傳播數學知識．促進數學教育

Interview with Prof. Constantine M. Dafermos

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

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

**Interviewee:** Constantine Dafermos (CD)

**Date:** October 25th, 2013

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

Prof. Constantine Dafermos was born on May 26, 1941 at Athens, Greece. He obtained a Diploma in Civil Engineering from the National Technical University of Athens in 1964 and Ph.D. in Mechanics from Johns Hopkins University in 1967. He was a faculty at Cornell University, and since 1975, has been at the Division of Applied Mathematics at Brown University. His broad research interests include nonlinear hyperbolic systems of conservation laws and mechanics. He was awarded Norbert Wiener Prize in Applied Mathematics and elected to the National Academy of Sciences in 2016.

TPL: First, it is very nice that you have traveled such a distance to come here.

CD: It’s my pleasure, as you can imagine.

TPL: I would like to start with a special question for you, namely, the ancient Greeks as we know are just extraordinary, Euclid and others: why is that? What made that to happen?

CD: This question has been asked by many people and there are theories, even theories that are politically or ideologically motivated. Contributing factors include that the economic conditions in Greece were improved and that technology had opened the way for Greeks to travel to Egypt and Babylonia, meeting people who had their own civilizations. However, in my opinion, the principal factor was the high level of intellectual energy that prevailed in the country. This phenomenon is often encountered in history—think of 16th- and 17th-century England, a small fraction of the world population that produced giants like Shakespeare and Newton.

TPL: Darwin.

CD: Darwin came later, when England had already become an empire. However, in the 17th century, England was still small, isolated and not very wealthy, yet that high intellectual energy somehow emerged and shone. I am certain that you have had many similar periods in Chinese history.

TPL: Yes, the Confucius time.

CD: So this explains what happened in Greece, in particular, with Euclid in mathematics, as you mentioned. Of course this started in Greece proper in the classical period, but people like Euclid and Archimedes did not operate inside Greece but in places like Alexandria and Sicily. Alexander the Great had established an empire and it seems the most energetic and creative Greeks left the mainland and moved to the colonies.

TPL: You were comparing with 17th-century England, it’s true that they produced great writers, scientists; I am just thinking that in Greek mathematics, it was not only the depth and breadth, but it was unique.

CD: Yes, but of course the world was smaller then. Also there is some bias in our evaluation because for various reasons, historical and otherwise, we have embraced the classical Greek civilization in particular and have built our scientific culture on it. There is no reason that that should necessarily have been the case, but we glorify the classical Greek mathematicians because we imitated their values, we have adopted their style for doing mathematics.

TPL: I see, we are their descendants. Take for example Confucius, who did similar things as Euclid did. Confucius synthesized and then brought out certain core values in ethics. But Greeks did it in that abstract way—ethics are one thing, but the axiomatic approach is another. There was nothing like this axiomatic approach before the Greeks.

CD: That’s right.

TPL: So that’s fantastic, isn’t this unique?

CD: Perhaps we feel it is fantastic because it is compatible with what we ourselves are doing today. Alternative legitimate civilizations care only about the facts and consider axiomatics a waste of time.

TPL: Could I bifurcate from this to rational mechanics--which is in some sense an attempt to axiomatize mechanics. Is that the right way to put it?

CD: Yes, but I think that the more precise statement would be that the leaders of a movement—because it was a movement—they named it rational mechanics, opted for axiomatization. The history behind this is as follows. Throughout the 19th century and before, theoretical mechanics was considered a branch of mathematics. However, in the 20th century, mathematicians and physicists gradually abandoned mechanics, as they considered it an exhausted, mature theory. On the other hand, engineers kept doing research in mechanics, without necessarily adhering to mathematical rigor. So then Truesdell 1 appears on the scene. He felt that his life’s mission was to restore mechanics as a branch of mathematics. He named this movement “rational mechanics,” reviving a term used earlier in France and in Italy. This irritated the engineers who interpreted it as a statement that their own research was “irrational.” Of course this was not what Truesdell had in mind. In their effort to place mechanics in the framework of pure, rather than applied, mathematics, Truesdell and his associates overemphasized axiomatics and this contributed to the eventual decline of the movement.

TPL: Mechanics within the mathematics community of course is alive and well. There is a journal that Truesdell started, Archive for Rational Mechanics and Analysis. The journal is also doing very well. Can you comment on the evolution of this journal? You are very much involved.

CD: Absolutely. In the beginnings, when Truesdell was professor at Indiana University, he founded a journal with the title Journal of Rational Mechanics and Analysis. After Truesdell’s departure from Indiana, this journal changed names a number of times and is now called Indiana University Mathematics Journal. Subsequently, in the late 1950’s, Truesdell founded the Archive for Rational Mechanics and Analysis. The original mission of both these journals was to promote the program of incorporating rational mechanics into the body of traditional mathematics by publishing side by side papers in analysis and papers in mechanics, and if possible papers that combined both subjects. Papers in rational mechanics dominated the early issues of the Archive, because at that time this journal provided the only venue for publication of such papers. However, gradually analysis gained the upper hand in the Archive as well, with involvement of analysts such as Serrin2. The current editors of the Archive are trying to restore the balance between the two subjects by encouraging the submission of more papers in mechanics.

TPL: You have a background in engineering. Your talk this morning was very analytical. I enjoyed it a lot, it was very nice analysis. So personally, how do you view your own research, in terms of mechanics and analysis?

CD: Of course what happened to me was not atypical. In the 1950s, as Europe was developing after the Second World War, engineering was a very profitable profession and there was high prestige associated with being admitted as a student by an engineering school. As a result, there was pressure from families on their children to attend engineering departments. In high school, my dream was to study physics, but since I was a loyal son I followed my parents’ advice and entered a technical university. Frankly I did not enjoy engineering courses, but as I was a proud and conscientious student, I put in a lot of effort and did well. However, as soon as I graduated I grabbed the opportunity to come to the US for graduate studies in more theoretical fields. Naturally I had to pick an area that had some connection with my undergraduate background, and continuum mechanics fit that bill. This kind of path was very common for European students of my generation, and it is still common for young students coming from developing countries. They typically settle in areas on the interface between engineering and mathematics, such as fluid dynamics, control theory, theoretical computer science, etc.

TPL: But then when you came to the U.S., you gradually became more mathematical, right?

CD: Yes. I enrolled in the small, graduate-only Department of Mechanics at the Johns Hopkins University. There were few courses offered in continuum, solid and fluid mechanics, so the graduate students were encouraged to take courses in the mathematics department. I loved the idea, but I naïvely believed that my mathematical background from the engineering school was adequate, so I rushed to register for a number of advanced courses. I had to immediately drop the course on group theory. I did survive the course on differential geometry, but I did not learn much, as I lacked the prerequisites. On the other hand, I loved the course on complex analysis, taught by Kodaira3, and learned a lot from it. The following year, I took a course on real analysis, taught by Philip Hartman4. He was a great teacher and he is probably responsible for my falling in love with analysis. I also sat in on a number of lectures on linear elliptic PDEs given by Gaetano Fichera (

TPL: Ah, the mixed type guy), who was visiting from Rome, and I was fascinated by the subject. The above set my future course, as I selected, by myself, a thesis topic in the PDEs area (on the system of equations of linear thermoelasticity). Unfortunately, none of the professors in my department had interest or expertise in PDEs, so I was left on my own devices and had to reinvent the wheel. For instance, I was astounded and proud to have discovered an argument in a proof, until I later realized that it was just the standard notion of compactness!

TPL: That is very interesting, please explain it in a bit more detail.

CD: Well, in my thesis I was dealing with sequences of almost periodic functions and it was crucial to construct convergent subsequences. I was very proud to be able to do that, by using very primitive tools, but belatedly I realized that it was just a consequence of the standard compactness theorem for almost periodic functions.

TPL: Another person besides Truesdell, whose name is….

CD: Jerald Ericksen6?

TPL: Yes, yes.

CD: Jerry Ericksen was my formal thesis advisor, but in a sense Truesdell served as co-advisor. Ericksen wrote a thesis under the direction of Gilbarg7 (he was a classmate and academic brother of Jim Serrin) at Indiana University, during the time that Truesdell was a professor there. He worked for a number of years at the Office of Naval Research before reuniting with Truesdell at Johns Hopkins. Truesdell and Ericksen were very different: Truesdell’s style was precise and formal, while Ericksen’s approach was more intuitive. I feel that I learned a lot from both attitudes.

TPL: I don’t know Ericksen well, I was in his house once. He is very gentle, looks like a philosopher.

CD: He is not philosophical, but he has the appearance of a sage. He is gentle in the sense of never raising his voice, he is laconic, in that sense he resembles the mythical oracle of Delphi in ancient Greece. According to tradition, the oracle always made ambiguous prophesies, which could be interpreted however people wished. Similarly, Ericksen avoided making sharp scientific statements. When comparing A with B, he would say that most likely A equals B but then there may be valid arguments pointing to A being bigger than B, and, who knows, in the end it may turn out that A is less than B! He never made definitive statements.

TPL: Ericksen has done fundamental work.

CD: Absolutely! He made fundamental contributions to the theory of nonlinear elasticity, non-Newtonian fluids, and liquid crystals. In particular, it is his work on liquid crystals that is currently popular and will probably survive the longest.

TPL: He wrote down the functional?

CD: He wrote down the conservation laws of liquid crystals theory in equilibrium state.

TPL: I would like to trace back, during your years as a student growing up in Greece: what happened to you in that period? You grew up in a nice family, as I understand, and you were a good student. What did your father do?

CD: Both my parents were high school teachers of classical Greek, so perhaps in the back of their minds was that I should follow in their footsteps. Thus as a child I was heavily exposed to classical Greek, Latin, and history, while mathematics was de-emphasized. In addition, my math teachers in elementary school were not that good, so mathematics used to be my weakest subject. It was only when I was first exposed, at the age of 14, to Euclidean geometry that I fell in love with mathematics. So my early background was more in the classics and history rather than in the physical sciences and mathematics.

TPL: That’s very interesting. Once you wrote a paper “Elasticity is special”; I would say you are special, so this background may shed some light on that. Over the years, mathematics has developed so much--for instance Analysis, over the past decades, even before your time, has developed to a rather high degree of sophistication. How do you view the current state of Analysis?

CD: To tell you the truth, I am worried not only about the state of Analysis but also about the state of mathematics as a whole, for the following reason. As it becomes more technical, it resembles the biblical tower of Babel in which the builders started speaking different languages, communication among them ceased, and the project was abandoned. In mathematics, are we going to break into clusters of people who are doing fascinating, sophisticated work, but in such a narrow field that it is incomprehensible to other clusters? In earlier generations of mathematicians, a lot of progress resulted from cross-fertilization among different fields. Are we going to lose that? On the other hand, I am fascinated by the work of the young generation of mathematicians who are moving into our field.

TPL: Back to our conversation about your journey. So after your graduate studies you went to Cornell and then Brown. Brown in particular has been an important center of applied mathematics. In fact, it seems to me that the center of applied mathematics was there even before the Courant Institute.

CD: Both institutions emerged at about the same time. Of course the name applied mathematics was not attached officially to the Courant Institute—it was the full mathematics department at NYU. By contrast, the Division of Applied Mathematics was totally independent of the mathematics department at Brown. To a certain extent Brown had a monopology on the title “applied mathematics.” At the same time, the topics covered at Brown in the 1950s were mainly elasticity and plasticity, which were popular among engineers but did not have an impact on the mathematical community comparable to the research on PDEs conducted at Courant, during the same period, by Friedrichs, John, Lax, Nirenberg, etc.

TPL: Lefschetz 8 was there.

CD: Here is the history of Lefschetz’s connection to Brown. He was primarily a topologist but late in his career he got interested in ODEs and control theory. He was of Russian extraction and he was aware of the Russian contributions to that field. In the 1950s, after retiring from Princeton, he persuaded the Martin-Marietta corporation to set up a research institute in Baltimore for the purpose of contributing to the space effort. There he built up a group of young mathematicians, some of whom, like Bellman9, Kalman10, Hale11, Peixoto12, etc. eventually became leading figures. In 1963 there was an internal fight within the Division of Applied Mathematics at Brown and most faculty members left. The university then invited the Lefschetz group to fill the void. This changed the research emphasis in the Division from mechanics to dynamical systems and control theory. Lefschetz himself did not take up residence at Brown but was visiting regularly in order to provide leadership and guidance to the group.

TPL: So external factors, Russian and so on, were important impetuses for that development.

CD: Most importantly, Brown was the first place that brought into the United States famous Russian mathematicians like Pontryagin13 at a time when it was very difficult to have scientific exchanges between the two countries.

TPL: Oh, Pontryagin came.

CD: Yes, and many other mathematicians, like Mitropolskiy14, working in dynamical systems and control theory.

TPL: At that time the Russians were really doing the best work in dynamical systems, weren’t they?

CD: Yes, especially on the applied side.

TPL: Let me make a digression. You know that culture is important. Every Greek elementary school student must have been told about Greece’s glorious past. How does this affect modern education in Greece?

CD: Very interesting question! The Greek state has made a concerted effort to infuse Greeks with pride in their glorious past. This has generated three classes of Greeks: those who study the ancient history and are sincerely proud of the accomplishments of their forefathers; those who, out of reaction to the indoctrination by the state, totally reject, and make fun of, the old glory of Greece; and finally those who exploit, often hypocritically, the connection with ancient Greece, for political gain. I don’t know whether there are similar phenomena in Chinese society.

TPL: I am sure an analogy can be drawn for Chinese society. Let us move a little bit toward your more recent activities. You have written this book [Hyperbolic Conservation Laws in Continuum Physics, Springer Verlag]: I would say it is heroic, your book on shock waves. You have spent a tremendous amount of time on it.

CD: Absolutely! Over the decade 1990 to 2000, I did almost nothing else but work on the book. To begin with, I had to read, understand and set in the proper frame a very large number of papers. I also tried to incorporate some original research, not necessarily something major that would warrant writing a full paper, but an interesting complement to the extant literature. There are many such examples scattered throughout the book. This required considerable reflection. In addition to that, I don’t type, so I wrote everything by hand and a secretary typed it. Since I am not a natural writer, who can produce a perfect draft in a single try, but at the same time I aspire to write as best I can, I had to produce numerous drafts. I estimate that for the 600 to 700 page book I must have written 10,000 pages by hand. That took a tremendous amount of effort.

TPL: That’s heroic, the amount of material there. But tell me, in this process, you read every 19th century paper. Is there something that struck you and really changed your perspective on the subject?

CD: I am sure that many things struck me but cannot point to anything in particular.

TPL: I would imagine that you are like a hermit in the middle ages, you read the 19th century papers, you read the original papers of important people, and then outside there are people going on doing their research and you meet them. In this state of being a hermit, do you have something to tell us, some small instance?

CD: I was particularly impressed by the work of Stokes15, who conceived, for the first time, the idea of a shock. It was tremendously original, for the time, to interpret discontinuous functions as solutions of PDEs. Stokes did not get the credit he deserved because it was pointed out by Kelvin16 and by Rayleigh17 that his shocks were isothermal and failed to conserve energy. Intimidated by this criticism, Stokes abandoned and denounced his own theory. As a result, the creation of the theory of shocks is commonly attributed to Riemann18. However, the original idea is due to Stokes.

TPL: Was Riemann aware of Stokes?

CD: Of course, of course, he referred to Stokes.

TPL: You continue to work on this book project, to update it—you are still working on it?

CD: Well, to tell you the truth, I am thinking of producing another iteration, as a swan song. Of course this would mean years of hard work and loss of sleep.

TPL: While you are looking over the subject, the research is going on, we are going to have the next meeting in Brazil. The meeting itself has changed gradually, don’t you think?

CD: What has changed in these meetings is that they have become broader. They have lost the original spirit, in which, as you remember, the participants knew each other like family. They have now become established official meetings with many participants coming from different places, parallel sessions, etc. Still one gets many lectures on topics in traditional hyperbolic conservation laws but also lectures on the periphery of our field. I believe that in the final analysis this broadening is a good thing.

TPL: Last time in Padova, there were over 300 people. Do you still have students now?

CD: No, I no longer take students, for a number of reasons. To begin with, considering the nature and state of the theory of hyperbolic conservation laws, I would not encourage anyone to get into this subject, unless he/she has the right personality, stamina and analytical strength. Next, if you take a student you have the obligation to see him or her through for the next four years, and there is no assurance that I will be intellectually active, or even alive, four years from now. Finally, after I hit the age of 65 I stopped applying for grants to any agency, so if I were to take a student I would have to go back to writing proposals for financial support.

TPL: Now with the current situation, a lot of people think we have overproduced PhDs anyway. So I have seen a number of people, there are quite a number here even in this meeting, they become pure researchers, scholars, some do that extremely well after retirement, actually we have very good examples here. So maybe the academic world is changing a little bit. Earlier we saw people who produced so many PhDs. How many PhD students have you had?

CD: About 20.

TPL: You have produced about 20. Other good people have produced more and less, but producing more than 5 is not an exception for good people. In the future, that does not seem to be sustainable.

CD: It’s true. On the other hand, relatively few mathematicians with a PhD write papers beyond their thesis, and a large portion of those who remain active never produce PhD students of their own. This acts like a sort of population control. I would not be inclined to impose quotas on the number of PhD students per advisor any more than I would restrict families to having a single child.

TPL: Have you ever thought of this question: what do you think will be the future of shock wave theory?

CD: I wish I knew the answer! As you know there are many fascinating open problems. On the other hand, these problems are very hard. Sometimes I get discouraged, but I want to be optimistic. I am hoping that someone, probably a young person, will make decisive progress and this will generate enthusiasm for the field.

TPL: I would like to dwell on this narrow thinking. Over the past few decades you’ve been in the business, were there instances when you felt that we seemed to have reached the end of the aisle, and then it turned out that was wrong, people made interesting discoveries and things went on?

CD: To tell you the truth, I never felt that. In retrospect, there were moments when one would have thought that the field had hit a roadblock, but since I loved the subject and wanted to work in it, I ignored these warnings and kept going.

TPL: You seem to give me the impression that this time is a little bit different, you feel this field is getting rather sophisticated and is harder for young people to enter.

CD: I think so. Also, for practical purposes, I don’t see open positions, at least in the U.S.; the area is not considered sexy at the moment. This weights in the minds of students and turns them away from the field.

TPL: One could ask the same question about mathematical sciences in general.

CD: That is true, but the nice thing about mathematics is that it is useful to many people. The danger is that mathematical research may become an exotic subject, limited to relatively few people in elite universities, while mathematical service teaching may be left to cheap adjunct professors.

TPL: Yes, I have heard people talk about this scenario becoming more of a reality.

CD: To tell you the truth, ten years ago I was worried that something like that might happen, but my fear has not materialized. Perhaps, since mathematicians are cheap in that we don’t need labs, etc., the universities will keep on hiring mathematics instructors as regular faculty members.

TPL: On the other hand, talking about the relevance of mathematics to society, I think it has increased. You must have seen the statistics on what are considered the best professions. At the bottom are taxi driver, lumberjack, garbage collector, although some people think these are interesting professions, but in any case the top five are all mathematical--I don’t know exactly what, but they include mathematician, statistician, actuary, accountant, so it seems to me for the well-being of society it is important to make the general public more mathematical.

CD: It is a fact we notice every day at the university. Enrollment in math courses is increasing, students come from biology, economics and other departments that did not use to require lots of mathematics of their students. More and more things are done mathematically, so this requires more people to develop mathematical ability and to think mathematically. I believe the general public is aware of that. On the other hand, the general public thinks that mathematics has been exhausted as a research discipline—biology is discovering new things, physics is discovering new things, but mathematicians just solve equations and teach what is already known.

TPL: This is an old topic. People say that in order to be a good mathematics teacher you have to feel excited about the subject, and to feel excited about the subject you need from time to time to try to do some research. Do you buy that kind of thinking?

CD: Yes, in fact that was also Truesdell’s philosophy not only on teaching mathematics but also on doing research in the history of mathematics. The history of mathematics has undergone a great transition over the past fifty years. In the past, the emphasis was on mathematical creation—why did Gauss prove such and such a theorem, what was the program of the French school, etc. By contrast, recently the emphasis lies on the social effects of mathematics—how social conditions affect mathematical research. Truesdell rejected that trend. He professed that the only people who can do legitimate research in the history of mathematics are mathematicians who have done mathematical research themselves—not necessarily great researchers, but people with first-hand experience of what research on mathematics is all about. I think he was right.

TPL: He started a journal.

CD: Yes, it still exists, called the Archive for the History of Exact Sciences.

TPL: Is that journal doing as well as the Archive for Rational Mechanics and Analysis?

CD: Well, of course, it is much smaller, because its audience is much smaller. I don’t know exactly, it is not considered as the leading journal on the history of mathematics, perhaps because professional historians felt offended by Truesdell’s pronouncements. I used to read it but after Truesdell passed away I stopped reading it so I don’t know how it has evolved.

TPL: Education is a complex subject and for a leading mathematician to talk about mathematics education may be more difficult because for them mathematics is so easy. When we talk about mathematics education we mostly should be thinking about how to teach students who are not so talented mathematically.

CD: From time to time I see in the Notices of the AMS articles by certain research mathematicians who late in their careers have got involved in teaching mathematics to high school students. However, I have not followed this literature so I don’t know.

TPL: Perhaps I have exploited you kindness too much, let’s stop here and continue another time.

CD: Thank you very much, it is wonderful to be here with you.

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