傳播數學知識．促進數學教育

Interview with Prof. François Golse

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

**Interviewer:** Tai-Ping Liu (TPL)

**Interviewee:** François Golse (FG)

**Date:** March 26th, 2009

**Venue:** Institute of Mathematics, Academia Sinica

Prof. François Golse was born on September 10, 1962 in Talence. He received his Ph.D. in1986 from Université Paris VIII. In 1987, he became a researcher at the Centre National de la Recherche Scientifique (CNRS) and conducted his research at Ecole Normale Superieure in Paris. Since 1993, he has been a faculty at Université Paris VI and at Ecole Polytechnique. For his distinct contributions in mathematical physics, he has won Louis Armand Prize from French Academy of Sciences, Claude- Antoine Peccot Award from Académie française and the first SIAG-APDE Prize of SIAM (for the best work on partial differential equations).

TPL: This workshop is only for three days but I was surprised to see there is so much going on in the kinetic theory. Your research started with kinetic theory or not?

FG: Well, actually, I would say that my very first interest, domain in research, was conservation laws. I remember that I attended a course by Luc Tartar when I was a student, in l'Ecole Normale Supérieure in Paris, in 1983, I guess. And it was a fascinating course, you know how Luc is. Then I asked him to be my Ph.D. advisor, but at the time, although still living in France, he was quitting the Orsay university. First, he told me to work on discrete kinetic models and, later on, he advised me to work with Claude Bardos. That’s how I started in kinetic theory. It was Claude Bardos who had me work on the Boltzmann equation itself.

TPL: At one time I was around Paris for some reason, and you got a prize when you were very young.

FG: Yes, I got a prize from French Academy in 1988 I think. It’s a prize for very young researchers. I think you have to be under the age of 30 - I’m not quite sure.

TPL: And how old were you?

FG: Hum … 26 if it really was in 1988. As I said it’s a prize reserved for young people.

TPL: But you did this Averaging Lemma before that?

FG: Yes, it was just before that. Actually, how we arrived at this averaging lemma is quite funny.

TPL: O.K.

FG: We did it with Benoît Perthame while we were both doing our national service (military service in France still existed at the time). I was lucky enough to do the national service at the same time as Benoît Perthame, and we were sharing the same office and, working under the supervision of Rémi Sentis. We were in a group of very interesting people, doing a lot of good work on radiative transfer and neutron transport. The head of this national lab was Robert Dautray - more precisely I think that Robert Dautray was the scientific director of the French Atomic Energy Commission at the time. He and Henri Cabannes were really influential in the development of kinetic models in France and among French mathematicians. At the time, kinetic theory was quite new in the French applied mathematics community, but in the labs operated by the Atomic Energy Commission, of course, there were many specialists of kinetic theory, especially for neutron transport and radiative transfer.

TPL: This Averaging Lemma is such a fundamental thing. It initiated the whole development, including DiPerna-Lions and, your work with Saint-Raymond, right? So can you say a few more words about how the Lemma came about?

FG: Oh, well, as I said, this is a funny story. Let’s see, as far as I remember, during my national service with Benoît Perthame, somebody asked us a question about continuity of temperature in radiative transfer models. The question was whether or not there were shocks for temperature, or whether the temperature was smoother than the radiation field. Now temperature can be viewed as some kind of average, more precisely it is a macroscopic quantity, computed in terms of the average over angles of the radiation field. I remember that Benoît showed me a quick proof of continuity in 1D based on this observation. So that was the problem. Some time later Rémi Sentis asked me a question about ... well, I think it was a question about the principal eigenvalue of the transport operator, more precisely the convergence to the principal eigenvalue of the diffusion operator under the usual diffusion scaling regime. Then I thought of generalizing to the higher dimensional setting this argument that Benoît showed me. Benoît was not there at the time for some reason, which I don’t remember, and when he returned, we worked on that question, and it was the beginning of our interest in the Averaging Lemma, which was published afterwards. I think it was included in the appendix of last chapter of this series of books by Robert Dautray and Jacques-Louis Lions, and in a note at the Comptes Rendus at the French Academy of Sciences. What was funny was that, the book was in the process of being published, but they noticed one proof that was incomplete, and that lemma was needed to conclude. You know, I think it was the nicest way to do one’s national service in France at that time, probably. In that sense, I have been very lucky; most people could be sent to boot camp doing regular national service or otherwise work on not so interesting stuff, but I had the good fortune of being in a very good environment.

TPL: I recall that DiPerna was actually in the last couple of years of his life when he worked on Boltzmann equations. He gave his every ounce of his energy pushed through that work with Pierre-Louis Lions. And then there was a meeting in L’Aquila and Arkreyd was there. He was reading the DiPerna-Lions. DiPerna arrived later in the meeting. Arkreyd made the following statement: he said that “if I knew of the Averaging Lemma, I should be able to obtain the DiPerna-Lions result,” and he went on to say that “why it should be such a long and winding argument?” I suppose you don’t want to make comment on his statement.

FG: Say it again? Do you mean, why is the DiPerna-Lions proof so long once you use the Averaging Lemma? You see, there is an intrinsic difficulty. After you know the Averaging Lemma, then you can pass to the limit in the collision operator, renormalized by some averaged quantity. But this renormalization by an averaged quantity doesn’t mix with the left-hand side; the only thing by which you can renormalize nicely is by a quantity that is not averaged, right? In other words, a function of $F$, not a functional of $F$. So there is this question of exchanging the averaged and the non-averaged renormalization, which is actually a quite technical part of the work of DiPerna-Lions. They managed to solve this difficulty by solving along characteristics. I think in the process of working out this part of their theory of renormalized solutions, they also discovered the theory for treating vector fields with low regularity. So I think it would be unfair to neglect this part of the construction of the solution of the Boltzmann equation. This part of the DiPerna-Lions paper, maybe you can say it’s less spectacular because, you know, it looks like a classical argument involving super- and subsolutions, maybe you don’t see where are the new ingredients. But really I think it was the seed for their theory of renormalized solutions of transport equations for vector field with low Sobolev regularity, which has produced lots of magnificent results. I mean, you have their paper in Inventiones, and the newer paper by Luigi Ambrosio in the case of BV flows, also appeared in Inventiones. Of course, the Averaging Lemma is fundamental in the DiPerna-Lions paper, but even after using the Averaging Lemma, important ideas are needed to conclude the proof.

TPL: That’s very beautiful. You could not possibly imagine that this Averaging Lemma has such a great impact, right?

FG: Not really. We understood that it was something useful but to which extend it was useful was not clear at that time. OK.

TPL: There are so many great mathematicians in Paris. I’d like to ask you to say something about that. Take for example Jean Leray, what kind of person is he?

FG: I was very young when I met him. Once, Claude Bardos had arranged an interview with Leray13. Certainly, Leray was a fascinating person. When I was a student in l'Ecole Normale, even before I met Claude Bardos, I think, I indicated to Louis Boutet de Monvel, who was the director of math in Ecole Normale at the time, that I was interested in hyperbolic PDEs. And he said: “OK. There is a series of papers by Leray on hyperbolic problems which you should absolutely read, because Leray is one of the greatest mathematicians in the 20st century.” This sounded a bit strange to me: indeed, I had heard of Leray, because of his work in topology - for instance, during my first year at Ecole Normale, I had attended a couple of talks about sheaves and cohomology, where the name of Leray was mentioned of course, for instance about acyclic covers. Anyway, at the time, I thought of Leray as a topologist, I had no idea that Leray was an analyst. Later on, I discovered that Leray was attending the Lions Seminar in Collège de France, and sometimes would make very interesting comments about the talks. He would not talk so often, but he was somebody with an impressive intellect, and that showed when I talked to him with Bardos, I remember that. I have had the chance of meeting some great mathematicians, but Leray seemed to be moving in a different dimension. His work had renewed several branches of mathematics - sheaves, spectral sequences and more generally homological methods are the fundamental tools of topology, algebraic geometry, microlocal analysis following Sato; on the other hand, Leray's work on nonlinear PDEs, with the importance he gave to the notion of weak solutions, and to the problem of regularity and uniqueness, pretty much defined the field as we see it today. So I fully appreciated that Leray was a great historical figure in the world of mathematics, that's why I was very impressed.

TPL: Who are the other important people around?

FG: Oh, there were lots of important people, already among my professors. For instance, Pierre-Louis Lions was giving an introductory course on nonlinear analysis at Ecole Normale Supérieure. As a student at Ecole Normale, I was also taking courses in Paris VI, and most of my professors were really famous mathematicians - although at the time I did not realize that. For instance, Paul Malliavin was a professor in Paris VI at the time, and I took his course on integration and probability. He had a very interesting, a really remarkable book, covering that course. There was also Brezis, of course … Haïm Brezis was a wonderful teacher, he was teaching his course after his book on functional analysis and applications. Both are retired from Paris VI, but extremely active in research. For instance, Paul Malliavin has written since 2006 a couple of very interesting papers on incompressible flows.

TPL: How old is he now?

FG: I think he was born in 1925. He’s very impressive, not only for the depth of his own mathematics, but also for his enthusiasm and interest in other people's seminar talks.

TPL: Why are there so many good French mathematicians?

FG: I really don’t know, I think that there always was a tradition. Perhaps one reason is that mathematics was viewed as a convenient way of selecting good students.

TPL: So, it is emphasized in the high school education.

FG: Yes, it certainly was emphasized in the high school education when I was a student in high school in the 1970s. Being good at math at the time was really “the thing”. I mean, it was really a good investment for your future. From the historic viewpoint, I think that mathematics was always viewed as useful in the French education system. I read that Monge started his course on descriptive geometry at Ecole Normale Supérieure in 1794 with the sentence "Pour tirer la Nation Française de la dépendance où elle a été jusqu'à présent de l'industrie étrangère..." (to withdraw the French Nation from the dependence on foreign industry, where it has stayed until now...) Mathematics was also important for artillery, and Napoleon himself thought highly of mathematics, it seems...

TPL: Yeah, but this reminds me that in the US a lot of researches are done through Department of Energy, basically with military applications in mind. Then they are spinned-off to a lot of industry.

FG: Right. In France I don’t think it was oriented so much towards military per se. After all, before the French Revolution, there had been important French mathematicians (Pascal for instance) and I don't think their work was geared towards military applications.

TPL: Coming back to the topic of kinetic theory, I know for many years you have been trying to study the hydrodynamic limits of the Boltzmann equation.

FG: Well, you told me I could say things a little bit outrageous or funny during our interview. So maybe it’s a good time to do so.

TPL: Yes, yes.

FG: OK. So… after Luc Tartar told me to go see Claude Bardos and ask him for a Ph.D. subject, Claude invited me to his office and he showed me two problems. One problem was about conservation laws, multi-phase flows in the context of oil recovery. He told me “OK, I have a meeting with some people from ELF (a French oil company) tomorrow morning. Why don't you join us and see what it looks like.” The second problem was about the Boltzmann equation. Claude wrote the Boltzmann equation on the board, and he mentioned the problem of going from the Boltzmann equation to the compressible Euler system. So he said “you know in this project there is theorem 0, which would be to prove global existence of solutions for Boltzmann equations, independently of the size of initial data; and theorem infinity would be to show that these solutions converge to solutions of the compressible Euler system, which would in particular provide a way constructing global solutions of Euler.” There was a set of lecture notes by Nishida published by the University of Orsay in the 1970s, which are actually quite interesting. Claude told me “read that, come tomorrow to see the guys from ELF, and then you choose whichever subject you prefer.” So the next morning I attended the lecture by the engineers doing oil recovery. Now, the only thing I knew about conservation laws was the Tartar-DiPerna compensated compactness method for $2\times 2$ systems in space dimension 1. In other words, I was desperately ignorant and naive. At some point in the talk I was lost, the guys were manipulating solutions of their monstrous hyperbolic system, involving hydrodynamic fields for oil, gas, water and what not… I didn't have the faintest idea of what was going on, so I asked them: “can you explain to me what the boundary conditions are?” To which one of the guys replied "Boundary conditions?! haha, we don't even know the domain..." So, I thought to myself: I'd better work on the Boltzmann equation. At least it seemed to me to be a much better defined problem. In retrospect, it was quite a strange way to be introduced to one's future field of research.

TPL: But oil recovery equation also can be very ill-posed, right?

FG: Yes, but you know at that time, I could not appreciate that … as I said, I was too ignorant to appreciate that.

TPL: But the Boltzmann equation, of course, is not a piece of cake, right?

FG: As I soon discovered...

TPL: So how, how did it go? This lasted over ten years, this engagement.

FG: More than that, I started working on my Ph.D. in 1984, 85 … I don’t remember, but maybe around 1984 …

TPL: You were very young at that time. When did you finish your Ph.D.?

FG: 24. I don’t know if you call that a PhD, but my first thesis in France was in 86.

TPL: OK, so when you were 24.

FG: 24, yes. Well, at the time we were under pressure to defend Ph.D. rather quickly because there were no post-doc positions in France at the time and so it was better for you to be ready to apply for a real position in university. So, there was some pressure to finish quickly.

TPL: I remember the first paper on the hydrodynamic limits. The theorem has many hypotheses, quite a number of hypotheses, then gradually dropped one by one, eventually it finishes with a definitive result with Saint-Raymond. How would you describe the process, such a long process?

FG: Hum … I think at the beginning I was inspired by an idea which I read I think in a paper by Henry McKean, suggesting that it was possible to relate the incompressible Navier-Stokes to kinetic theory. At the time I didn’t know the work of Sone sensei (Yoshio Sone) on that problem. But it’s Dave Levermore who explained to us – us meaning Claude Bardos and I – this idea of scaling the Mach number and Knudsen number together. At the time with Dave and Claude we were really convinced that, with this scaling, the formal theory, and the DiPerna-Lions framework, applying the velocity Averaging Lemmas in the way that we had used it on other problems like radiative transfer problems, it was possible to relate DiPerna-Lions solutions to Leray solutions of Navier-Stokes. Because these two theories … they look very similar, in particular the role of dissipation in both theories is really very much the same. So Claude was firmly convinced that these theories were, so to speak, sister theories. Then, when we tried to prove things, we saw that there were several difficulties that would resist us, you know, but after some time we reduced this problem to a list of three assumptions. This was our first paper, which came out in the Communications on Pure and Applied Mathematics. Afterwards, these assumptions were gradually removed… perhaps not in the order that we thought they would be removed initially, but they were eventually removed.

TPL: That must be a great satisfaction.

FG: Yes … yes of course. But I was never working full time on this project… so I never had any frustrations waiting for so long. I think between the first time we thought about this problem with Claude and Dave, which must be back in 1988, and the time that we understood with Laure Saint-Raymond how to remove the last obstacle on the road... which must be around the beginning of 2001, it’s more than 10 years. If I have worked on that problem constantly for more than 10 years, I would become mad, at least fell discouraged. But I was never working constantly on that problem, so I never had negative feelings about it.

TPL: Actually, I’ve been quite impressed with the breath of your research ... you worked on many different things … as I said it the other day... I’ve heard you given two excellent talks just within the last few months, one in shockwave theory on large-time behavior and this talk here. The French mathematicians start from rather young, they try to acquire certain breadth in their research interest.

FG: I think having broad interests in research is something valued very much by Claude Bardos … He’s himself mathematically broad. Being his student, I think it was natural form to try to imitate him in that respect. So I tried my hand at different kinds of problems. Besides that, I enjoy doing mathematics with different people. For me, doing mathematics is some kind of social activity. So, I like working with other people. I don’t like so much working by myself... more precisely; I don't get the same kind of enthusiasm as when I work with other people. Perhaps working with many different people is also a way to work on very different problems.

TPL: In fact, you are one of the very few people working with this group in Kyoto … Sone Sensei, Aoki and so forth. They are not exactly in the same scientific components as most analysts. You are one of the few people able to communicate deeply with them.

FG: I think Sone sensei visited Paris in the late 80’s… He was invited by Professor Cabannes, with whom Claude Bardos was in very friendly terms. That’s how I met Sone sensei. At the time there was the Hermes (European space shuttle) project in France, which still existed in the 80’s, but was eventually abandoned in the early 90’s. Thus at that time, lots of people were working on the Boltzmann equation. We also were interested in this project, and cooperating with some researchers from the Dassault company. It’s at that period which I met with Sone sensei who explained his theory of half-space problem with evaporation or condensation at the boundary. At the beginning I could not understand the essence of the problem. Professor Sone was very patient, repeated the story to me for many years. I have a slow mind because I want to understand things in depth, and for me it could take many years. Okay. But I knew from the beginning it was something very important. So it was clear to me that it was worth investing in that direction. Since I've known Professor Sone, I've had the impression that, whatever I could tell him about the Boltzmann Equation, he had known it for more than 30 years.

TPL: This is a very nice way you put it … of course your admiration of Sone Sensei is a mutual one.

FG: I am much honored, of course.

TPL: Now, the part of the analysis that I know of French schools, every generation there are some very bright young researchers coming up, every so often. This is amazing to me, it has a constant renewal.

FG: Let’s hope you are right on that.

TPL: Yes, so far…

FG: So far so good …

TPL: Just in this workshop there is this young Clement Mouhot, whose former advisor Cedric Villani was here in Academia Sinica some years ago, when he was also very young.

FG: But nowadays, I think there the interest in science is decaying among the students, in the younger generation. So I don’t know if what you are describing will continue; but I do hope so.

TPL: Perhaps a generation before you would say the same thing about the young generation. This seems to be a common worry. In spite of this worry, through the generations, French mathematics stays strong.

FG: I hope it’s so…

TPL: I remembered that there was a European alliance on conservation laws around 1986.

FG: Yes.

TPL: Some French mathematicians said that they should catch up with US on conservation laws. But now Europe is strong on conservation laws. Also before you and others get involved in Kinetic Theory, Kinetic Theory was done first after the war in Courant Institute and then in Japan, then the French were really the ones doing it since then.

FG: I think that Professor Cabannes after WWII was probably one of the first French mathematicians working in Kinetic Theory, in discrete Kinetic Theory. I say "mathematician" although formally, he was a professor in a department of mechanical engineering. On the other hand, there was also Yvon… you know Jacques Yvon is the Y of the BBGKY hierarchy. But Yvon was a physicist; I don't know how much he did in Kinetic Theory per se. But I would say that among people close to the French mathematical community perhaps Cabannes was the first, and also Jean-Pierre Guiraud26 who was a professor in Paris VI, in the same department as professor Cabannes. However, in spite of their efforts, it took some time before kinetic models became familiar objects in the Lions school, for instance.

TPL: There are … all over the world people are talking about biology…

FG: Yes.

TPL: France, of course, has been very strong in biology. It goes all the way back to the 19th Century. Is the mathematical community making serious effort in this?

FG: Yes, for the applied mathematics community, biology is now one important source of inspiration. You can imagine that with someone like Perthame working on this field, it will attract very good young people and working on very interesting problems. I am not being involved in that myself. But one of my former students, Marie Doumic, is working in Perthame's group. I think it’s a very active and very interesting field of research.

TPL: There is an article by Freeman Dyson, recently appeared in AMS notices, he talked about the French tradition and the English tradition. The French tradition is the Descartes’ tradition which is more global and more abstract. So he classifies this as a bird which flies high and see far. Then there is the English tradition, the Bacon’s tradition, emphasizing the experiences and the concrete problems. Dyson says this is like a frog which stays between the bushes but goes deep and walk on one problem at a time. Never mind about this, I mean, you can classify anything into two kinds. But people have certain impressions when we talk about French mathematics, for example, the Bourbaki. Is there anything you can say about the French mathematics, certain characteristics of it?

FG: It’s a complicated question... well, for one thing, I think that Bourbaki corresponds with a period of the history of French mathematics. Bourbaki was started between the two World Wars by a group of young people. I think that even Leray was among the first people of the project, though he dropped immediately. At the time, I think there was an interest in formalizing mathematics and in diffusing in the French mathematical community the kind of mathematics that was practiced in Germany. The French mathematical community suffered a lot from the loss of many young men who died during WWI. Perhaps the same could be said on the German side, I don't know really... The fact is, there was a very impressive school of Mathematics in Germany between the two Wars, with people like Hilbert, Weyl, Hasse, Siegel, Courant... I think the German school of mathematics was a source of inspiration to André Weil, who was one of the founders of Bourbaki. Now, you know, Bourbaki was not all of the French mathematics. When people think of French mathematics as being exclusively Bourbaki, I think it’s very misleading. French mathematics existed before Bourbaki. Think of Hadamard for instance: he had incredibly wide interests in mathematics ranging from PDEs - we all know his work on the wave equation, and more generally on hyperbolic equations - to number theory, and including the geometry of surfaces with negative curvature, which play such an important role in ergodic theory.

TPL: There is a recently published biography on Hadamard.

FG: So, the French mathematics as being too abstract because of Bourbaki, maybe it's a misrepresentation. But it’s true that Bourbaki was extremely influential, in the 50s, 60s and 70s. There is also the Lions school, which established applied mathematics in France. It was extremely successful, but maybe some people viewed the Lions school of applied mathematics as being some kind of a Bourbaki variant of applied mathematics. I mean, it’s true that there are certainly differences between applied mathematics in the sense of Lions and what you would call applied mathematics in the UK for instance. There is a different choice of method, and there is clearly a different taste. Coming back to French vs. British science, I'd like to mention Voltaire's Philosophical Letters which he wrote in England in the 1730's. In letters XIV-XVII, he compares the scientific achievements of Descartes and Newton. Although Newton's theories were based on careful experiments - think of Newton's Opticks, for instance - Voltaire makes it very clear that it would be unfair to think of Descartes and Newton in the terms suggested by Dyson's paper (the eagle and the frog). As a matter of fact, the great level of generality in Newton's differential and integral calculus, as well as in his principles of dynamics have greatly contributed to the "unreasonable effectiveness of mathematics in the natural sciences", to quote Wigner's famous title.

TPL: You said that when you talk to Sone sensei you immediately sensed importance of it, and so you are taking a very slow approach. The breadth of your perspective is impressive. One thing I find quite remarkable is that young French mathematicians all give excellent talks. This is rather amazing. Why is this?

FG: Well, first of all, I shall say it’s not true: there are famous counter-examples, which obviously I am not going to disclose... Besides, the ability to give excellent talks is not confined to French mathematicians.

TPL: But I think that the average norm is higher there

FG: Well, going back to Sone sensei. For instance, I wish to say that one of the persons who really helped me a lot in understanding what was being done in Kyoto school was Aoki sensei. Aoki sensei is one example of somebody who gives excellent talks.

TPL: Yes, indeed.

FG: Aoki Sensei has a very clear and deep mind, when he says something, it’s always important. Now about French mathematicians, I think it’s part of the training in France that mathematics should be crystal clear. Maybe it’s a more recent feature of the French mathematical education. I said I'm reluctant to give counter-examples of French mathematicians being poor lecturers... yet there's at least one example that I feel comfortable mentioning. Since I became a professor at Ecole polytechnique, I have had to produce lecture notes. At first, I was furious about that because it's very painful: I had to write more than 300 pages in a few weeks... At some point I thought it was because the students' revolt in 1968. I assumed that before 68, maybe writing these lecture notes was not mandatory. But apparently it was. The story (legend?) is that Cauchy was very hard to follow as a professor of analysis in Ecole polytechnique, and so the students forced him to provide a write-up of his lectures.

TPL: So Cauchy was in Ecole polytechnique?

FG: Yes. So you see, even outstanding mathematicians like Cauchy are sometimes not extremely good at giving lectures. What can I say? I believe that there is no general rule, no correlation between having a talent for creating or discovering important mathematics, and being good at explaining your work to other people. Some mathematicians are blessed with both talents: Jacques-Louis Lions was an outstanding lecturer, and so is Pierre-Louis Lions.

TPL: There are many very good lecturers, Cedric Villani, for instance.

FG: Absolutely. Cedric is very enthusiastic also. He’s not only a good lecturer, he’s also extremely enthusiastic, and able to convey his enthusiasm to his audience. That's capital.

TPL: This time you sort of are squeezing your time to come to Taipei. Next time you come for a longer period. Sorry that you caught a cold, but you are okay now?

FG: Yes, yes… it’s not a big deal… just a cold which I caught on the plane.

TPL: So maybe we would stop here… and you would come back and stay a bit longer …

FG: Yes …. I most sincerely hope so.

- Tai-Ping Liu is a faculty member at the Institute of Mathematics, Academia Sinica.