1.1. The Lester R Ford Award.

This Mathematical Association of America prize was established in 1964 as the Lester R Ford Award to honour the contributions of mathematician and former Mathematical Association of America president Lester R Ford. It is warded annually by to authors of articles of expository excellence published in The American Mathematical Monthly or Mathematics Magazine. The prize was renamed the Paul R Halmos -Lester R Ford Award in 2012.

1.2. Peter Lax wins the Lester R Ford Award.

Peter D Lax received the 1966 award for his paper Numerical solutions of partial differential equations published in The American Mathematical Monthly 72 (2) (1965), 78-84.

The paper has the following Introduction:

No one will quarrel with the statement that computing is bound to revolutionise applied mathematics by providing new and powerful tools for getting numerical answers, the life and blood of the subject. The versatility of the new computers has already stimulated the systematic study of numerical methods; this has attracted a number of pure mathematicians who, before, had been repelled by the ad hoc nature of the methods used by the hapless applied mathematician.

This brief article is about the numerical solution of partial differential equations. A very general (although by no means the only) method for solving these is to convert them into difference equations through the replacement of derivatives by difference quotients of some sort. The effectiveness of this method is attested to by the vast literature on it which has sprung up during the last 15 years - a literature so extensive that it is hopeless to try to summarise it. Instead of trying to present some of the highlights of the new developments I shall try to convey its flavour. My aim is to convince a sceptical reader who may regard using finite differences as the last resort of a scoundrel that the theory of difference equations is a rather sophisticated affair, more sophisticated than the corresponding theory of partial differential equations. My argument will be based on two contentions:

1) In order to prove that solutions of a sequence of difference equations converge one needs estimates for difference operators which are analogous to the estimates needed in the existence and uniqueness theory for solutions of differential equations.

2) Estimates for difference operators are much harder to derive than the corresponding estimates for differential operators.

2.1. John von Neumann Lecture Prize.

The Society for Industrial and Applied Mathematics (SIAM), with support from IBM and other industry corporations, founded the John von Neumann Lecture Prize in 1959 for:-

... outstanding and distinguished contributions to the field of applied mathematical sciences and for the effective communication of these ideas to the community.

The first award was made in 1960. The Prize winner receives a monetary award and presents a survey lecture at the SIAM Annual Meeting.

2.2. Peter Lax wins the John von Neumann Lecture Prize.

Peter D Lax delivered the ninth John von Neumann Lecture at the 1968 SIAM Summer Meeting at the University of Wisconsin on 28 August 1968. His lecture was Nonlinear partial differential equations and computing and it began as follows:

I seem to belong to just about the youngest generation of mathematicians who still knew von Neumann. I owe that privilege to two fortuitous circumstances: One was our common Hungarian background; blood, after all, is thicker than water. Before I embarked to the United States, Professor D König wrote from Budapest to von Neumann asking him to take an interest in my mathematical education. Within a few days of my arrival in New York von Neumann, with great kindness, called, spent an afternoon sizing me up, and then advised me what to read and where to go to school.

The second coincidence was that both of us spent part of the war years at Los Alamos, he as one of a galaxy of glittering stars which included Fermi, Bohr, Oppenheimer, Bethe and the young but already legendary Feynman, and I as one of a group of young soldiers, along with John Kemeny, Richard Bellman, Alex Heller, all of us addicted to hero-worship. Von Neumann frequently favoured the mathematical contingent with lectures, perfectly organised around the logical core of the subject; the technicalities were described and commented on, but not presented in detail. The lectures were delivered with great fluency; the speaker seldom fumbled for the right expression, and his sentences, in spite of their almost Germanic length, were always completed with grammatical perfection. One would have thought that these lectures were carefully prepared - yet we knew that they were delivered impromptu, off the cuff.

In the early fifties von Neumann returned each summer to Los Alamos for a number of weeks. That landscape, such a strange mixture, and the seclusion of the spot seemed to soothe his usually restless spirit. Also he took a keen interest in the variety of problems with which the Laboratory was then grappling. He took the lead in meeting current computing needs: he concerned himself with the technical engineering aspects of computer design, with the organisation of large computer programs, and in equal measure numerical procedures for computing large-scale motions of continuous media. He took the lead in devising new, sophisticated computing techniques; I would like to report on these and some further developments which grew out of them.

In addition to realising the importance of powerful computing techniques for obtaining accurate approximations of specific solutions of nonlinear equations, von Neumann was fascinated by the possibility that patterns disclosed by numerical calculations might reveal entirely unsuspected properties of solutions. In the second part of my talk I would like to describe a recent striking instance of this

3.1. The Lester R Ford Award.

The Award is described under 1.1 above.

3.2. Peter Lax wins the Lester R Ford Award.

Peter D Lax received the 1973 award for his paper The formation and decay of shock waves published in The American Mathematical Monthly 79 (3) (1972), 227-241.

The paper has the following Introduction:

The theory of propagation of shock waves is one of a small class of mathematical topics whose basic problems are easy to explain but hard to resolve. This article is a brief introduction to the subject: we shall describe the origin of the governing equations, some of the striking phenomena, and a few of the mathematical tools used to analyse them.

3.3. Note on Peter D Lax.

The award winning paper contains the following Note:

Peter Lax received his Ph.D. at New York University under K. Friedrichs and has spent most of his academic career at New York University, where he is presently a professor. He is a frequent summer visitor at Stanford and the Los Alamos Scientific Lab. His research contributions in partial differential equations, linear and non-linear problems of mathematical physics, computing, and functional analysis have had a profound impact. He was a Fullbright lecturer in 1958, he is a Vice-President of the AMS, he is an elected member of the National Academy of Sciences, he was an AMS Gibbs lecturer, and he received an MAA Lester Ford Award. He is co-author with R Phillips of Scattering Theory (Academic Press, 1967).

4.1. The Chauvenet Prize.

The Chauvenet Prize is awarded at the Annual January Meeting of the Mathematical Association of America to the author of an outstanding expository article on a mathematical topic. First awarded in 1925, the Prize is named for William Chauvenet, a professor of mathematics at the United States Naval Academy. It was established through a gift in 1925 from J L Coolidge, then MAA President. Winners of the Chauvenet Prize are among the most distinguished of mathematical expositors.

4.2. Peter Lax wins the Chauvenet Prize.

Peter D Lax received the 1974 prize for his paper The formation and decay of shock waves published in The American Mathematical Monthly 79 (3) (1972), 227-241.

We have given details of this paper under 3.2 and 3.3 above.

5.1. The Norbert Wiener Prize.

The AMS-SIAM Norbert Wiener Prize in Applied Mathematics was established in 1967 in honour of Professor Norbert Wiener and was endowed by a fund from the Department of Mathematics of the Massachusetts Institute of Technology. The endowment was further supplemented by a generous donor. The first award was made in 1970 and until the year 2000 it was awarded every five years. It was awarded in 2004 and from then on every three years.

5.2. Peter Lax wins the Norbert Wiener Prize.

The second award of the Norbert Wiener Prize was made to Peter D Lax in 1975:

... for his broad contributions to applied mathematics, in particular, for his work on numerical and theoretical aspects of partial differential equations and on scattering theory.

6. National Academy of Sciences Award in Applied Mathematics and Numerical Sciences (1983).

The National Academy of Sciences Award in Applied Mathematics and Numerical Analysis - a prize of $10,000 - was awarded every three years beginning in 1972. It was given for outstanding work in applied mathematics and numerical analysis by a candidate whose research has been carried out in an institution in North America. The last award was made in 2002 and the National Academy of Sciences discontinued the award in 2005.

6.2. Peter Lax wins the NAS Award in Applied Mathematics and Numerical Analysis.

The 1983 National Academy of Sciences Award in Applied Mathematics and Numerical Analysis was given to Peter D Lax:

... for his penetrating, variegated, and fundamental contributions to mathematical theory and its applications to problems in functional analysis, numerical analysis, linear and non linear partial differential equations, wave propagation, and scattering theory.

7.1. The National Medal of Science.

Established in 1959 by the U.S. Congress, the National Medal of Science is the highest recognition the nation can bestow on scientists and engineers. The presidential award is given to individuals deserving of special recognition by reason of their outstanding contributions to knowledge in the physical, biological, mathematical, engineering, or social and behavioural sciences, in service to the Nation. These broad areas include such disciplines as astronomy, chemistry, computer and information science and engineering, geoscience, materials research, and research on STEM education.

A committee of distinguished scientists and engineers is appointed by the president of the United States to evaluate the nominees for the award. Medals are presented to recipients by the president during an awards ceremony at the White House.

Since its establishment, the National Medal of Science has been awarded to 529 distinguished scientists and engineers whose careers span decades of research and development.

7.2. Peter Lax wins the National Medal of Science.

In 1986 Peter D Lax was awarded the National Medal of Science for Mathematics. The award was made:

For his outstanding, innovative and profound contributions to the theory of partial differential equations, applied mathematics, numerical analysis and scientific computation.

The National Medal of Science for Mathematics was presented to Peter Lax by President Reagan at a White House Ceremony on 12 March 1986.

8.1. The Wolf Prize.

The Wolf Prize is awarded annually and honours exceptional individuals who transcend barriers of religion, gender, race, geography, and political stance. In the scientific domain, the awards are conferred in Medicine, Agriculture, Mathematics, Chemistry, and Physics. The awards in the arts recognise excellence in Painting and Sculpture, Music, and Architecture.

The Wolf Prize acknowledges scientists and artists worldwide for their outstanding achievements in advancing science and the arts for the betterment of humanity. By awarding the prize, the Wolf Foundation salutes leaders and pioneers in these fields who have contributed to a better world.

The first awards were made in 1978.

8.2. Peter Lax wins the Wolf Prize.

The 1987 Wolf Prize in Mathematics was awarded to Peter D Lax:-

... for his outstanding contributions to many areas of analysis and applied mathematics.

The citation states:

Professor Peter D Lax, a graduate of the Courant Institute, embodies the best traditions of D Hilbert as continued by R Courant. Among his many contributions are the solution of the Cauchy problem with oscillatory data, the clarification of the role of stability of a numerical scheme, the comprehensive development of scattering theory, the theory of non-linear conservation laws and a deep insight into the Korteweg-de Vries equation. Professor Lax's influence has been profound and decisive in both pure and applied mathematics.

9.1. The Leroy P Steele Prize.

The Leroy P Steele Prize was established by the American Mathematical Society in 1970 in honour of George David Birkhoff, William Fogg Osgood, and William Caspar Graustein. It was endowed under the terms of a bequest from Leroy P Steele. From 1970 to 1976 one or more prizes were awarded each year for outstanding published mathematical research; most favourable consideration was given to papers distinguished for their exposition and covering broad areas of mathematics. In 1977 the Council of the American Mathematical Society modified the terms under which the prizes were awarded. In 1993, the Council formalised three categories of the prize by naming each of them: (1) The Leroy P Steele Prize for Lifetime Achievement; (2) The Leroy P Steele Prize for Mathematical Exposition; and (3) The Leroy P Steele Prize for Seminal Contribution to Research.

9.2. Peter Lax wins the Leroy P Steele Prize.

The 1992 Leroy P Steele Prize was awarded to Peter D Lax:

... for his numerous and fundamental contributions to the theory and applications of linear and nonlinear partial differential equations and functional analysis, for his leadership in the development of computational and applied mathematics, and for his extraordinary impact as a teacher.

Since there wasn't an American Mathematical Society-Mathematical Association of America Summer Meeting in 1992, this award was made at the January 1993 AMS-MAA Annual Meeting held in San Antonio Texas.

10.1. The Abel Prize.

The Abel Prize was awarded for the first time in 2003 but it was first suggested over 100 years earlier. Sophus Lie, when he saw that Nobel's plans for annual prizes did not include one for mathematics, proposed the setting up of an Abel Prize which would be awarded every five years. He contacted mathematicians world-wide and gathered wide support. However he had not set up any machinery to carry the idea forward and when he died soon after this, in 1899, nothing further happened.

The year 1902 was one in which the centenary of Abel's birth was celebrated. A decision was again taken to establish an international Abel Prize but again the plan did not come to fruition. With the bicentenary of Abel's birth approaching, Arild Stubhaug, who had written a major new biography of Abel, made another attempt to set up an Abel Prize.

A committee was set up which gathered support both within Norway and also international support. They put their proposals before the Norwegian government in May 2001 and in a speech on the campus of the University of Oslo in August 2001, the Norwegian Prime Minister announced that the Government would establish an Abel Fund.

The Norwegian Academy of Science and Letters announces the winners of the Abel prize, the first being awarded in 2003.

10.2. Peter D Lax - 2005 Abel Prize Citation.

The Norwegian Academy of Science and Letters has decided to award the Abel Prize for 2005 to Peter D Lax, Courant Institute of Mathematical Sciences, New York University:

... for his ground-breaking contributions to the theory and application of partial differential equations and to the computation of their solutions.

Ever since Newton, differential equations have been the basis for the scientific understanding of nature. Linear differential equations, in which cause and effect are directly proportional, are reasonably well understood. The equations that arise in such fields as aerodynamics, meteorology and elasticity are nonlinear and much more complex: their solutions can develop singularities. Think of the shock waves that appear when an airplane breaks the sound barrier.

In the 1950s and 1960s, Lax laid the foundations for the modern theory of nonlinear equations of this type (hyperbolic systems). He constructed explicit solutions, identified classes of especially well-behaved systems, introduced an important notion of entropy, and, with Glimm, made a penetrating study of how solutions behave over a long period of time. In addition, he introduced the widely used Lax-Friedrichs and Lax-Wendroff numerical schemes for computing solutions. His work in this area was important for the further theoretical developments. It has also been extraordinarily fruitful for practical applications, from weather prediction to airplane design.

Another important cornerstone of modern numerical analysis is the "Lax Equivalence Theorem". Inspired by Richtmyer, Lax established with this theorem the conditions under which a numerical implementation gives a valid approximation to the solution of a differential equation. This result brought enormous clarity to the subject.

A system of differential equations is called "integrable" if its solutions are completely characterised by some crucial quantities that do not change in time. A classical example is the spinning top or gyroscope, where these conserved quantities are energy and angular momentum.

Integrable systems have been studied since the 19th century and are important in pure as well as applied mathematics. In the late 1960s a revolution occurred when Kruskal and co-workers discovered a new family of examples, which have "soliton" solutions: single-crested waves that maintain their shape as they travel. Lax became fascinated by these mysterious solutions and found a unifying concept for understanding them, rewriting the equations in terms of what are now called "Lax pairs". This developed into an essential tool for the whole field, leading to new constructions of integrable systems and facilitating their study.

Scattering theory is concerned with the change in a wave as it goes around an obstacle. This phenomenon occurs not only for fluids, but also, for instance, in atomic physics (Schrödinger equation). Together with Phillips, Lax developed a broad theory of scattering and described the long-term behaviour of solutions (specifically, the decay of energy). Their work also turned out to be important in fields of mathematics apparently very distant from differential equations, such as number theory. This is an unusual and very beautiful example of a framework built for applied mathematics leading to new insights within pure mathematics.

Peter D Lax has been described as the most versatile mathematician of his generation. The impressive list above by no means states all of his achievements. His use of geometric optics to study the propagation of singularities inaugurated the theory of Fourier Integral Operators. With Nirenberg, he derived the definitive Gårding-type estimates for systems of equations. Other celebrated results include the Lax-Milgram lemma and Lax's version of the Phragmén-Lindelöf principle for elliptic equations.

Peter D Lax stands out in joining together pure and applied mathematics, combining a deep understanding of analysis with an extraordinary capacity to find unifying concepts. He has had a profound influence, not only by his research, but also by his writing, his lifelong commitment to education and his generosity to younger mathematicians.

10.3. Peter D Lax - 2005 Abel Prize Biography.

Peter D Lax was born on 1 May 1926 in Budapest, Hungary. He was on his way to New York with his parents on 7 December 1941 when the US joined the war.

Peter D Lax received his PhD in 1949 from New York University with Richard Courant as his thesis advisor. Courant had founded the Courant Institute of Mathematical Sciences at NYU where Lax served as Director from 1972-1980. In 1950 Peter D. Lax went to Los Alamos for a year and later worked there several summers as a consultant, but already in 1951 he returned to New York University to begin his life work at the Courant Institute. Lax became professor in 1958. At NYU he has also served as Director of the AEC (Atomic Energy Commission) Computing and Applied Mathematics Center.

In nominating Lax as a member of the US National Academy of Sciences in 1962, Courant described him as "embodying as few others do, the unity of abstract mathematical analysis with the most concrete power in solving individual problems".

Peter D Lax is one of the greatest pure and applied mathematicians of our times and has made significant contributions, ranging from partial differential equations to applications in engineering. His name is connected with many major mathematical results and numerical methods, such as the Lax-Milgram Lemma, the Lax Equivalence Theorem, the Lax- Friedrichs Scheme, the Lax-Wendroff Scheme, the Lax Entropy Condition and the Lax-Levermore Theory.

Peter D Lax is also one of the founders of modern computational mathematics. Among his most important contributions to High Performance Computing and Communications community was his work on the National Science Board from 1980 to 1986. He also chaired the committee convened by the National Science Board to study large scale computing in science and mathematics - a pioneering effort that resulted in the Lax Report.

Professor Lax's work has been recognised by many honours and awards. He was awarded the National Medal of Science in 1986, presented by President Ronald Reagan at a White House ceremony. Lax received the Wolf Prize in 1987 and the Chauvenet Prize in 1974 and shared the American Mathematical Society's Steele Prize in 1992. He was also awarded the Norbert Wiener Prize in 1975 from the American Mathematical Society and the Society for Industrial and Applied Mathematics. In 1996 he was elected a member of the American Philosophical Society.

Peter D Lax has been both president (1977-80) and vice president (1969-71) of the American Mathematical Society.

Professor Peter D Lax is a distinguished educator who has mentored a large number of students. He has also been a tireless reformer of mathematics education and his work with differential equations has for decades been a standard part of the mathematics curriculum worldwide.

Peter D Lax has received many Honorary Doctorates from universities all over the world. When he was honoured by the University of Technology in Aachen, Germany in 1988, both his deep contribution to mathematics and the importance his work has had in the field of engineering were emphasised. He was also honoured for his positive attitude toward the use of computers in mathematics, research and teaching.

10.4. Peter D Lax - 2005 Abel Prize Autobiography.

Like most mathematicians, I became fascinated with mathematics early, about age ten. I was fortunate that my uncle could explain matters that puzzled me, such as why minus times minus is plus - it follows from the laws of algebra.

Mathematics had a deep tradition in Hungary, going back to the epoch-making invention of non-Euclidean geometry by János Bolyai, an Hungarian genius in the early 19th century. To this day, the Hungarian mathematical community seeks out mathematically talented students through contests and a journal for high school students. Winners are then nurtured intensively. I was tutored by Rózsa Péter, an outstanding logician and pedagogue; her popular book on mathematics, "Playing with Infinity" is still the best introduction to the subject for the general public.

At the end of 1941 I came to the US with my family, sailing from Lisbon on 5 December 1941. It was the last boat to America; I was 15 years old. My mentors wrote to Hungarian mathematicians who had already settled in the US, asking them to take an interest in my education. They were very supportive.

I finished my secondary education at Stuyvesant High School, one of the elite public schools in New York City; its graduates include many distinguished mathematicians and physicists. For me the important thing was to learn English, and the rudiments of American history. In the meanwhile I visited from time to time Paul Erdös at the Institute for Advanced Study at Princeton. He was extremely kind and supportive; he would give me problems, some of which I managed to solve. My first publication, in 1944, was "Proof of a conjecture of P Erdös on the derivative of a polynomial".

At the suggestion of Gabor Szegö I enrolled at New York University in the Spring of 1943 to study under the direction of Richard Courant, widely renowned for nurturing young talent. It was the best advice I ever received. But my studies came to a temporary halt in June 1944, when I was drafted into the US Army. I became an American citizen during my basic training.

I was then sent to Texas A & M to study engineering. There I passed the preliminary test in calculus with flying colours, and together with another soldier with similar background was excused from the course. Professor Klipple, a former student of R L Moore, generously offered to conduct just for the two of us a real variables course in the style of R L Moore. I learned a lot.

In June, 1945 I was posted to Los Alamos, the atomic bomb project. It was like living science fiction; upon arrival I was told that the whole town of 10,000 was engaged in an effort to build an atomic bomb out of plutonium, an element that does not exist in nature but was manufactured in a reactor at Hanford, WA. The project was led by some of the most charismatic leaders of science. I was assigned to the Theoretical Division; I joined a number of bright young soldiers, Dick Bellman, John Kemeny, Murray Peshkin and Sam Goldberg.

Von Neumann was a frequent consultant; each time he came, he gave a seminar talk on mathematics. He had little time to prepare these talks, but they were letter perfect. Only once did he get stuck; he excused himself by saying that he knew three ways of proving the theorem, but unfortunately chose a fourth.

Enrico Fermi and Niels Bohr were also consultants; since they were closely associated with nuclear physics, Security insisted that they use code names; Fermi became Henry Farmer, Bohr, Nicholas Baker. At a party a woman who had spent some time in Copenhagen before the war recognised Bohr, and said "Professor Bohr, how nice to see you". Remembering what Security had drilled into him, Bohr said "No, I am Nicholas Baker", but then immediately added "You are Mrs Houtermans". "No", she replied, "I am Mrs Placzek". She had divorced and remarried in the meantime.

I returned to New York University in the Fall of 1946 to get my undergraduate degree, and simultaneously to continue my graduate studies. I joined an outstanding class of graduate students in mathematics, Avron Douglis, Eugene Isaacson, Joe Keller, Martin Kruskal, Cathleen Morawetz, Louis Nirenberg and Anneli Kahn.

I got my PhD in 1949 under the direction of K O Friedrichs, a wonderful mathematician and a delightful, idiosyncratic person. He kept his life on a strict schedule; he knew that this was absurd, but it worked for him. When a graduate student of his repeatedly delayed finishing his dissertation, I explained to Friedrichs that the student was very neurotic. That seemed to him no excuse; "Am I not just as neurotic?," he said, "yet I finish my work."

In the course of some research we did jointly, I found a reference in a Russian journal. Friedrichs said that he knew the alphabet, a couple of hundred words, and the rudiments of Russian grammar, and was willing to read the paper. I was worried about the language barrier, but Friedrichs said: "That it is in Russian is nothing; the difficulty is that it is mathematics".

Anneli and I married in 1948; our first child, John, was born in 1950. We spent part of 1950/51 at Los Alamos. We returned to New York in the fall of 1951 to take up an appointment as Research Assistant Professor, in the mathematics department of New York University. I remained in the department for nearly fifty years, basking in the friendly collegial atmosphere of the place.

In the fifties I spent most of my summers at Los Alamos. At that time under the leadership of von Neumann, Los Alamos was the world leader in numerical computing and had the most up-to-date computers. I became, and remained, deeply involved in problems of the numerical solutions of hyperbolic equations, in particular the equations of compressible flow.

In 1954 the Atomic Energy Commission placed a Univac computer at the Courant Institute; it was the first supercomputer, with a thousand words of memory. Our first task was to calculate the flood stages on the Columbia river in case the Grand Cooley dam were destroyed by sabotage. The AEC wanted to know if the Hanford reactor would be flooded. Originally the Corps of Engineers was charged with this task, but it was beyond their capabilities. The team at the Courant Institute, led by Jim Stoker and Eugene Isaacson, found that the reactor would be safe.

The Univac was manufactured by the Remington-Rand Corporation. The official installation of the computer at New York University was a sufficiently important event for James Rand, the CEO, to attend and bring along some members of his Board of Directors, including the chairman, General Douglas McArthur, and General Leslie Groves. In his long career General McArthur had been Commandant of the Corps of Engineers; he was keenly interested in our calculations, and grasped the power of modern computers.

The post-war years were a heady time for mathematics, in particular for the theory of partial differential equations, one of the main lines of research at the Courant Institute as well as other institutes here and abroad. The subject is a wonderful mixture of applied and pure mathematics; most equations describe physical situations, but then take on a life of their own.

Richard Courant retired as Director of the Institute in 1958, at the age of 70. His successor was Jim Stoker who accomplished the crucial task of making the Institute part of New York University by securing tenure for its leading members.

After Stoker retired, the younger generation took over, Jürgen Moser, Louis Nirenberg, the undersigned, Raghu Varadhan, Henry McKean, Cathleen Morawetz, and others; our present Director is Leslie Greengard. Significant changes took place during these years, but the Institute adhered to the basic principle that had guided Richard Courant: not to pursue the mathematical fashion of the day ("I am against panic buying in an inflated market") but to hire promising young people. Also, for Courant mathematics was a cooperative enterprise, not competitive.

In the sixties Ralph Phillips and I embarked on a project to study scattering theory. Our cooperation lasted 30 years and led to many new results, including a reformulation of the theory. This reformulation was used by Ludvig Faddeev and Boris Pavlov to study automorphic functions; they found a connection between automorphic scattering and the Riemann hypothesis.

Also in the sixties Martin Kruskal and Norman Zabusky, guided by extensive numerical computations, found remarkable properties of solutions of the Korteweg-de Vries equation. These eventually led to the discovery that the KdV equation is completely integrable, followed by the discovery, totally unsuspected, of a whole slew of completely integrable systems. I had the pleasure and good luck to participate in this development.

In 1970 a mob protesting the Vietnam war invaded the Courant Institute and threatened to blow up our CDC computer. They left after 48 hectic hours; my colleagues and I in the Computing Center smelled smoke and rushed upstairs just in time to disconnect a burning fuse. It was a foolhardy thing to do, but we were too angry to think.

In 1980 I was appointed to a six year term on the National Science Board, the policy making body of the National Science Foundation. It was an immensely gratifying experience; I learned about issues in many parts of science, as well as about the politics of science. My colleagues were outstanding scientists and highly colourful characters.

By the time the eighties rolled around, the Government no longer placed supercomputers at universities, severely limiting the access of academic scientists to computing facilities. My position on the Science Board gave me a chance to remedy this intolerable situation. A panel I chaired recommended that the NSF set up regional Computing Centers, accessible to distant users through high capacity lines. The Arpanet Project, the precursor of the Internet, demonstrated the practicality of such an arrangement.

I have always enjoyed teaching at all levels, including introductory calculus. At the graduate level my favourite courses were linear algebra, functional analysis, and partial differential equations. The notes I have prepared while teaching formed the basis of the books I have written on these subjects.

I supervised the PhD dissertations of 55 graduate students; many have become outstanding mathematicians. Some became close personal friends.

I retired from teaching in 1999, shortly after reaching the age 70. According to a US law passed in 1994, nobody can be forced to retire on account of age in any profession, including teaching at a university. This sometimes had unwelcome consequences, such as the case of a professor at a West Coast university who stayed on the faculty well into his seventies. Eventually his colleagues petitioned the administration to retire him on the ground that he is a terrible teacher. At a hearing he had a chance to defend himself; his defence was, "I have always been a terrible teacher".

In retirement I occupy myself by writing books, and by continuing to puzzle over mathematical problems. I receive invitations to visit and lecture at mathematical centres. I attend the annual meeting of the American Mathematical Society. I spend a lot of time with my friends and my family, including three rapidly growing grandsons. Anneli died in 1999; Lori Courant and I were fortunate to find each other, and we are living happily ever after.

Mathematics is sometimes compared to music; I find a comparison with painting better. In painting there is a creative tension between depicting the shapes, colours and textures of natural objects, and making a beautiful pattern on a flat canvas. Similarly, in mathematics there is a creative tension between analysing the laws of nature, and making beautiful logical patterns.

Mathematicians form a closely knit, world wide community. Even during the height of the Cold War, American and Soviet scientists had the most cordial relations with each other. This comradeship is one of the delights of mathematics, and should serve as an example for the rest of the world.

10.5. Peter D Lax - 2005 Abel Prize Interview.

This interview, by Martin Raussen and Christian Skau, was on 24 May 2005 in Oslo prior to Peter Lax being awarded the Abel Prize.

Raussen & Skau: On behalf of the Norwegian and Danish Mathematical Societies we would like to congratulate you on winning the Abel Prize for 2005. You came to the U.S. in 1941 as a fifteen-year old kid from Hungary. Only three years later, in 1944, you were drafted into the U.S Army. Instead of being shipped overseas to the war front, you were sent to Los Alamos in 1945 to participate in the Manhattan Project, building the first atomic bomb. It must have been awesome as a young man to come to Los Alamos to take part in such a momentous endeavour and to meet so many legendary famous scientists: Fermi, Bethe, Szilard, Wigner, Teller, Feynman, to name some of the physicists, and von Neumann and Ulam, to name some of the mathematicians. How did this experience shape your view of mathematics and influence your choice of a research field within mathematics?

Lax: In fact, I returned for a year's stay at Los Alamos after I got my Ph.D. in 1949 and then spent many summers as a consultant. The first time I spent in Los Alamos, and especially the later exposure, shaped my mathematical thinking. First of all, it was the experience of being part of a scientific team - not just of mathematicians, but people with different outlooks - with the aim being not a theorem, but a product. One cannot learn that from books, one must be a participant, and for that reason I urge my students to spend at least a summer as a visitor at Los Alamos. Los Alamos has a very active visitor's program. Secondly, it was there - that was in the 1950s - that I became imbued with the utter importance of computing for science and mathematics. Los Alamos, under the influence of von Neumann, was for a while in the 1950s and the early 1960s the undisputed leader in computational science.

Research Contributions

R & S: May we come back to computers later? First some questions about some of your main research contributions to mathematics: You have made outstanding contributions to the theory of nonlinear partial differential equations. For the theory and numerical solutions of hyperbolic systems of conservation laws your contribution has been decisive, not to mention your contribution to the understanding of the propagation of discontinuities, so-called shocks. Could you describe in a few words how you were able to overcome the formidable obstacles and difficulties this area of mathematics presented?

Lax: Well, when I started to work on it I was very much influenced by two papers. One was Eberhard Hopf's on the viscous limit of Burgers' equation, and the other was the von Neumann-Richtmyer paper on artificial viscosity. And looking at these examples I was able to see what the general theory might look like.

R & S: The astonishing discovery by Kruskal and Zabusky in the 1960s of the role of solitons for solutions of the Korteweg-de Vries (KdV) equation, and the no less astonishing subsequent explanation given by several people that the KdV equation is completely integrable, represented a revolutionary development within the theory of nonlinear partial differential equations. You entered this field with an ingenious original point of view, introducing the so-called Lax-pair, which gave an understanding of how the inverse scattering transform applies to equations like the KdV, and also to other nonlinear equations which are central in mathematical physics like the sine-Gordon and the nonlinear Schrödinger equation. Could you give us some thoughts on how important you think this theory is for mathematical physics and for applications, and how do you view the future of this field?

Lax: Perhaps I should start by pointing out that the astonishing phenomenon of the interaction of solitons was discovered by numerical calculations, as was predicted by von Neumann some years before, namely that calculations will reveal extremely interesting phenomena. Since I was a good friend of Kruskal, I learned early about his discoveries, and that started me thinking. It was quite clear that there are infinitely many conserved quantities, and so I asked myself: How can you generate all at once an infinity of conserved quantities? I thought if you had a transformation that preserved the spectrum of an operator then that would be such a transformation, and that turned out to be a very fruitful idea, applicable quite widely.

Now you ask how important is it? I think it is pretty important. After all, from the point of view of technology for the transmission of signals, signalling by solitons is very important and a promising future technology in trans-oceanic transmission. This was developed by Linn Mollenauer, a brilliant engineer at Bell Labs. It has not yet been put into practice, but it will be some day. The interesting thing about it is that classical signal theory is entirely linear, and the main point of soliton signal transmission is that the equations are nonlinear. That's one aspect of the practical importance of it.

As for the theoretic importance: the KdV equation is completely integrable, and then an astonishing number of other completely integrable systems were discovered. Completely integrable systems can really be solved in the sense that the general population uses the word solved. When a mathematician says he has solved the problem he means he knows the solution exists, that it's unique, but very often not much more.

Now the question is: Are completely integrable systems exceptions to the behaviour of solutions of non-integrable systems, or is it that other systems have similar behaviour, only we are unable to analyse 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 revealing things. So I do think that studying completely integrable systems will give a clue to the behaviour of more general systems as well.

Who could have guessed in 1965 that completely integrable systems would become so important?

R & S: The next question is about your seminal paper "Asymptotic solutions of oscillating initial value problems" from 1957. This paper is considered by many people to be the genesis of Fourier Integral Operators. What was the new viewpoint in the paper that proved to be so fruitful?

Lax: It is a micro-local description of what is going on. It combines looking at the problem in the large and in the small. It combines both aspects, and that gives it its strengths. The numerical implementation of the micro-local point of view is by wavelets and similar approaches, which are very powerful numerically.

R & S: May we touch upon your collaboration with Ralph Phillips - on and off over a span of more than thirty years - on scattering theory, applying it in a number of settings. Could you comment on this collaboration, and what do you consider to be the most important results you obtained?

Lax: That was one of the great pleasures of my life! Ralph Phillips is one of the great analysts of our time and we formed a very close friendship. We had a new way of viewing the scattering process with incoming and outgoing subspaces. We were, so to say, carving a semi-group out of the unitary group, whose infinitesimal generator contained almost all the information about the scattering process. So we applied that to classical scattering of sound waves and electromagnetic waves by potentials and obstacles. Following a very interesting discovery of Faddeev and Pavlov, we studied the spectral theory of automorphic functions. We elaborated it further, and we had a brand new approach to Eisenstein series for instance, getting at spectral representation via translation representation. And we were even able to contemplate - following Faddeev and Pavlov - the Riemann hypothesis peeking around the corner.

R & S: That must have been exciting!

Lax: Yes! Whether this approach will lead to the proof of the Riemann hypothesis, stating it, as one can, purely in terms of decaying signals by cutting out all standing waves, is unlikely. The Riemann hypothesis is a very elusive thing. You may remember in Peer Gynt there is a mystical character, the Boyg, which bars Peer Gynt's way wherever he goes. The Riemann hypothesis resembles the Boyg!

R & S: Which particular areas or questions are you most interested in today?

Lax: I have some ideas about the zero dispersion limit.

Pure and Applied Mathematics

R & S: May we raise a perhaps contentious issue with you: pure mathematics versus applied mathematics. Occasionally one can hear within the mathematical community statements that the theory of nonlinear partial differential equations, though profound and often very important for applications, is fraught with ugly theorems and awkward arguments. In pure mathematics, on the other hand, beauty and aesthetics rule. The English mathematician G H Hardy is an extreme example of such an attitude, but it can be encountered also today. How do you respond to this? Does it make you angry?

Lax: I don't get angry very easily. I got angry once at a dean we had, terrible son of a bitch, destructive liar, and I got very angry at the mob that occupied the Courant Institute and tried to burn down our computer. Scientific disagreements do not arouse my anger. But I think this opinion is definitely wrong. I think Paul Halmos once claimed that applied mathematics was, if not bad mathematics, at least ugly mathematics, but I think I can point to those citations of the Abel Committee dwelling on the elegance of my works!

Now about Hardy: When Hardy wrote A Mathematician's Apology he was at the end of his life, he was old, I think he had suffered a debilitating heart attack, he was very depressed. So that should be taken into account. About the book itself: There was a very harsh criticism by the chemist Frederick Soddy, who was one of the co-discoverers of the isotopes - he shared the Nobel Prize with Rutherford. He looked at the pride that Hardy took in the uselessness of his mathematics and wrote: "From such cloistral clowning the world sickens." It was very harsh because Hardy was a very nice person.

My friend Joe Keller, a most distinguished applied mathematician, was once asked to define applied mathematics and he came up with this: "Pure mathematics is a branch of applied mathematics." Which is true if you think a bit about it. Mathematics originally, say after Newton, was designed to solve very concrete problems that arose in physics. Later on, these subjects developed on their own and became branches of pure mathematics, but they all came from applied background. As von Neumann pointed out, after a while these pure branches that develop on their own need invigoration by new empirical material, like some scientific questions, experimental facts, and, in particular, some numerical evidence.

R & S: In the history of mathematics, Abel and Galois may have been the first great mathematicians that one may describe as "pure mathematicians", not being interested in any "applied" mathematics as such. However, Abel did solve an integral equation, later called "Abel's integral equation", and Abel gave an explicit solution, which incidentally may have been the first time in the history of mathematics that an integral equation had been formulated and solved. Interestingly, by a simple reformulation one can show that the Abel integral equation and its solution are equivalent to the Radon Transform, the mathematical foundation on which modern medical tomography is based. Examples of such totally unexpected practical applications of pure mathematical results and theorems abound in the history of mathematics - group theory that evolved from Galois' work is another striking example. What are your thoughts on this phenomenon? Is it true that deep and important theories and theorems in mathematics will eventually find practical applications, for example in the physical sciences?

Lax: Well, as you pointed out, this has very often happened: Take for example Eugene Wigner's use of group theory in quantum mechanics. And this has happened too often to be just a coincidence. Although, one might perhaps say that other theories and theorems which did not find applications were forgotten. It might be interesting for a historian of mathematics to look into that phenomenon. But I do believe that mathematics has a mysterious unity which really connects seemingly distinct parts, which is one of the glories of mathematics.

R & S: You have said that Los Alamos was the birthplace of computational dynamics, and I guess it is safe to say that the U.S. war effort in the 1940s advanced and accelerated this development. In what way has the emergence of the high-speed computer altered the way mathematics is done? Which role will high-speed computers play within mathematics in the future?

Lax: It has played several roles. One is what we saw in Kruskal's and Zabusky's discovery of solitons, which would not have been discovered without computational evidence. Likewise the Fermi-Pasta-Ulam phenomenon of recurrence was also a very striking thing which may or may not have been discovered without the computer. That is one aspect.

But another is this: in the old days, to get numerical results you had to make enormously drastic simplifications if your computations were done by hand, or by simple computing machines. And the talent of what drastic simplifications to make was a special talent that did not appeal to most mathematicians. Today you are in an entirely different situation. You don't have to put the problem on a Procrustean bed and mutilate it before you attack it numerically. And I think that has attracted a much larger group of people to numerical problems of applications - you could really use the full theory. It invigorated the subject of linear algebra, which as a research subject died in the 1920s. Suddenly the actual algorithms for carrying out these operations became important. It was full of surprises, like fast matrix multiplication. In the new edition of my linear algebra book I will add a chapter on the numerical calculation of the eigenvalues of symmetric matrices.

You know it's a truism that due to increased speed of computers, a problem that took a month forty years ago can be done in minutes, if not seconds today. Most of the speed-up is attributed, at least by the general public, to increased speed of computers. But if you look at it, actually only half of the speed-up is due to this increased speed. The other half is due to clever algorithms, and it takes mathematicians to invent clever algorithms. So it is very important to get mathematicians involved, and they are involved now.

R & S: Could you give us personal examples of how questions and methods from applied points of view have triggered "pure" mathematical research and results? And conversely, are there examples where your theory of nonlinear partial differential equations, especially your explanation of how discontinuities propagate, have had commercial interests? In particular, concerning oil exploration, so important for Norway!

Lax: Yes, oil exploration uses signals generated by detonations that are propagated through the earth and through the oil reservoir and are recorded at distant stations. It's a so-called inverse problem. If you know the distribution of the densities of materials and the associated waves' speeds, then you can calculate how signals propagate. The inverse problem is that if you know how signals propagate, then you want to deduce from it the distribution of the materials. Since the signals are discontinuities, you need the theory of propagation of discontinuities. Otherwise it's somewhat similar to the medical imaging problem, also an inverse problem. Here the signals do not go through the earth but through the human body, but there is a similarity in the problems. But there is no doubt that you have to understand the direct problem very well before you can tackle the inverse problem.

Hungarian Mathematics

R & S: Now to some questions related to your personal history. The first one is about your interest in, and great aptitude for, solving problems of a type that you call "Mathematics Light" yourself. To mention just a few, already as a seventeen-year-old boy you gave an elegant solution to a problem that was posed by Erdös and is related to a certain inequality for polynomials, which was earlier proved by Bernstein. Much later in your career you studied the so-called Pólya function which maps the unit interval continuously onto a right-angled triangle, and you discovered its amazing differentiability properties. Was problem solving specifically encouraged in your early mathematical education in your native Hungary, and what effect has this had on your career later on?

Lax: Yes, problem solving was regarded as a royal road to stimulate talented youngsters, and I was very pleased to learn that here in Norway they have a successful high-school contest, where the winners were honoured this morning. But after a while one shouldn't stick to problem solving, one should broaden out. I return to it every once in a while, though.

Back to the differentiability of the Pólya function: I knew Pólya quite well having taken a summer course with him in 1946. The differentiability question came about this way: I was teaching a course on real variables, and I presented Pólya's example of an area-filling curve, and I gave as homework to the students the problem of proving that it's nowhere differentiable. Nobody did the homework, so then I sat down and I found out that the situation was more complicated.

There was a tradition in Hungary to look for the simplest proof. You may be familiar with Erdös' concept of The Book. That's The Book kept by the Lord of all theorems and the best proofs. The highest praise that Erdös had for a proof was that it was out of The Book. One can overdo that, but shortly after I had gotten my Ph.D., I learned about the Hahn-Banach theorem, and I thought that it could be used to prove the existence of Green's function. It's a very simple argument - I believe it's the simplest - so it's out of The Book. And I think I have a proof of Brouwer's Fixed Point Theorem, using calculus and just change of variables. It is probably the simplest proof and is again out of The Book. I think all this is part of the Hungarian tradition. But one must not overdo it.

R & S: There is an impressive list of great Hungarian physicists and mathematicians of Jewish background that had to flee to the U.S. after the rise of fascism, Nazism and anti-Semitism in Europe. How do you explain this extraordinary culture of excellence in Hungary that produced people like de Hevesy, Szilard, Wigner, Teller, von Neumann, von Karman, Erdös, Szegö, Pólya, yourself, to name some of the most prominent ones?

Lax: There is a very interesting book written by John Lukacs with the title "Budapest 1900: A Historical Portrait of a City and its Culture", and it chronicles the rise of the middle class, rise of commerce, rise of industry, rise of science, rise of literature. It was fuelled by many things: a long period of peace, the influx of mostly Jewish population from the East eager to rise, and intellectual tradition. You know in mathematics, Bolyai was a cultural hero to Hungarians, and that's why mathematics was particularly looked upon as a glorious profession.

R & S: But who nurtured this fantastic flourishing of talent, which is so remarkable?

Lax: Perhaps much credit should be given to Julius König, whose name is probably not known to you. He was a student of Kronecker, I believe, but he also learned Cantor's set theory and made some basic contribution to it. I think he was influential in nurturing mathematics. His son was a very distinguished mathematician, Denes König, really the father of modern graph theory. And then there arose extraordinary people. Leopold Fejér, for instance, had enormous influence. There were too many to fill positions in a small country like Hungary, so that's why they had to go abroad. Part of it was also anti-Semitism.

There is a charming story about the appointment of Leopold Fejér, who was the first Jew proposed for a professorship at Budapest University. There was opposition to it. At that time there was a very distinguished theologian, Ignatius Fejér, in the Faculty of Theology. Fejér's original name was Weiss. So one of the opponents, who knew full well that Fejér's original name had been Weiss, said pointedly: This professor Leopold Fejér that you are proposing, is he related to our distinguished colleague Father Ignatius Fejér? And Eötvös, the great physicist who was pushing the appointment, replied without batting an eyelash: "Illegitimate son." That put an end to it.

R & S: And he got the job?

Lax: He got the job.

Scribbles That Changed the Course of Human Affairs

R & S: The mathematician Stanislaw Ulam was involved with the Manhattan Project and is considered to be one of the fathers of the hydrogen bomb. He wrote in his autobiography Adventures of a Mathematician: "It is still an unending source of surprise for me to see how a few scribbles on a blackboard, or on a sheet of paper, could change the course of human affairs." Do you share this feeling? And what are your feelings about what happened to Hiroshima and Nagasaki, to the victims of the explosions of the atomic bombs that brought an end to World War II?

Lax: Well, let me answer the last question first. I was in the army, and all of us in the army expected to be sent to the Pacific to participate in the invasion of Japan. You remember the tremendous slaughter that the invasion of Normandy brought about. That would have been nothing compared to the invasion of the Japanese mainland. You remember the tremendous slaughter on Okinawa and Iwo Jima. The Japanese would have resisted to the last man. The atomic bomb put an end to all this and made an invasion unnecessary. I don't believe reversionary historians who say: "Oh, Japan was already beaten, they would have surrendered anyway." I don't see any evidence for that.

There is another point which I raised once with someone who had been involved with the atomic bomb project. Would the world have had the horror of nuclear war if it had not seen what one bomb could do? The world was inoculated against using nuclear weaponry by its use. I am not saying that alone justifies it, and it certainly was not the justification for its use. But I think that is a historical fact.

Now about scribbles changing history: Sure, the special theory of relativity, or quantum mechanics, would be unimaginable today without scribbles. Incidentally, Ulam was a very interesting mathematician. He was an idea man. Most mathematicians like to push their ideas through. He preferred throwing out ideas. His good friend Rota even suggested that he did not have the technical ability or patience to work them out. But if so, then it's an instance of Ulam turning a disability to tremendous advantage. I learned a lot from him.

R & S: It is amazing for us to learn that an eighteen-year-old immigrant was allowed to participate in a top-secret and decisive weapon development during WWII.

Lax: The war created an emergency. Many of the leaders of the Manhattan Project were foreigners, so being a foreigner was no bar.

Collaboration. Work Style

R & S: Your main workplace has been the Courant Institute of Mathematical Sciences in New York, which is part of New York University. You served as its director for an eight-year period in the 1970s. Can you describe what made this institute, which was created by the German refugee Richard Courant in the 1930s, a very special place from the early days on, with a particular spirit and atmosphere? And is the Courant Institute today still a special place that differs from others?

Lax: To answer your first question, certainly the personality of Courant was decisive. Courant saw mathematics very broadly, he was suspicious of specialisation. He wanted it drawn as broadly as possible, and that's how it came about that applied topics and pure mathematics were pursued side by side, often by the same people. This made the Courant Institute unique at the time of its founding, as well as in the 1940s, 1950s, and 1960s. Since then there are other centres where applied mathematics is respected and pursued. I am happy to say that this original spirit is still present at the Courant Institute. We still have large areas of applied interest, meteorology and climatology under Andy Majda, solid state and material science under Robert Kohn and others, and fluid dynamics. But we also have differential geometry as well as some pure aspects of partial differential equations, even some algebra.

I am very pleased how the Courant Institute is presently run. It's now the third generation that's running it, and the spirit that Courant instilled in it - kind of a family feeling - still prevails. I am happy to note that many Norwegian mathematicians received their training at the Courant Institute and later rose to become leaders in their field.

R & S: You told us already about your collaboration with Ralph Phillips. Generally speaking, looking through your publication list and the theorems and methods you and your collaborators have given name to, it is apparent that you have had a vast collaboration with a lot of mathematicians. Is this sharing of ideas a particularly successful, and maybe also joyful, way of advancing for you?

Lax: Sure, sure. Mathematics is a social phenomenon after all. Collaboration is a psychological and interesting phenomenon. A friend of mine, Vera John-Steiner, has written a book (Creative Collaboration) about it. Two halves of a solution are supplied by two different people, and something quite wonderful comes out of it.

R & S: Many mathematicians have a very particular work style when they work hard on certain problems. How would you characterise your own particular way of thinking, working, and writing? Is it rather playful or rather industrious? Or both?

Lax: Phillips thought I was lazy. He was a product of the Depression, which imposed a certain strict discipline on people. He thought I did not work hard enough, but I think I did!

R & S: Sometimes mathematical insights seem to rely on a sudden unexpected inspiration. Do you have examples of this sort from your own career? And what is the background for such sudden inspiration in your opinion?

Lax: The question reminds me of a story about a German mathematician, Schottky, when he reached the age of seventy or eighty. There was a celebration of the event, and in an interview like we are having, he was asked: "To what do you attribute your creativity and productivity?" The question threw him into great confusion. Finally he said: "But gentlemen, if one thinks of mathematics for fifty years, one must think of something!" It was different with Hilbert. This is a story I heard from Courant. It was a similar occasion. At his seventieth birthday he was asked what he attributed his great creativity and originality to. He had the answer immediately: "I attribute it to my very bad memory." He really had to reconstruct everything, and then it became something else, something better. So maybe that is all I should say. I am between these two extremes. Incidentally, I have a very good memory.

Teaching

R & S: You have also been engaged in the teaching of calculus. For instance, you have written a calculus textbook with your wife Anneli as one of the co-authors. In this connection you have expressed strong opinions about how calculus should be exposed to beginning students. Could you elaborate on this?

Lax: Our calculus book was enormously unsuccessful, in spite of containing many excellent ideas. Part of the reason was that certain materials were not presented in a fashion that students could absorb. A calculus book has to be fine-tuned, and I didn't have the patience for it. Anneli would have had it, but I bullied her too much, I am afraid. Sometimes I dream of redoing it because the ideas that were in there, and that I have had since, are still valid.

Of course, there has been a calculus reform movement and some good books have come out of it, but I don't think they are the answer. First of all, the books are too thick, often more than 1,000 pages. It's unfair to put such a book into the hands of an unsuspecting student who can barely carry it. And the reaction to it would be: "Oh, my God, I have to learn all that is in it?" Well, all that is not in it! Secondly, if you compare it to the old standards, Thomas, say, it's not so different - the order of the topics and concepts, perhaps.

In my calculus book, for instance, instead of continuity at a point, I advocated uniform continuity. This you can explain much more easily than defining continuity at a point and then say the function is continuous at every point. You lose the students; there are too many quantifiers in that. But the mathematical communities are enormously conservative: "Continuity has been defined pointwise, and so it should be!"

Other things that I would emphasise: To be sure there are applications in these new books. But the applications should all stand out. In my book there were chapters devoted to the applications, that's how it should be - they should be featured prominently. I have many other ideas as well. I still dream of redoing my calculus book, and I am looking for a good collaborator. I recently met someone who expressed admiration for the original book, so perhaps it could be realised, if I have the energy. I have other things to do as well, like the second edition of my linear algebra book, and revising some old lecture notes on hyperbolic equations. But even if I could find a collaborator on a calculus book, would it be accepted? Not clear. In 1873, Dedekind posed the important question: "What are, and what should be, the real numbers?" Unfortunately, he gave the wrong answer as far as calculus students are concerned. The right answer is: infinidecimals. I don't know how such a joke will go down.

Heading Large Institutions

R & S: You were several times the head of large organisations: director of the Courant Institute in 1972-1980, president of the American Mathematical Society in 1977-1980, leader of what was called the Lax Panel on the National Science Board in 1980-1986. Can you tell us about some of the most important decisions that had to be taken in these periods?

Lax: The president of the American Mathematical Society is a figurehead. His influence lies in appointing members of committees. Having a wide friendship and reasonable judgement are helpful. I was very much helped by the secretary of the American Mathematical Society, Everett Pitcher.

As for being the director of the Courant Institute, I started my directorship at the worst possible time for New York University. They had just closed down their School of Engineering, and that meant that mathematicians from the engineering school were transferred to the Courant Institute. This was the time when the Computer Science Department was founded at Courant by Jack Schwartz. There was a group of engineers that wanted to start activity in informatics, which is the engineers' word for the same thing. As a director I fought very hard to stop that. I think it would have been very bad for the university to have had two computing departments - it certainly would have been very bad for our Computer Science Department. Other things: Well, I was instrumental in hiring Charlie Peskin at the recommendation of Alexander Chorin. I was very pleased with that. Likewise, hiring Sylvain Cappell at the recommendation of Bob Kohn. Both were enormous successes.

What were my failures? Well, maybe when the Computer Science Department was founded I should have insisted on having a very high standard of hiring. We needed people to teach courses, but in hindsight I think we should have exercised more restraint in our hiring. We might have become the number one computer science department. Right now the quality has improved very much - we have a wonderful chairwoman, Margaret Wright.

Being on the National Science Board was my most pleasant administrative experience. It's a policy-making body for the National Science Foundation (NSF), so I found out what making policy means. Most of the time it just means nodding "yes", and a few times saying "no". But then there are sometimes windows of opportunity, and the Lax Panel was a response to such a thing. You see, I noticed through my own experience and that of my friends who are interested in large scale computing (in particular, Paul Garabedian, who complained about it), that university computational scientists had no access to the supercomputers. At a certain point the government, which alone had enough money to purchase these supercomputers, stopped placing them at universities. Instead they went to national labs and industrial labs. Unless you happened to have a friend there with whom you collaborated, you had no access. That was very bad from the point of view of the advancement of computational science, because the most talented people were at the universities. At that time accessing and computing at remote sites became possible thanks to ARPANET, which then became a model for the Internet. So the panel that I established made strong recommendation that the NSF establish computing centres, and that was followed up. My quote on our achievement was a paraphrase of Emerson: "Nothing can resist the force of an idea that is ten years overdue."

R & S: A lot of mathematical research in the U.S. has been funded by contracts from DOD (Department of Defense), DOE (Department of Energy), the Atomic Energy Commission, the NSA (National Security Agency). Is this dependence of mutual benefit? Are there pitfalls?

Lax: I am afraid that our leaders are no longer aware of the subtle but close connection between scientific vigour and technological sophistication.

Personal Interests

R & S: Would you tell us a bit about your interests and hobbies that are not directly related to mathematics?

Lax: I love poetry. Hungarian poetry is particularly beautiful, but English poetry is perhaps even more beautiful. I love to play tennis. Now my knees are a bit wobbly, and I can't run anymore, but perhaps these can be replaced - I'm not there yet. My son and three grandsons are tennis enthusiasts so I can play doubles with them. I like to read. I have a knack for writing. Alas, these days I write obituaries - it's better to write them than being written about.

R & S: You have also written Japanese haikus?

Lax: You're right. I got this idea from a nice article by Marshall Stone - I forget exactly where it was - where he wrote that the mathematical language is enormously concentrated, it is like haikus. And I thought I would take it one step further and actually express a mathematical idea by a haiku.

R & S: Professor Lax, thank you very much for this interview on behalf of the Norwegian, the Danish, and the European Mathematical Societies!

Lax: I thank you.

11.1. SIAM Prize for Distinguished Service to the Profession.

The SIAM Prize for Distinguished Service to the Profession was established in 1985 and originally intended to be awarded periodically. It is now awarded annually for contributions to the advancement of applied mathematics on the national or international level.

11.2. Peter Lax wins the SIAM Prize for Distinguished Service to the Profession.

Peter D Lax was the eighth recipient of the SIAM Prize for Distinguished Service to the Profession when he received the award in 2006.

12.1. The Lomonosov Gold Medal.

The Lomonosov Gold Medal is named after the Russian scientist Mikhail Lomonosov. It is awarded each year since 1959 for outstanding achievements in the natural sciences and the humanities by the USSR Academy of Sciences and later by the Russian Academy of Sciences. Since 1967, two medals have been awarded annually: one to a Russian and one to a foreign scientist. It is the Russian Academy of Sciences' highest accolade.

12.2. Peter Lax wins the Lomonosov Gold Medal.

The two Lomonosov Gold Medals awarded in 2013 were to the Russian mathematician Ludwig Dmitrievich Faddeev and to the Peter David Lax, a professor in the United States of America. Lax received the award

... for outstanding contribution to the theory of hydrodynamic solitons.