Interview with Peter D. Lax

Martin Raussen and Christian Skau

Peter D. Lax is the recipient of the 2005 Abel Prize of the Norwegian Academy of Science and Letters. On May 24, 2005, prior to the Abel Prize celebrations in Oslo, Lax was interviewed by Martin Raussen of Aalborg University and Christian Skau of the Norwegian University of Science and Technology. This interview originally appeared in the European Mathematical Society Newsletter, September 2005, pages 24–31.

Raussen & Skau: On behalf of the Norwegian and bued with the utter importance of computing for Danish Mathematical Societies we would like to con- science and mathematics. Los Alamos, under the gratulate you on winning the Abel Prize for 2005. influence of von Neumann, was for a while in the You came to the U.S. in 1941 as a fifteen-year- 1950s and the early 1960s the undisputed leader old kid from Hungary. Only three years later, in in computational science. 1944, you were drafted into the U.S Army. Instead of being shipped overseas to the war front, you Research Contributions were sent to Los Alamos in 1945 to participate in R & S: May we come back to computers later? First the Manhattan Project, building the first atomic some questions about some of your main research bomb. It must have been awesome as a young man contributions to mathematics: You have made out- to come to Los Alamos to take part in such a mo- standing contributions to the theory of nonlinear mentous endeavor and to meet so many legendary partial differential equations. For the theory and nu- famous scientists: Fermi, Bethe, Szilard, Wigner, merical solutions of hyperbolic systems of conser- Teller, Feynman, to name some of the physicists, and vation laws your contribution has been decisive, von Neumann and Ulam, to name some of the math- not to mention your contribution to the under- ematicians. How did this experience shape your standing of the propagation of discontinuities, so- view of mathematics and influence your choice of called shocks. Could you describe in a few words how a research field within mathematics? you were able to overcome the formidable obstacles Lax: In fact, I returned for a year’s stay at Los and difficulties this area of mathematics presented? Alamos after I got my Ph.D. in 1949 and then spent Lax: Well, when I started to work on it I was very many summers as a consultant. The first time I much influenced by two papers. One was Eberhard spent in Los Alamos, and especially the later ex- Hopf’s on the viscous limit of Burgers’ equation, posure, shaped my mathematical thinking. First of and the other was the von Neumann-Richtmyer all, it was the experience of being part of a scien- paper on artificial viscosity. And looking at these tific team—not just of mathematicians, but people examples I was able to see what the general the- with different outlooks—with the aim being not a ory might look like. theorem, but a product. One cannot learn that from R & S: The astonishing discovery by Kruskal and books, one must be a participant, and for that Zabusky in the 1960s of the role of solitons for so- reason I urge my students to spend at least a sum- lutions of the Korteweg-deVries (KdV) equation, and mer as a visitor at Los Alamos. Los Alamos has a the no less astonishing subsequent explanation given very active visitor’s program. Secondly, it was by several people that the KdV equation is com- there—that was in the 1950s—that I became im- pletely integrable, represented a revolutionary de- velopment within the theory of nonlinear partial dif- Martin Raussen is associate professor of mathematics at ferential equations. You entered this field with an Aalborg University, Denmark. His email address is ingenious original point of view, introducing the [email protected]. so-called Lax-pair, which gave an understanding of Christian Skau is professor of mathematics at Norwegian how the inverse scattering transform applies to University of Science and Technology, Trondheim, Norway. equations like the KdV, and also to other nonlinear His email address is [email protected]. equations which are central in mathematical physics,

FEBRUARY 2006 NOTICES OF THE AMS 223 Now the question is: Are completely integrable systems exceptions to the behavior of solutions of non-integrable systems, or is it that other systems have similar behavior, only we are unable to ana- lyze it? And here our guide might well be the Kolmogorov-Arnold-Moser theorem which says that a system near a completely integrable system behaves as if it were completely integrable. Now, what near means is one thing when you prove theorems, another when you do experiments. It’s another aspect of numerical experimentation re- vealing things. So I do think that studying com- pletely integrable systems will give a clue to the be-

(Photo: Knut Falch/Scanpix) Copyright: Det Norske Videnskaps-Akademi/Abelprisen. havior of more general systems as well. Peter D. Lax was interviewed by Martin Raussen and Who could have guessed in 1965 that completely Christian Skau at the Hotel Continental in Oslo. integrable systems would become so important? R & S: The next question is about your seminal like the sine-Gordon and the nonlinear Schrödinger paper “Asymptotic solutions of oscillating initial equation. Could you give us some thoughts on how value problems” from 1957. This paper is consid- important you think this theory is for mathemati- ered by many people to be the genesis of Fourier In- cal physics and for applications, and how do you tegral Operators. What was the new viewpoint in the view the future of this field? paper that proved to be so fruitful? Lax: Perhaps I should start by pointing out that Lax: It is a micro-local description of what is the astonishing phenomenon of the interaction of going on. It combines looking at the problem in the solitons was discovered by numerical calculations, large and in the small. It combines both aspects, as was predicted by von Neumann some years be- and that gives it its strengths. The numerical im- fore, namely that calculations will reveal extremely plementation of the micro-local point of view is by interesting phenomena. Since I was a good friend wavelets and similar approaches, which are very of Kruskal, I learned early about his discoveries, and powerful numerically. that started me thinking. It was quite clear that R & S: May we touch upon your collaboration with there are infinitely many conserved quantities, and Ralph Phillips—on and off over a span of more than so I asked myself: How can you generate all at once thirty years—on scattering theory, applying it in a an infinity of conserved quantities? I thought if you number of settings. Could you comment on this col- had a transformation that preserved the spectrum laboration, and what do you consider to be the most of an operator then that would be such a trans- important results you obtained? formation, and that turned out to be a very fruit- Lax: That was one of the great pleasures of my life! Ralph Phillips is one of the great analysts of ful idea, applicable quite widely. our time and we formed a very close friendship. We Now you ask how important is it? I think it is had a new way of viewing the scattering process pretty important. After all, from the point of view with incoming and outgoing subspaces. We were, of technology for the transmission of signals, sig- so to say, carving a semi-group out of the unitary nalling by solitons is very important and a promis- group, whose infinitesimal generator contained al- ing future technology in trans-oceanic transmission. most all the information about the scattering This was developed by Linn Mollenauer, a brilliant process. So we applied that to classical scattering engineer at Bell Labs. It has not yet been put into of sound waves and electromagnetic waves by po- practice, but it will be some day. The interesting tentials and obstacles. Following a very interesting thing about it is that classical signal theory is en- discovery of Faddeev and Pavlov, we studied the tirely linear, and the main point of soliton signal spectral theory of automorphic functions. We elab- transmission is that the equations are nonlinear. orated it further, and we had a brand new approach That’s one aspect of the practical importance of it. to Eisenstein series for instance, getting at spectral As for the theoretic importance: the KdV equa- representation via translation representation. And tion is completely integrable, and then an aston- we were even able to contemplate—following Fad- ishing number of other completely integrable sys- deev and Pavlov—the Riemann hypothesis peeking tems were discovered. Completely integrable around the corner. systems can really be solved in the sense that the R & S: That must have been exciting! general population uses the word solved. When a Lax: Yes! Whether this approach will lead to the mathematician says he has solved the problem he proof of the Riemann hypothesis, stating it, as one means he knows the solution exists, that it’s unique, can, purely in terms of decaying signals by cutting but very often not much more. out all standing waves, is unlikely. The Riemann

224 NOTICES OF THE AMS VOLUME 53, NUMBER 2 hypothesis is a very elusive thing. You may re- experimental facts, and, in particular, some nu- member in Peer Gynt there is a mystical character, merical evidence. the Boyg, which bars Peer Gynt’s way wherever he R & S: In the history of mathematics, Abel and goes. The Riemann hypothesis resembles the Boyg! Galois may have been the first great mathematicians R & S: Which particular areas or questions are that one may describe as “pure mathematicians”, you most interested in today? not being interested in any “applied” mathematics Lax: I have some ideas about the zero dispersion as such. However, Abel did solve an integral equa- limit. tion, later called “Abel’s integral equation”, and Abel gave an explicit solution, which incidentally Pure and Applied Mathematics may have been the first time in the history of math- R & S: May we raise a perhaps contentious issue with ematics that an integral equation had been for- you: pure mathematics versus applied mathemat- mulated and solved. Interestingly, by a simple re- ics. Occasionally one can hear within the mathe- formulation one can show that the Abel integral matical community statements that the theory of equation and its solution are equivalent to the Radon nonlinear partial differential equations, though pro- Transform, the mathematical foundation on which found and often very important for applications, is modern medical tomography is based. fraught with ugly theorems and awkward argu- Examples of such totally unexpected practical ments. In pure mathematics, on the other hand, applications of pure mathematical results and beauty and aesthetics rule. The English mathe- theorems abound in the history of mathematics— matician G.H. Hardy is an extreme example of such group theory that evolved from Galois’ work is an- an attitude, but it can be encountered also today. other striking example. What are your thoughts on How do you respond to this? Does it make you this phenomenon? Is it true that deep and impor- angry? tant theories and theorems in mathematics will Lax: I don’t get angry very easily. I got angry once eventually find practical applications, for example at a dean we had, terrible son of a bitch, destruc- in the physical sciences? tive liar, and I got very angry at the mob that oc- Lax: Well, as you pointed out, this has very often cupied the Courant Institute and tried to burn happened: Take for example Eugene Wigner’s use down our computer. Scientific disagreements do of group theory in quantum mechanics. And this not arouse my anger. But I think this opinion is def- has happened too often to be just a coincidence. initely wrong. I think Paul Halmos once claimed that Although, one might perhaps say that other theo- applied mathematics was, if not bad mathematics, ries and theorems which did not find applications at least ugly mathematics, but I think I can point were forgotten. It might be interesting for a histo- to those citations of the Abel Committee dwelling rian of mathematics to look into that phenomenon. on the elegance of my works! But I do believe that mathematics has a mysteri- Now about Hardy: When Hardy wrote A Mathe- ous unity which really connects seemingly distinct matician’s Apology he was at the end of his life, he parts, which is one of the glories of mathematics. was old, I think he had suffered a debilitating heart R & S: You have said that Los Alamos was the attack, he was very depressed. So that should be birthplace of computational dynamics, and I guess taken into account. About the book itself: There was it is safe to say that the U.S. war effort in the 1940s a very harsh criticism by the chemist Frederick advanced and accelerated this development. In what Soddy, who was one of the co-discoverers of the way has the emergence of the high-speed computer isotopes—he shared the Nobel Prize with Ruther- altered the way mathematics is done? Which role will ford. He looked at the pride that Hardy took in the high-speed computers play within mathematics in uselessness of his mathematics and wrote: “From the future? such cloistral clowning the world sickens.” It was Lax: It has played several roles. One is what we very harsh because Hardy was a very nice person. saw in Kruskal’s and Zabusky’s discovery of soli- My friend Joe Keller, a most distinguished ap- tons, which would not have been discovered with- plied mathematician, was once asked to define ap- out computational evidence. Likewise the Fermi- plied mathematics and he came up with this: “Pure Pasta-Ulam phenomenon of recurrence was also a mathematics is a branch of applied mathematics.” very striking thing which may or may not have Which is true if you think a bit about it. Mathematics been discovered without the computer. That is one originally, say after Newton, was designed to solve aspect. very concrete problems that arose in physics. Later But another is this: in the old days, to get nu- on, these subjects developed on their own and be- merical results you had to make enormously dras- came branches of pure mathematics, but they all tic simplifications if your computations were done came from applied background. As von Neumann by hand, or by simple computing machines. And the pointed out, after a while these pure branches that talent of what drastic simplifications to make was develop on their own need invigoration by new a special talent that did not appeal to most math- empirical material, like some scientific questions, ematicians. Today you are in an entirely different

FEBRUARY 2006 NOTICES OF THE AMS 225 situation. You don’t have to put the problem on a Erdo˝s and is related to a certain inequality for Procrustean bed and mutilate it before you attack polynomials, which was earlier proved by Bern- it numerically. And I think that has attracted a stein. Much later in your career you studied the so- much larger group of people to numerical problems called Pólya function which maps the unit interval of applications—you could really use the full the- continuously onto a right-angled triangle, and you ory. It invigorated the subject of linear algebra, discovered its amazing differentiability properties. which as a research subject died in the 1920s. Sud- Was problem solving specifically encouraged in your denly the actual algorithms for carrying out these early mathematical education in your native Hun- operations became important. It was full of sur- gary, and what effect has this had on your career prises, like fast matrix multiplication. In the new later on? edition of my linear algebra book I will add a chap- Lax: Yes, problem solving was regarded as a ter on the numerical calculation of the eigenvalues royal road to stimulate talented youngsters, and I of symmetric matrices. was very pleased to learn that here in Norway they You know it’s a truism that due to increased have a successful high-school contest, where the speed of computers, a problem that took a month winners were honored this morning. But after a forty years ago can be done in minutes, if not sec- while one shouldn’t stick to problem solving, one onds today. Most of the speed-up is attributed, at should broaden out. I return to it every once in a least by the general public, to increased speed of while, though. computers. But if you look at it, actually only half Back to the differentiability of the Pólya func- of the speed-up is due to this increased speed. The tion: I knew Pólya quite well having taken a sum- other half is due to clever algorithms, and it takes mer course with him in 1946. The differentiability mathematicians to invent clever algorithms. So it question came about this way: I was teaching a is very important to get mathematicians involved, course on real variables, and I presented Pólya’s ex- and they are involved now. ample of an area-filling curve, and I gave as home- R & S: Could you give us personal examples of work to the students the problem of proving that how questions and methods from applied points of it’s nowhere differentiable. Nobody did the home- view have triggered “pure” mathematical research work, so then I sat down and I found out that the and results? And conversely, are there examples situation was more complicated. where your theory of nonlinear partial differential There was a tradition in Hungary to look for the equations, especially your explanation of how dis- simplest proof. You may be familiar with Erdo˝s’ con- continuities propagate, have had commercial in- cept of The Book. That’s The Book kept by the terests? In particular, concerning oil exploration, Lord of all theorems and the best proofs. The high- so important for Norway! est praise that Erdo˝s had for a proof was that it was Lax: Yes, oil exploration uses signals generated out of The Book. One can overdo that, but shortly by detonations that are propagated through the after I had gotten my Ph.D., I learned about the earth and through the oil reservoir and are recorded Hahn-Banach theorem, and I thought that it could at distant stations. It’s a so-called inverse problem. be used to prove the existence of Green’s function. If you know the distribution of the densities of ma- It’s a very simple argument—I believe it’s the sim- terials and the associated waves’ speeds, then you plest—so it’s out of The Book. And I think I have can calculate how signals propagate. The inverse a proof of Brouwer’s Fixed Point Theorem, using problem is that if you know how signals propagate, calculus and just change of variables. It is proba- then you want to deduce from it the distribution bly the simplest proof and is again out of The of the materials. Since the signals are discontinu- Book. I think all this is part of the Hungarian tra- ities, you need the theory of propagation of dis- dition. But one must not overdo it. continuities. Otherwise it’s somewhat similar to R & S: There is an impressive list of great Hun- the medical imaging problem, also an inverse prob- garian physicists and mathematicians of Jewish lem. Here the signals do not go through the earth background that had to flee to the U.S. after the rise but through the human body, but there is a simi- of fascism, Nazism and anti-Semitism in Europe. How larity in the problems. But there is no doubt that do you explain this extraordinary culture of excel- you have to understand the direct problem very well lence in Hungary that produced people like de before you can tackle the inverse problem. Hevesy, Szilard, Wigner, Teller, von Neumann, von Karman, Erdo˝s, Szeg˝o, Pólya, yourself, to name Hungarian Mathematics some of the most prominent ones? R & S: Now to some questions related to your personal Lax: There is a very interesting book written by history. The first one is about your interest in, and John Lukacs with the title “Budapest 1900: A His- great aptitude for, solving problems of a type that you torical Portrait of a City and its Culture”, and it call “Mathematics Light” yourself. To mention just a chronicles the rise of the middle class, rise of com- few, already as a seventeen-year-old boy you gave merce, rise of industry, rise of science, rise of lit- an elegant solution to a problem that was posed by erature. It was fueled by many things: a long period

226 NOTICES OF THE AMS VOLUME 53, NUMBER 2 of peace, the influx of mostly Jewish population remember the tremendous slaughter on Okinawa from the East eager to rise, and intellectual tradi- and Iwo Jima. The Japanese would have resisted to tion. You know in mathematics, Bolyai was a cul- the last man. The atomic bomb put an end to all tural hero to Hungarians, and that’s why mathe- this and made an invasion unnecessary. I don’t be- matics was particularly looked upon as a glorious lieve reversionary historians who say: “Oh, Japan profession. was already beaten, they would have surrendered R & S: But who nurtured this fantastic flourish- anyway.” I don’t see any evidence for that. ing of talent, which is so remarkable? There is another point which I raised once with Lax: Perhaps much credit should be given to someone who had been involved with the atomic Julius Ko˝nig, whose name is probably not known bomb project. Would the world have had the hor- to you. He was a student of Kronecker, I believe, ror of nuclear war if it had not seen what one bomb but he also learned Cantor’s set theory and made could do? The world was inoculated against using some basic contribution to it. I think he was influ- nuclear weaponry by its use. I am not saying that ential in nurturing mathematics. His son was a alone justifies it, and it certainly was not the jus- very distinguished mathematician, Denes Ko˝nig, re- tification for its use. But I think that is a historical ally the father of modern graph theory. And then fact. there arose extraordinary people. Leopold Fejér, for Now about scribbles changing history: Sure, the instance, had enormous influence. There were too special theory of relativity, or quantum mechanics, many to fill positions in a small country like Hun- would be unimaginable today without scribbles. In- gary, so that’s why they had to go abroad. Part of cidentally, Ulam was a very interesting mathe- it was also anti-Semitism. matician. He was an idea man. Most mathematicians There is a charming story about the appointment like to push their ideas through. He preferred of Leopold Fejér, who was the first Jew proposed throwing out ideas. His good friend Rota even sug- for a professorship at Budapest University. There gested that he did not have the technical ability or was opposition to it. At that time there was a very patience to work them out. But if so, then it’s an distinguished theologian, Ignatius Fejér, in the Fac- instance of Ulam turning a disability to tremendous ulty of Theology. Fejér’s original name was Weiss. advantage. I learned a lot from him. So one of the opponents, who knew full well that R & S: It is amazing for us to learn that an Fejér’s original name had been Weiss, said point- eighteen-year-old immigrant was allowed to par- edly: This professor Leopold Fejér that you are ticipate in a top-secret and decisive weapon devel- proposing, is he related to our distinguished col- opment during WWII. league Father Ignatius Fejér? And Eo˝tvo˝s, the great Lax: The war created an emergency. Many of the physicist who was pushing the appointment, replied leaders of the Manhattan Project were foreigners, without batting an eyelash: “Illegitimate son.” That so being a foreigner was no bar. put an end to it. R & S: And he got the job? Collaboration. Work Style Lax: He got the job. R & S: Your main workplace has been the Courant Institute of Mathematical Sciences in New York, Scribbles That Changed the Course of which is part of New York University. You served as Human Affairs its director for an eight-year period in the 1970s. R & S: The mathematician Stanislaw Ulam was in- Can you describe what made this institute, which volved with the Manhattan Project and is considered was created by the German refugee Richard Courant to be one of the fathers of the hydrogen bomb. He in the 1930s, a very special place from the early days wrote in his autobiography Adventures of a Math- on, with a particular spirit and atmosphere? And is ematician: “It is still an unending source of surprise the Courant Institute today still a special place that for me to see how a few scribbles on a blackboard, differs from others? or on a sheet of paper, could change the course of Lax: To answer your first question, certainly the human affairs.” Do you share this feeling? And personality of Courant was decisive. Courant saw what are your feelings about what happened to Hi- mathematics very broadly, he was suspicious of spe- roshima and Nagasaki, to the victims of the explo- cialization. He wanted it drawn as broadly as pos- sions of the atomic bombs that brought an end to sible, and that’s how it came about that applied World War II? topics and pure mathematics were pursued side Lax: Well, let me answer the last question first. by side, often by the same people. This made I was in the army, and all of us in the army expected the Courant Institute unique at the time of its to be sent to the Pacific to participate in the inva- founding, as well as in the 1940s, 1950s, and 1960s. sion of Japan. You remember the tremendous Since then there are other centers where applied slaughter that the invasion of Normandy brought mathematics is respected and pursued. I am happy about. That would have been nothing compared to to say that this original spirit is still present at the the invasion of the Japanese mainland. You Courant Institute. We still have large areas of

FEBRUARY 2006 NOTICES OF THE AMS 227 applied interest, meteorology and climatology these two extremes. Incidentally, I have a very good under Andy Majda, solid state and material science memory. under Robert Kohn and others, and fluid dynam- ics. But we also have differential geometry as well Teaching as some pure aspects of partial differential equa- R & S: You have also been engaged in the teaching tions, even some algebra. of calculus. For instance, you have written a calcu- I am very pleased how the Courant Institute is lus textbook with your wife Anneli as one of the co- presently run. It’s now the third generation that’s authors. In this connection you have expressed running it, and the spirit that Courant instilled in strong opinions about how calculus should be ex- it—kind of a family feeling—still prevails. I am posed to beginning students. Could you elaborate happy to note that many Norwegian mathemati- on this? cians received their training at the Courant Insti- Lax: Our calculus book was enormously unsuc- tute and later rose to become leaders in their field. cessful, in spite of containing many excellent ideas. R & S: You told us already about your collabo- Part of the reason was that certain materials were ration with Ralph Phillips. Generally speaking, look- not presented in a fashion that students could ab- ing through your publication list and the theorems sorb. A calculus book has to be fine-tuned, and I and methods you and your collaborators have given didn’t have the patience for it. Anneli would have name to, it is apparent that you have had a vast col- had it, but I bullied her too much, I am afraid. laboration with a lot of mathematicians. Is this shar- Sometimes I dream of redoing it because the ideas ing of ideas a particularly successful, and maybe also that were in there, and that I have had since, are joyful, way of advancing for you? still valid. Lax: Sure, sure. Mathematics is a social phe- Of course, there has been a calculus reform nomenon after all. Collaboration is a psychologi- movement and some good books have come out of cal and interesting phenomenon. A friend of mine, it, but I don’t think they are the answer. First of all, Vera John-Steiner, has written a book (Creative the books are too thick, often more than 1,000 Collaboration) about it. Two halves of a solution are pages. It’s unfair to put such a book into the hands supplied by two different people, and something of an unsuspecting student who can barely carry quite wonderful comes out of it. it. And the reaction to it would be: “Oh, my God, I R & S: Many mathematicians have a very par- have to learn all that is in it?” Well, all that is not ticular work style when they work hard on certain in it! Secondly, if you compare it to the old stan- problems. How would you characterize your own dards, Thomas, say, it’s not so different—the order particular way of thinking, working, and writing? of the topics and concepts, perhaps. Is it rather playful or rather industrious? Or both? In my calculus book, for instance, instead of Lax: Phillips thought I was lazy. He was a prod- continuity at a point, I advocated uniform conti- uct of the Depression, which imposed a certain nuity. This you can explain much more easily than strict discipline on people. He thought I did not defining continuity at a point and then say the work hard enough, but I think I did! function is continuous at every point. You lose the R & S: Sometimes mathematical insights seem to students; there are too many quantifiers in that. But rely on a sudden unexpected inspiration. Do you the mathematical communities are enormously have examples of this sort from your own career? conservative: “Continuity has been defined point- And what is the background for such sudden in- wise, and so it should be!” spiration in your opinion? Other things that I would emphasize: To be sure Lax: The question reminds me of a story about there are applications in these new books. But the a German mathematician, Schottky, when he applications should all stand out. In my book there reached the age of seventy or eighty. There was a were chapters devoted to the applications, that’s celebration of the event, and in an interview like how it should be—they should be featured promi- we are having, he was asked: “To what do you at- nently. I have many other ideas as well. I still dream tribute your creativity and productivity?” The ques- of redoing my calculus book, and I am looking for tion threw him into great confusion. Finally he a good collaborator. I recently met someone who said: “But gentlemen, if one thinks of mathemat- expressed admiration for the original book, so per- ics for fifty years, one must think of something!” haps it could be realized, if I have the energy. I have It was different with Hilbert. This is a story I heard other things to do as well, like the second edition from Courant. It was a similar occasion. At his sev- of my linear algebra book, and revising some old entieth birthday he was asked what he attributed lecture notes on hyperbolic equations. But even if his great creativity and originality to. He had the I could find a collaborator on a calculus book, answer immediately: “I attribute it to my very bad would it be accepted? Not clear. In 1873, Dedekind memory.” He really had to reconstruct everything, posed the important question: “What are, and what and then it became something else, something bet- should be, the real numbers?” Unfortunately, he ter. So maybe that is all I should say. I am between gave the wrong answer as far as calculus students

228 NOTICES OF THE AMS VOLUME 53, NUMBER 2 are concerned. The right answer is: infinidecimals. had no access to the supercomputers. At a certain I don’t know how such a joke will go down. point the government, which alone had enough money to purchase these supercomputers, stopped Heading Large Institutions placing them at universities. Instead they went to R & S: You were several times the head of large or- national labs and industrial labs. Unless you hap- ganizations: director of the Courant Institute in pened to have a friend there with whom you col- 1972–1980, president of the American Mathemat- laborated, you had no access. That was very bad ical Society in 1977–1980, leader of what was called from the point of view of the advancement of com- the Lax Panel on the National Science Board in putational science, because the most talented peo- 1980–1986. Can you tell us about some of the most ple were at the universities. At that time accessing important decisions that had to be taken in these and computing at remote sites became possible periods? thanks to ARPANET, which then became a model Lax: The president of the American Mathemat- for the Internet. So the panel that I established ical Society is a figurehead. His influence lies in ap- made strong recommendation that the NSF estab- pointing members of committees. Having a wide lish computing centers, and that was followed up. friendship and reasonable judgement are helpful. My quote on our achievement was a paraphrase of I was very much helped by the secretary of the Emerson: “Nothing can resist the force of an idea American Mathematical Society, Everett Pitcher. that is ten years overdue.” As for being the director of the Courant Insti- R & S: A lot of mathematical research in the U.S. tute, I started my directorship at the worst possi- has been funded by contracts from DOD (Depart- ble time for New York University. They had just ment of Defense), DOE (Department of Energy), the closed down their School of Engineering, and that Atomic Energy Commission, the NSA (National Se- meant that mathematicians from the engineering curity Agency). Is this dependence of mutual bene- school were transferred to the Courant Institute. fit? Are there pitfalls? This was the time when the Computer Science De- Lax: I am afraid that our leaders are no longer partment was founded at Courant by Jack Schwartz. aware of the subtle but close connection between There was a group of engineers that wanted to scientific vigor and technological sophistication. start activity in informatics, which is the engineers’ word for the same thing. As a director I fought very Personal Interests hard to stop that. I think it would have been very R & S: Would you tell us a bit about your interests bad for the university to have had two computing and hobbies that are not directly related to mathe- departments—it certainly would have been very matics? bad for our Computer Science Department. Other Lax: I love poetry. Hungarian poetry is particu- things: Well, I was instrumental in hiring Charlie larly beautiful, but English poetry is perhaps even Peskin at the recommendation of Alexander Chorin. more beautiful. I love to play tennis. Now my knees I was very pleased with that. Likewise, hiring are a bit wobbly, and I can’t run anymore, but per- Sylvain Cappell at the recommendation of Bob haps these can be replaced—I’m not there yet. My Kohn. Both were enormous successes. son and three grandsons are tennis enthusiasts so What were my failures? Well, maybe when the I can play doubles with them. I like to read. I have Computer Science Department was founded I a knack for writing. Alas, these days I write obitu- should have insisted on having a very high stan- aries—it’s better to write them than being written dard of hiring. We needed people to teach courses, about. but in hindsight I think we should have exercised R & S: You have also written Japanese haikus? more restraint in our hiring. We might have become Lax: You’re right. I got this idea from a nice ar- the number one computer science department. ticle by Marshall Stone—I forget exactly where it Right now the quality has improved very much— was—where he wrote that the mathematical lan- we have a wonderful chairwoman, Margaret Wright. guage is enormously concentrated, it is like haikus. Being on the National Science Board was my And I thought I would take it one step further and most pleasant administrative experience. It’s a pol- actually express a mathematical idea by a haiku. icy-making body for the National Science Founda- (See Peter Lax’s haiku below.) tion (NSF), so I found out what making policy R & S: Professor Lax, thank you very much for means. Most of the time it just means nodding this interview on behalf of the Norwegian, the Dan- “yes”, and a few times saying “no”. But then there ish, and the European Mathematical Societies! are sometimes windows of opportunity, and the Lax Lax: I thank you. Panel was a response to such a thing. You see, I no- ticed through my own experience and that of my Speed depends on size friends who are interested in large scale comput- Balanced by dispersion ing (in particular, Paul Garabedian, who complained Oh, solitary splendor. about it), that university computational scientists

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