go to the university and you just go for the “Licenciatura” which is like the masters. This is a serious problem of Argentina (general laughter), there is not a middle ground. So, if you survive the system, you do well. If you don’t survive the system you leave the University empty handed. I think it’s a problem, but unfortunately that’s how the system works. For me, I think it was the atmosphere of the university…when I was a student, we spent the day at the University…we had many open spaces, there were many places students can go and work on tables and so. We basically lived there and shared the problems and ideas …I think it was more just the group, that we were all part of a common enterprise…

CSL: In that period, the relationship between professors and students was very intimate?

LAC: This was mostly between students…, professors were nice. They will answer your questions, but mostly it was teaching assistants and students, and mostly the students…The university was not close to our homes, and most of us had to travel an hour to get their, by bus. So once you were there you stayed and spent the day there doing homework. And this was a great atmosphere. I think it’s the same everywhere. What matters is the group surrounding you. For instance in high school. When you have high schools which are very high quality, the professors are good, but it is the children that make the atmosphere fun, because they all compete in a friendly way

CSL: Have you gone back to your university?

LAC: I do. I go back to Argentina every year, sometimes twice a year.

CSL: So is it the same?

LAC: Well, in some sense it’s not the same. The 60s in Argentina like with many parts of the world were very exciting times.

CKL: In politics?

LAC: Well in the 60s there were no bounds. Young people felt that there were no bounds, in a positive way. Throughout the United States, and in France there was this sense that…

CKL: So you mean influenced by the student movement, which I think start from France?

LAC: Probably because this was finally a time of prosperity after the war… This was a time in Argentina when everyone could sort of find its place. You could study or find a nice job…you could find your place. So young people were confident.

TPL: I understand in Mexico, the 60s were a good time also, right?

LAC: Probably true, yes.

TPL: Optimism, there was a lot of optimism.

LAC: Optimism, a lot of faith in the future, and a lot of faith in science for instance.

TPL: Then you came to the U.S.

LAC: I came to the U.S. Yes, I came to the U.S. for a year with a fellowship from Argentina, and then, I never went back …(laughs).

TPL: So such a long period in the U.S. So maybe you can say something about which part was steep in your research or learning.

LAC: Oh I did. I enjoy all the places in different ways. I went to Minnesota having done my thesis in Argentina in special polynomials, did not know any PDE’s and I discovered them there. So Minnesota was a time where I learned a lot, but all the periods were different. I was very happy with my work in Chicago. It was mainly a time to get ideas. Courant was a wonderful activity at the time when I went, with Nirenberg, Varadhan. Each institution has its value, and I found that each institute has its value in a different way.

CKL: So when did you meet Nirenberg?

LAC: Well, I met him a couple of times briefly, and all of a sudden he called me when I was visiting Italy and just asked me: “Would you be willing to come to Courant?” That’s where I really met him, when I went to Courant for the first time. it was 1980 to ’82 when I went to Courant for the first time and started collaborating.

CKL: So you had the paper for the partial regularity?

LAC: This was at that time, and there were a bunch of papers with him and Joel Spruck, using the continuity method for fully non-linear equations, Monge Ampére, …

TPL: Now that you mention Nirenberg. Do you want to say something about some of the people you encountered…De Giorgi, in particular? What kind of guy is he?

LAC: De Giorgi? He was a great guy. I believe he was the president of Amnesty International for Italy, apart from being a great mathematician. He was person who was very worried about human rights, he was a very special person, very sweet and dedicated. All the Italian mathematicians loved him.

TPL: But is it possible to say in what way De Giorgi was a great mathematician? Is it possible to talk about what mode he was working in?

LAC: Well, I didn’t know him personally that much. I didn’t go to Pisa that much, I just met him a week at a time, three or four times. Mathematically, I think I was telling you, he solved the Hilbert problem when he was 26 or 27. Build the generalized theory of minimal surfaces when he was thirty something and he did homogenization, like ’57 and ’58. In fact, something I’m really annoyed with is, when you read the book “A Beautiful mind” about Nash, everybody is very sorry that they didn’t give the Field’s medal to Nash, because of De Giorgi, but nobody regrets they didn’t give the Field’s medal to De Giorgi who did the work earlier in much more isolated conditions. (general laughter)

TPL: But I think neither of them needed that.

Caffarelli (agreeing): No, no, not at the end…he was a very sweet man, very congenial…if people ask him about it, and he would said “Don’t worry about it.”

JSG: So what particular subject do you like best?

LAC: Well…All are beautiful in some sense. What I like is that at some moment you realize you have a grasp of the problem, and this is the beautiful part of mathematics. I don’t know, it’s a different feeling, for instance, when I did my first papers I felt I was discovering more, at the time, my work was more technical…now ,instead it’s like you have more of a vision, you age and see more the big picture…you know there’s always this theory that a mathematician has to do everything before he’s 22, or something like that (general laughter), and I think it’s not true. What happens is that people notice if someone has done something at 22 because it’s unexpected. But I think all the times are nice, it’s a different view of mathematics.

CSL: Actually, I particularly like the idea of Professor Caffarelli. I think he does mathematics in a very natural way. He grasps a problem, the point of view is very natural. For example, actually, when I came back to Taiwan and after I spent three years in this university, after that time, I decided to move to a remote part of Taiwan—I almost retired to society, working on some math problems. The first problem I picked up was to study the singularity problem.

LAC: I know, I remember.

CSL: My work was actually inspired by Professor Caffarelli’s paper. The paper is actually concerned with how to classify the entire solution of the equation for a critical exponent and there you can actually transform an infinity point to the finite point, and you just deal with the equation with singularities. So this part, the idea is very simple, I think it’s natural…well, in any case, I think this is the first time we PDE people try to treat problems with singularities. I’m also very fascinated with your joint works with Nirenberg and others on Monge-Amp\`ere equations by using some kind of convexity, and the problem is the starting point for your works on fully non-linear equation. Well, I’m wondering, can you describe your working with Louis Nuremberg, when you’re doing work with him, what kind of inspiration between you two?

LAC: It’s an intense collaboration. Working with him, it’s very hard to say how things appear, you start discussing… and it is like ideas bounce back and forth. I think collaboration is a very nice process. Sometimes collaborations happen simply where somebody knows one piece and the other person knows another piece and they sit down and put it together. But there are collaborations where things start from scratch and you have to go discovering for ideas. Louis ( Nirenberg) is great in that sense, we get together…and chat…

CSL: Your work on this non-linear, elliptic PDE is a very geometric way, I think this kind of thinking is actually quite unique amongst PDE people. (general laughter) Very geometric.

LAC: Well, that may be because I learned PDE pretty much by myself. When I went to Minnesota, I followed some lectures by Hans Lewy, and asked him for some problems, I did not know PDE…he asked me about two or three problems…and there was not a functional analysis program behind them… About harmonic functions winding around curves and so on. There was no literature on it, so I kind of went back and looked at old papers and, I had to discover… and I think that gave me an edge…

TPL: So it’s nice to begin as an outsider?

LAC: Yes…many times yes, many times yes.

TPL: (Laughs) Uh-huh. Now, when you give a lecture, there’s a very strong impression that you’re in a world, like a beautiful garden, and that you totally enjoy yourself. That’s the impression we get. Is it true?

LAC: Yeah, yeah! I like to talk about math, yeah. All of our lives, we convey, discuss with people, and I like that.

TPL: Now you know PDE, that’s an understatement. Can you mention some of the PDE methods, techniques or ideas, or activities or personalities?

LAC: Well, I think I wanted to do…now it’s too late since two of them died. But years ago I really wanted to do a meeting where the three analysts that I admire most would explain their ideas… Calderon, Carleson and De Giorgi To describe them, I like an expression that Gene Fabes had about Stein, an analyst I also admire. He said: I think he can hear the music and I think this really conveys how they do mathematics. They did things that changed the view of their fields. When I was still in Argentina, I studied Carleson theorem on point-wise convergence of Fourier series, it was a theorem of theorems.

TPL: Carleson came to Taiwan in the early 80s, and he talked about Komolgorov, the counter example of L1 functions which is almost everywhere divergent, we were talking about that. He said something parallel to what you were saying about Louis namely that Komolgorov is also very geometrical. And that comes from the master Carleson. (laughs) And the same, Carleson doesn’t talk about himself, so…

LAC: Uh-huh.

TPL:You have collaborated with quite a number of people. We mentioned Louis Nuremberg…

LAC: I worked a lot with Louis, Avner Friedman, Joel Spruck , Yanyan Li. Yeah, I think I have many collaborations

CSL: So do you have Chinese students?

LAC: Yes, I have two that graduated, Lihe Wang and Peiyong Wang. And I have one more, Lan Tang in the middle of his PhD. Yeah, three.

TPL: Now, these days the academic environment is very different. It keeps changing. Definitely different from the 60s. We seem to have for example, this SCI business... In Asia in particular, it's sort of emphasized. But in the U.S. people don't pay as much attention to that.

LAC: Well in science in the U.S. it kind of works, right, because we import a lot of foreigners. More than half of the PhDs in mathematics are foreigners. For the United States, it's a great deal. In Argentina, they train 30 people, ten succeed and 5 go to the United States. If you take an American kid, right? He goes into mathematics. What does he need to succeed in mathematics? If he works really hard in four years and gets a PhD, in a very technical and difficult area. If he's really good, he will go as a post-doc, at a good institution, across the country right? If he works very hard, after two or three years he gets an assistant professorship in some other institution again across the US. And if he works very hard for three or four more years, he can become a professor. He sort of postpones his income for ten years...(laughs).

TPL: They are much better off than the biology majors, though. In biology they have to be post-doc forever. And it's much harder to get tenure track after that.

LAC: And a person who's getting his masters in business administration after two or three years is making a full professor's salary. (laughs) And a master in business administration is much less demanding: you don't have to write a thesis...you know, rational curves, Euler equation… I understand why American children are very reluctant to go into science.

TPL: That's true.

LAC: We are successful, we enjoy our life, but that's a very small percentage.

JSG: Another question is, now in Taiwan, the Ministry of Education is evaluating all the universities in Taiwan. So particularly, mathematics, also science, are talking about what kind of standards you want to put, to evaluate each department. For example the “Science Citation Index.” What do you think about such criteria being applied to mathematics?

LAC: I think the evaluation of a department is much more complicated than just some indexes.

JSG: Sure.

LAC: You have to evaluate, you know, what is the benefit to the general society. An evaluation of a department involves also its value to society and not just how many big shots.

TPL: We do a lot more service in teaching.

LAC: Yeah. It's a very complex thing. Somebody did a ranking in the United States, sort of about the quality of the department, by the value of the degree. Texas A&M came in second. (general laughter). When you read it in detail it was very reasonable. They produced, at low cost for the students, well trained people that took all kind of occupations. So this is something you have to measure someway or another.

TPL: Most of the students entering Texas A&M are local.

LAC: Are local students; it's a very inexpensive place, really helps improve their education. He felt that this was more valuable than a top private university that has very bright kids who can succeed no matter what. I think science is extremely cheap, very efficient, for what they give. It is my impression...if you take another aspect like the National Science Foundation. It must be, in the US, by far, better spent for taxpayer’s money than most programs. People with NSF grants are very proud of it, there's very little waste.

CSL: Well, in my mind you are one of the greatest analysts. I just want to talk a bit about the Navier-Stokes equations. It seems right now that the most important PDE is the Navier-Stokes equations. It seems like the elliptic method still occupies a major part. Can you talk about the Navier-Stokes equation, anything about it.

LAC: O - K. (general laughter). Let me start off by saying that the Navier-Stokes equations are a very simple model, right, to describe some complex phenomenon within some range of parameters. So you know, I don’t know what is the real value to know that solutions of Navier-d of Stokes are bounded or not, or smooth or not, mathematically is very challenging, but I think the actual value for science is not that great. After all, if you have oil at very high speed it will burn. So, obviously the model will not apply. This is a discussion that we have sometimes and is more general than just the Navier Stokes equation. I feel it was very bad idea of the mathematical community to put so much emphasis in a few old problems because it describes mathematics by mathematicians worrying about 100 year old problems that nobody can solve. And I think mathematics is much more dynamic than that. There are very beautiful things in probability, physics, material sciences, in biology whatever...Much, much more interesting! And it would be better instead of putting money into solving those 100 year old problems, let's put $200,000 in exciting problems for young people. I remember when the Fermat problem was solved, for the newspapers the impression was that mathematicians spend generation upon generation to solve $a^n+b^n=c^n$, right? (laughter)... Of course, it would be nice if someone comes and solves the Navier-Stokes problem, we all appreciate beautiful arguments.

TPL: So could I go after your response? This moment today, right now: What are the nice things to do, for you? What is something a mathematician can do--any concrete directions, what interest you?

LAC: Well, one thing is to understand wrought equations, different media, the interaction between scales. Science has become very mathematical. I remember, twenty years ago, when computers started to become really powerful, a lot of scientists and engineers, declared the end of mathematics. (laughter) I mean, why do you need mathematics, just put it in a computer, and get the results, right? Crunch the numbers, crunch the numbers and crunch the numbers, right? Then they realized that if your model wasn't good, or if you did not understood what the equation was telling you then there was a limit on what you could do and many scientists realized that there is more than crunching numbers. You see, mathematics is in a golden era. There are a lot of mathematical problems, and sometimes math departments are incapable to react to that. There is mathematics and mathematicians. Mathematics is in a golden age. There are a lot of people who want to find quantitative ways of describing what they see, population dynamics, cell growth and things like that. So all through science, there is a lot of mathematics to be done. So I think it's good to do mathematics, and mathematics from scratch. This is good for a young person because you don't have to know 40 years of history to tackle a problem.

TPL: Uh-huh.

LAC: The only problem is, sometimes it's hard to be rewarded for doing something in that area.

TPL: Let's hope that mathematics prospers, and that the mathematics department also prospers.

LAC: I really hope so.

TPL: But that is not automatic.

CSL: I think right now mathematics isn't being appreciated that much by the scientific world.

LAC: Yes.

CSL: Ok now we doing mathematic work always ask to prove something. Now is it possible that right now, this kind of mathematical work could be affected by the development of technology...

LAC: Well...I think it's something we don't know! Or even right now, how do we train our students? But the problem doesn't mean the opportunity is not there. The difficulty is that if I train a student kind of in a cross-field where I want him to learn new mathematics...I am very reluctant .If I train him in my area, teach him what I know and how to think he's going to have a good job, right? If I put him in an unknown area, it's sort of unfair. Because even if he does beautiful things it's going to be very hard to start his career. There is no underlying structure that will allow for that, and this is a problem. I mean, what do you do in order to build the bridges.

TPL: Uh-huh. Right now we define good mathematics as good analysis and good "capital T" theorems.

LAC: Yeah.

TPL: That's not the only definition.

LAC: Yeah, yeah! You can say, we call that mathematics and we call the others: "new". But we should try to produce new graduates, right? This is a problem in mathematics and mathematicians have to make an effort.

TPL: We need to think about that, and constantly discuss this issue and be open-minded. But you set a good example, you come to PDE as an outsider, and then you seem to maintain this attitude forever as an outsider.

LAC: (laughs) Well, right now, I'm in a comfortable position.

TPL: Right, right.

LAC: But when you are young, you are being judged by some parameters and have enormous risks to defy.

TPL: The way you talk about Navier-Stokes equation, for you to say that we need to start from scratch and from there go to new directions. Well, it's very nice for you to say that.

LAC: Well, it's also very safe for me to be telling you because, we are in a position to explore something new. But our duty is sort of to give young people the freedom. Which is like going back to the '60s...

TPL: Right, right.

LAC: The freedom to tackle science.

TPL: Uh-huh.

CKL: Can you tell us something about your experience, how do you start a new field?

LAC: Well, I didn’t start a new field, I just worked continuously. Everything I did was just kind of moving slowly from the previous field. At our age, we have some kind of portfolio of two or three things we are working on all the time, right, and tools are working well and you're working on that and in the mean time you make a risky investment and try to do a piece of something different.

TPL: But you do move around. You don't stay in one place, you do move around.

LAC: Yeah, but always so in an evolution...

TPL: But perhaps Chi-kun's question is how do you move around? (laughs)

LAC: (laughs) Well that's fine, that's the beauty of collaboration; some of them are doing something kind of different, but there's common ground, you move there and collaborate and you move naturally there with the people you talk to. You build bridges with the people you talk to.

JSG: I agree with this, I agree with your opinion...

LAC: Yeah...

JSG: It's very nice to have collaboration...

LAC: Yeah, you talk with people and you start to think together about other things and...

CKL: Do you think you start by solving the problem, not by studying the subject? You start a new field by solving the problem first...

LAC: Well I think it's a combination. You don't have to go and start a new field. You talk to people, you have a problem, you understand what is the issue, you work on it and you go and learn.

TPL: This series of lectures, honestly, is rather heavy. Ok, I cannot speak for anyone else, but I sat through four lectures here and the fifth one at Academia Sinica, and these are really not five lectures, but a short course on a very core subject in mathematics. Now you talked about education, how mathematics research is headed in a new direction. You have this freedom of thought..., I don't know how to put it. Yeah, you go onto the blackboard and draw those pictures and that is...very unique. That's very unique. And you keep all the interest of the people.

LAC: They came! (laughs)

TPL: (laughs) Usually the audience decays exponentially.

LAC: Yeah, yeah. That was very nice. A very nice audience.

TPL: I think you definitely should come back to Taiwan soon.

LAC: Oh yeah, I would visit a lot.

TPL: And as I said, next time when you come back, we will not work you so hard. (laughs)

LAC: (laughs) But I enjoyed it, I really enjoy it. Talking and...it's good to have an audience. We are like theater people, right?

TPL: (laughs)

LAC: You spend so many hours thinking alone and you have the opportunity to explain.

TPL: Yeah, and the students keep coming back and I need to ask some of them to pay.

LAC: (laughs)

TPL: (laughs)

LAC: Give me the good news only! (laughs)

TPL: That's very nice. That's very nice. Yeah, perhaps we could...end here, and just say that this is Part I. (laughs)

LAC: (laughs) Ok!

CKL: To be continued...

TPL: To be continued! (laughs)

LAC: Let's not go to 4, like this movie ???

TPL: CS has just come in, I was just saying we should stop here unless you have some more questions?

CSL: I think he may be tired. He just gave a lecture and we just dumped so many questions...(laughs)

TPL: No, Luis never gets tired (laughs), but maybe we get tired (general laughter). Ok. Thank you, thank you very much.

LAC: Oh, thank you!

- Tai-Ping Liu is a faculty member at the Institute of Mathematics, Academia Sinica.
- Chang-Shou Lin is a faculty at the department of Mathematics, National Taiwan University.
- Jong-Shenq Guo is a faculty at the department of Mathematics, Tamkang University.
- C. K. Lin is a faculty at the department of Applied Mathematics, National Yang Ming Chiao Tung University.