Preface by Martin Friedrichs (5/2001): This is the full transcript of the memorial service given for Kurt O. Friedrichs at the Courant Institute of Mathematics on February 14, 1983. The audience consisted of colleagues, friends, family, and students. As content to allow a reader of this document to skip to the parts they may be interested in.
S. Raghu Varadhan (Institute Director) – Short introduction James Stoker (Co founder of Institute) – Long history of Institute and personnel remembrances Louis Nirenberg (Colleague) – Mathematical contribution to partial differential equations Peter Lax (Colleague) – Mathematical topics and his approach to mathematics David Isaacson (Student) – Mathematical physics Harold Grad (Colleague) – Mathematics and fusion energy Jerry Berkowitz (Colleague) – Personal remembrances Martin Friedrichs – Personnel characteristics and impact of mathematics on Dad Walter Friedrichs – Relationship to Institute and family Cathleen Morawetz (Student and Colleague) – Short closing remarks
K. O. FRIEDRICHS
Memorial Service New York University February 14, 1983
S. RAGHU VARADHAN
This meeting is in honor of Professor Kurt Friedrichs who, all of you know, has been associated with the Institute for a long time. I want to thank all of you for coming here today, in spite of the bad weather. The way this meeting has been organized, several people will speak for 10 or 15 minutes each, reminiscing about Friedrichs and what he meant to them and his scientific contributions. I will list the speakers in the order in which they will speak. I will be followed by Professor James Stoker, then by Louis Nirenberg, Peter Lax, David Isaacson, Harold Grad and Jerry Berkowitz. They will be followed by Martin and Walter Friedrichs and then Cathleen Morawetz will close out the meeting.
I just want to add a couple of personal remarks summarizing my own feelings. The year I joined the faculty of the Institute, which was 1966, was the year that Friedrichs was the Director of the Institute and I’ve known him fairly well since that time. Although he is no longer with us, one should realize that he had a very fruitful career -- essentially two careers on two different continents spanning his lifetime and through his scholarship, his professional work, his students, his ideas and the tradition he leaves behind. I think as far as we’re concerned, he’ ll be immortal in our minds.
JAMES J. STOKER
I should have spoken at the big birthday celebration, but I will say some of the things I had intended to say there -- but that would have been a happy occasion -- this one I feel sad about and will for a long time. Friedrichs and I were friends for a long time, 45 years in fact, and I will confine my remarks to describing our friendship. Others may talk about his scientific doings and such things as that, but for one thing, I don’t want to talk too long, that’s risky for a man of my age -- particularly a professor of that age -- they are known to be loquacious and to want to go down pathways reminiscing at great length. I think of Courant when I say that. A favorite phrase of his, often repeated was “Old times in Göttingen.” Well, one of the dangers of doing what I have m mind to do is that describing a friendship means that I have to talk somewhat about myself and of course that’s dangerous since I don’t want to talk too long. I think it is right that I should describe the history, how it came about that such a strange trio, as Courant, Friedrichs, and I collaborated for so long. Three more different men it would be hard to find, I would say. It came about in this way. Courant came here in 1934 and, of course, he wanted to start a school. He had in mind a school that would function somewhat in a way he had made it go in Göttingen but he needed collaborators for that. One was obvious, that was to bring Friedrichs from Germany. He knew him well, after all; Friedrichs went to G6ttingen and I think his thesis was written under Courant if I’m not mistaken.
I came in to it by accident. I came from Pittsburgh -- from the Carnegie Institute of Technology -- and Courant had seen me only once and knew of me only because I was a pupil of Heinz Hopf in theory. Hopf wrote to Courant and recommended me to him, so Courant brought me on to meet me in New York and we talked together and he offered me a position at the engineering school at the Heights. I was somewhat dubious about it; it seemed a strange thing to be doing. I should say I would not have gone to Zurich for my degree if I could have gone to G6ttingen where Courant was. It was very obvious that at the time I went to Europe, that was in 1932, that it was just not going to work in Germany -- the conditions were utterly unfavorable for that -- so by accident I landed in Zurich. I knew no one there, not Hopf either. In fact, I intended to work for a man who taught mechanics because that was my main subject; I was in the Engineering Mechanics Department in Pittsburgh at the Carnegie Institute of Technology. I listened to Heinz Hopf in the first semester in the lectures. He taught complex variable and differential geometry. He was an extraordinarily capable lecturer and I was so taken with him that I went to him for a subject for a thesis which would have been not in applied mathematics at all. I am a peculiar mixture; I am an applied mathematician but my degree is in differential geometry. Anyway, it was through Hopf then that Courant knew of me.
I came then finally to New York -- that was in 1937, in the summer -- and Kurt Friedrichs arrived in the same month, and we became acquainted at once. We immediately started collaborating on a problem. I would have enjoyed describing that problem but that would take too much time. We worked on that problem for 2 years and I had a chance to see Friedrichs’ wonderful capability in analysis, his fertile inventions. That problem was very complicated -- non-linear and difficult -- and it was attacked from various directions and each time Friedrichs had a new fertile idea about what to do. We worked for 2 years, mostly weekends because we taught undergraduates at the Heights and had a graduate class -- so the weekends were for research.
Part of the things that we did was to compute numerically. Once the analytical formulas were there it was important to see what would actually happen if you solved a problem numerically. There were very surprising things that happened. We were constantly fascinated and wondered whether what we were doing could be right because it seemed so strange. One of the them was: What happens if you have a plane, of elastic circular plate and you put it under compression. If the pressure is too large it will become unstable and buckle or bend. One of the first things Friedrichs did was to make some preliminary calculations which indicated very strongly that in the course of buckling, the compression would be relieved gradually. Of course, that is from differential geometry, fairly obvious, but that with very high pressure the uniform compression initially was turned finally into a uniform tension everywhere. We were somewhat amazed at that. The computations were frightful. We hadn’t even a desk computer though we finally did get one, so we calculated using logarithms, for instance, for a while. We spent a tremendous amount of time in that rather strange enterprise but now it is good to remark.
Nowadays those things that took us a couple of years would be done in a couple of minutes, or even seconds, with modern computing equipment. Friedrichs and I from then on were together for 45 years. On the other hand, we never again had the luxury of spending weeks working ourselves, because we had to work with other people.
Courant had the object of creating a school according to his ideas of how that should be. The main ones were: teaching and research should go hand in hand. Also, he wanted to select people carefully so that it would be a congenial working group, cooperative, and on good terms with one another. That he succeeded in very well. Those principles we stuck to. Friedrichs and I both agreed with him thoroughly and did what we could to help him out, though it was slow progressing at first, principally because the University had no money and Courant felt that it was his job to find it. It was quite a while before much was found. Still, in a short time, Courant had collected some very good people around him, good students.
It was a peculiar collaboration between three very different people, but it worked and partly because Friedrichs and I were in sympathy with the aims but we were not always so sympathetic with the tactics of getting on. That meant that Fried-richs was in the middle, which he really detested, he didn’t like that kind of trouble. However, that’ s the way it was with us all. We disagreed at times, even rather strongly, but nevertheless we never had a quarrel in 45 years, never. If somehow we disagreed, then agreement was reached one way or another, then we would both accept it and try to go along with it. Eventually though, things changed so enormously and when I think back over it, Friedrichs was somewhat unlike me in that he believed very much in careful and thoughtful planning and he hoped that things would go according to plan. I have noticed that, in looking back, a great deal depended purely on accidents that you couldn’t foresee at all.
The big growth in our Institute and the first beginnings of a success for Courant came with our entrance into World War II. It was then that some money became available through what was called the Applied Mathematics Panel headed by Warren Weaver and Courant was closely associated with him. Weaver found some money to give Courant so that we could support students. Also, that had the effect that the three of us stayed together in New York. In the rest of the country, the people in mathematics who could work in applications mostly left their places to go to Aberdeen or Washington or other places where some kind of war related research was done. In that way we trained quite a few people. Quite a few of them are still with us. I think I should name some of those who are still here from those days: Louis Nirenberg, Harold Grad, Harold Shapiro, Anneli, Eugene Isaacson, and at that time Joe Keller -- he’s gone now but he was with us for most of the time. So the next generation already was on hand by the end of World War II. Of course Peter was in the army but as soon as the war ended he was back with us, and also Fritz John was with us for a time after the end of the war and then came to us permanently not so long afterward. So Courant had a big starting of what he wanted.
The big discontinuity in size and the way things went came when we got our first big computer, the Univac. That pretty much doubled the size of the enterprise in one blow and from then on there was no way that Friedrichs and I could disappear every weekend and work together. We had to work with other people which in a way is too bad, but it had its good side also. Later on I became the head of the enterprise, which I didn’t really like, but it lasted for 8 years and Friedrichs and I had very much to do with one another then because there were many difficulties. I had some differences with Courant about tactics and I wanted to change them radically but that was not so simple. In the end it worked out and Friedrichs was sometimes perturbed at it, nevertheless when he felt I was in the right he always backed me. I can really say that in 45 years of friendship we never had a quarrel of any consequence. I disagreed, disagreed rather violently some times, because I think many people have a false notion about Kurt.
Kurt was a man who basically loved danger. He liked to live in danger. I first noticed it in sports for instance, it was one of the cooperative things we did among our Institute people. Courant and Friedrichs first introduced me to skiing in the Adirondicks at Lake Placid which is a difficult enterprise. I floundered down those slopes for a couple of years before catching on to some of the tricks. I was sometimes aghast at watching Friedrichs. His way of taking those steep slopes with a right-angled bend at the bottom, was to go full speed and hope to make the turn. He was that kind of perfectionist, a professional in his attitude. I was sometimes aghast to seem him, in fact I watched him when he had his accident. It was at just such a place, I can see it very clearly, a very beautiful sunshiny day, very steep slope, the bottom a sharp right-angle turn. He couldn’t make it so he went straight ahead, put out his left arm to put the brakes on against a tree and broke his arm. From that time on his skiing declined -- not entirely but somewhat.
Friedrichs and I had a common interest in walking and looking at the landscape and observing what happens and identifying the birds and things like that. We both were fond of mountains and particularly skiing. Also, I shouldn’t really admit it, I once led Friedrichs and another man up a more or less vertical wall attached to me with a rope. Friedrichs loved to do it, he wanted to do it, but it is a mistake to think that because a man loves danger, that he isn’t afraid. That’s just not true. Friedrichs was quite excited about the thing. I had no business doing it, I hadn’t enough experience to secure those fellows on a rope on that cliff. Finally I brought the two of them up to a place where I could secure them but the rope was then tangled. There was an overhang there which I had to deal with, that usually causes a lot of trouble, but this one had lots of well-placed holes on it so that it wasn’t so dangerous. I was hanging on and couldn’t go ahead because they had the rope tangled. Friedrichs wanted to straighten out the principle -- what was wrong and who was at fault. I finally told him to separate it and let me go on and I’ll get you up to the top of this cliff without any further bother.
Friedrichs also, in his mathematics, always picked out the dangerous problems to work on, the really tough ones, in which there was no certainty of getting through and then he would push them through. I recall he told me once that someone had once made a remark about him in which he said, “Oh yes, then there is Friedrichs, that intrepid analyst.” That is a very good phrase for him. He seemed reserved but still there was that in him, strongly.
We had quite a few tastes in common but some not; for instance, a lifelong and very strong taste I have is for the reading of poetry. It began when I was a young boy and has continued all my life, and gradually in various languages besides English. I tried to persuade Friedrichs to take up that interest but he always said, “No, that’s not for me. The only poems I like to read are narrative poems in verse.” That is, he wanted to read a story. Friedrichs did not like lyrics with the hard and difficult issues in living.
I thought to end by inflicting my taste on you. It’s the first poem written and made known to the people in Italy; written by San Francesco d’ Assissi -- a beautiful poem -- “Cantico” about the creation, as in Genesis. He praises all the wonderful things that were created, in simple but beautiful language. It says, finally, “I praise our deaths which no living man can escape.” He was saying that it is a blessing also, and should be recognized as such, and in the end I think that Friedrichs did that. He would not see his old friends. In the end he did not want that any more. Nellie tells me that he ended without pain, peacefully and quietly as I would have expected. In his death he had the same kind of natural dignity and a sense of decorum and he didn’t want people around when he wasn’t at his best. I believe that is true.
Others will talk about his scientific accomplishments; I’ve indicated that I have a lot of feeling for them. I feel strongly that the wonderful work he’s done will remain in the memory of mathematicians for many years to come, along with a lot of first-class mathematicians of the last hundred years. He belongs among them.
I would like to talk about some aspects of Friedrichs work in partial differential equations. One of the aims of the gathering is to try to give some impression of the effect that Friedrichs had on the Courant Institute, on the research of people here, and on the points of view of people here. I know that many of the young people who are at the Institute are not fully aware of that. In the past few years the y have seen Friedrichs around. He would come once a week and they would have a little bit of contact but I don’t think that they fully appreciate what he did for the Institute and, in fact, it’s not at all easy to describe. I’d like to indicate a little of that. Of course people know that he worked in many fields of mathe-matics, both pure and applied. Everyone would at some point go to talk to Friedrichs to tell him about what he or she had done. This usually meant a three week appointment in advance, if you were lucky. He was an extremely busy man.
When I came here, just after the war as a graduate student, there were just a few of us at the time. We quickly learned where the action was. The action was wherever Friedrichs was. He was one of the main intellectual forces here. That remained so for years. His ideas and taste really shaped the taste and develop-ment of many people here; also his prejudices.4 perhaps. He was human, he had prejudices like everyone else, but some of his best friends were number theorists. I would like to say that Professor Jim Stoker was my thesis advisor but, in addition to him, I talked a great deal with Friedrichs. I feel that Friedrichs played an enormous role in molding my taste and view of mathematics.
For Friedrichs it was very important to stress methods. The methods were more important than the results. Very often when we worked on a problem, we didn’t care how the problem came out. We were trying to prove something and it didn’t matter if we proved that or we proved something exactly the opposite. What was interesting was to develop some methods that could be used not only on that prob-lem but on lots of other problems. The stress was on the methods. Friedrichs wanted to develop techniques which didn’t just solve one problem but which could then be applied to a host of others.
One of the main techniques was to find new inequalities. He was a lover of inequalities and I guess he infected rather than affected many of us with that love. Many of us simply became lovers of inequalities and somehow we almost view identities with contempt. Of course among the inequalities are energy estimates. These occur in his work in elliptic equations, hyperbolic equations, and mixed equations. Energy estimates were crucial in his work.
Another point of view that he felt very strongly about was that rough methods are better than refined methods. Very often some mathematician would be very proud of having done some very refined, delicate argument and Friedrichs, on the other hand, was proud for not having done a refined, delicate argument but having found some very crude argument which would do the job and being crude, could do other jobs too. It wasn’t just for one particular argument, but was adaptable. Some of that sank in on most of us: that the rougher the better, the cruder the better. This may come to a surprise to people who didn’t know him well, he was very, willing to make a mess. I don’t just refer to notation, this was well known.
Friedrichs was very willing to ruin any equation. I think it gave him particular joy if there was some problem that had a natural symmetry where people had blinder on and were tied to that symmetry. He would deliberately break the symmetry and get something new out of it. When people met him for the first time he made an impression of someone who doesn’t break the rules. My guess is, I will ask Nellie privately afterwards, that perhaps he got more pleasure out of breaking the rules than the rest of us do. We all break rules and we all get some joy; I conjecture that he got more joy out of breaking them, perhaps because he appreciated them more to begin with.
I want to give one or two of examples of his taking some nice equation and break-ing the niceness in order to achieve something else. For instance, one of the things we learned from him was that it pays to make a transformation which makes a nice domain at the cost of messing up the equations. In fluid dynamics, people use specific transformations to change the equation and in recent years, in so-called “free boundary problems”, people have been using some offbeat trans-formations which mess up the equation; Friedrichs already did that. One of the transformations that was used successfully just recently he had already done in 1934. He had seen that it pays to make that messy transformation.
It is very standard now in attacking partial differential equations and proving existence theorems to first obtain, so to speak, generalized solutions; that is, to work in some very broad’ class of functions. It has now become very traditional and you find it in all the books. Friedrichs was a pioneer in this progress. He was one of the first in the 1930’s to introduce the notions of generalized solutions.
He introduced things like Sobolev spaces. He derived many of the properties of these spaces. Sobolev was more systematic and derived a wider range of inequalities and somehow his name got attached to the spaces. Friedrichs more or less did them independently and this work was later picked up by Shalger and others in attacking non-linear problems. The work they had done was for linear problems.
He introduced notions of generalized functions having generalized derivatives, used them with some abstract functional analysis to solve boundary value problems or initial value problems for generalized solutions. The next step is to prove regularity of generalized solutions. He developed suitable techniques for that. He introduced very good smoothing procedures, operators they are called, which I normally call the Friedrichs mollifiers. In using those in a classic paper in 1953, he showed that generalized solutions of elliptic equations are smooth. Using the structure L2 theory, generalized solutions, energy, inequalities and smoothing, he proved the regularity theorem. That has now become sort of the classical way of proving regularity theorems. He adapted that procedure to redo the Hodge theory of harmonic forms on differential manifolds, in particular manifolds with boundary. One of the things that gave him pleasure in that, was that the techniques developed here applied very well there; and there you have invariant ways of writing forms and he found it very useful to use some noninvariant expressions. I think he took particular pleasure in pointing out that certain expressions which didn’t have a good invariant properties were particularly useful in proving the results.
I would like to call attention to one particular paper that I think is a kind of tour de force of Friedrichs; that’s his paper on the solitary wave, a paper with D. H. Hyers published in 1954. The paper is very striking. It was the first proof of the existence of such a wave and many people had tried. One of the striking things about it, is that there is a transformation of variable that’s made, a sort of stretching of variable. There is this stretching in the so-called shallow water theory that was done by Friedrichs and Hyers which ruins the equation. This is a perfect example of having a beautiful equation and messing it up by this horrible transformation of variable but that horrible transformation enables them to get through. It is a most striking, very remarkable paper to my mind. A few years ago a slightly simpler proof was found, but still Friedrichs’ result is used as a main ingredient in that paper. It is still a most remarkable paper.
PETER D. LAX
I plan to describe some of Friedrichs’ contributions to partial differential equations both linear and non-linear, but I too would like to start with some general observa-tions of how Friedrichs went about doing mathematics.
For Friedrichs mathematics was not a matter of proving theorems in a subject matter that was well established, but it was an effort to understand phenomena whose root was mostly but not always, in the physical world.’ The mathematics that he used, or mostly developed, was created as an aid in understanding. Therefore, the mathematics that he created was very strongly shaped by his own interest and taste and outlook. So much of what he has done has passed into the mathematical folklore and is referred merely as elliptic regularity or standard wave propagations.
The subjects that interested Friedrichs and on which he worked covered a very wide range. Nevertheless, Friedrichs was not a man to let this tremendous facility do things for him. To learn mathematics and to work out technical details, even after he has conceived the main lines of an attack, did not come easily to him. He had to slug it out and slug it out he did. I recall a time he was struggling with a mathematical problem that was in Russian and I asked him if he had difficulty reading it because it was in Russian. He said he had difficulty but not because it was in Russian but because it was mathematics.
I’d like to describe in a very sketchy way the work of Friedrichs in wave propaga-tion -- especially those guided by hyperbolic equations. Friedrichs worked both in the linear and non-linear theory of these equations. Friedrichs was deeply involved in the scattering theory at a time when that was a branch of physics not of mathematics. Friedrichs was drawn to it by his very early interest in operators with a continuous spectrum. That too has penetrated the folklore but at that time most people’s intuitions were formed more from the finite dimensional.
As Louis already remarked Friedrichs believed in proving the existence of solutions based on inequalities and in deriving these inequalities by the energy method. The energy method exploits a kind of symmetry in the governing equations and these symmetries are either inherent or are artificially created and Friedrichs was a master of creating such kind of artificial symmetry. In fact, when pseudo-differential operators made their way into mathematics, Friedrichs, who was in his 60’s, was very much involved in it. In fact, the name “pseudo-differential operator” was invented by Friedrichs according to strict principles with which he operated and he wrote lecture notes on it. He was drawn to it because they are such a flexible tool for creating unexpected symmetries. I presume he was also drawn to it because some of the ideas must have been familiar to him from the work of Weyl, the Weyl calculus, and Friedrichs was a great admirer of Herman Weyl and much, but not all, of his mathematics was influenced by it.
I would like to say a few words about a paper of Friedrichs which he published in 1958. It’s called “Symmetric Positive Systems”. This is an equation which embraces elliptic hyperbolic mixed. He was specifically interested in the Tricomi equation and he was very pleased and very proud at having found this general class which he could teach by his methods. He also invented a way of reducing these equations to difference equations so calculations could be made. Since the conditions for solvability of the equation made no explicit reference to speed of propagation of signals to sound speed, the difference equations, unlike the usual hyperbolic ones which have restrictions known as the Courant-Friedrichs restric-tion, had none in this setup. The inequalities he got came out of the zero order of terms rather than the first order of terms which contain such things as sound speed. So, Friedrichs was very pleased about that too. He called these difference equations, since they made no reference to sound speed, “deaf equa-tions,” but he added that they are not dumb. As all of Friedrichs mathematical jokes, there was a bit of self-reference to it since Friedrichs himself was a little bit deaf but not dumb.
There is not time left to talk about Friedrichs very great contributions to non-linear gas dynamics or the theory of shock waves. The book that he and Courant wrote “Supersonic Flow in Shock Waves” had a very great influence in the field and Friedrichs’ own work on decay of shock waves and the combustion theory have contributed a tremendous amount.
I would like to end by saying that in preparing these remarks I looked through reprints of Friedrichs’ work and not only was I impressed by their great worth, but I had all kinds of questions, “What are the interesting problems in shock waves today?” “What about combustion research?” “What did Friedrichs think about turbulence?” and it was very sad not to be able to ask him. On the other hand, it was very pleasant to recollect all the things that he had done, all the things that we learned from him.
I am going to speak about Friedrichs’ contributions to mathematical physics, and quantum mechanics in particular. Friedrichs contributed strongly to quantum theory and more generally to mathematical physics through his research and his teaching. I will describe briefly some of the research that contributed to quantum theory. I will almost completely neglect the enormous influence his lectures, writings, and conversations had on his students, friends, and colleagues.
Friedrichs’ work on spectral representation, spectral resolution, perturbation theory for continuous spectra, scattering theory, functional integration, quantum field theory, and the foundations of quantum mechanics is marked by several general features. The problems that he chose to study were basic to science as well as mathematics. The problems were, at the time he began his work, ill posed and the areas he worked in were frequently in a state of confusion. He took great pains to define the logical structure of these problems. He selected concrete examples that had most of the important features of these problems, which he then solved in detail with highly original methods. Finally, he took great pains in making his solutions look easy and natural and in pointing out where further difficulties lay.
Friedrichs’ first paper, in 1928, was in the general theory of relativity. Charac-teristically, this paper made clear the logical significance of Einstein’s general covariance postulate. By showing that Newton’s laws could be written in a generally covariance form, Friedrichs pointed out clearly that the postulate was a statement about the nature of the gravitational force law and not an invariance principle. This is discussed at length in Steven Weinberg’s recent book on rela-tivity with appropriate reference given to Friedrichs’ first paper.
Modern quantum theory began in 1920 with Schroedinger’s publication of his wave equation. As far as I can tell, Friedrichs’ first contributions to quantum theory were as a participant in a 1927 or 1926 seminar in Göttingen run by Max Born. Von Neumann, who had come to Göttingen to study with Hilbert, also participated in this seminar. One of Hilbert’s goals was to axiomatize physics. At this time he was attempting to axiomatize quantum mechanics. Von Neumann produced a mathematically sound axiomatization of quantum mechanics that he summarized in 1932 in his now famous book on the “Mathematical Foundations of Quantum Mechanics.” Friedrichs reviewed this book for the Zentralblatt. I’d like to quote for you from Friedrichs’ words, “This book contained introductory mathematical sections and sections involving quantum theory. They were rather abstract. We in the group around Courant were quite suspicious about the significance of such abstract work. Nevertheless, when in late 1930 I studied these abstract papers, I was dumbfounded. In fact, I had just handed to Courant, for publication, a manu- script on spectral theory. I asked him to return the manuscript. I then rewrote the paper from Von Neumann’s abstract language. That was the origin of the substantial part of my later work.”
The work that Friedrichs was referring to appeared first in 1934 in a series of papers on the spectral properties of semi-bounded operators. The first paper in the series contains what is undoubtedly Friedrichs’ most quoted result in the mathematics of quantum mechanics. This is his famous extension theory which says that “A densely defined symmetric operator on a Hilbert space, that is semi-bounded has a self-adjoint extension with the same bound.” This extension is now universally known as the “Friedrichs extension.” It is a basic tool in mathematical physics for two reasons. First, according to quantum mechanics, the observable values of the energy of a stable, conservative system should be given by the spectra of a semi—bounded self-adjoint operator, called quantum mechanical Hamiltonian. The existence of these self-adjoint operators follows immediately from Friedrichs’ theorem by establishing that on a dense set the operator in question is symmetric and semi-bounded. Some well— known examples where Friedrichs’ theorem applies are the harmonic oscillator, the hydrogen atom, and boson quantum field Hamiltonians. The second major use of the “Friedrichs extension” is more tech-nical. In estimating spectral properties of quantum mechanical and field theoretical Hamiltonians, one frequently makes use of differential operators with special boundary conditions. Again, the “Friedrichs extension’1 yields immediately the appropriate self-adjoint extension of these operators.
Friedrichs was ahead of his time with his next contribution. This was his paper of 1938 in which he developed a method for treating perturbations of operators with continuous spectra. This is precisely the type of perturbation problem that later arose in quantum mechanical scattering theory. Thus, when Heisenberg introduced his scattering matrix and Moller his wave operators in 1943 and 1946, respectively, Friedrichs already had developed a mathematical method for their study. His original perturbation method would nowadays be called “A time- independent approach to scattering theory.”
The other method used to study scattering theory is now called the “time-dependent method”. It was also explored and made clear by Friedrichs in 1948 in a paper in which he established the connection between Moller’s time- dependent approach and his earlier work on perturbation theory. In his earlier work, he showed that a special class of perturbations, that he called “gentle”, of an operator with a simple spectral representation, yielded an operator with an equivalent spectral representation. The equivalence was proven by what is now well known as the “Three Step Friedrichs Method.” He postulated the existence of intertwining operators, he gave equations now called the “Friedrichs Equations” that they should satisfy. He solved the equations and showed that the solutions were indeed intertwining and had the desired properties. The equations involved his now famous “gamma operator.”
In his later paper he showed that Moller’s wave operators were essentially his intertwining operators and that Heisenberg’s scattering matrix was given simply in terms of his gamma operator. lie considerably clarified the sense in which these scattering objects existed. His method and equations have been greatly extended to cover problems with Schroedinger operators and quantum field Hamiltonians. Discussions of some of these extensions can be found in the modern books of Dunford & Schwartz, Kato, Simon, and in Friedrichs’ classic “Perturbation of Spectra in Hilbert Space.”
Friedrichs next contributed to quantum field theory with a pioneering set of articles published in the early 1950’s. These articles were collected and published in 1953 in his book “Mathematical Aspects of the Quantum Theory of Fields.” This book contains an extensive treatment of linear quantum field problems. In particular, the book contains explicit examples of inequivalent representations of the cononical commutation relations. Friedrichs called these representations mereodic and amereodic fields. It contains the definition of an integral over a Hilbert space which is used to give the Schroedinger representation of the cononical commutation relations for the free boson field. This work on functional integration is studied by Friedrichs and his colleagues at greater depth in the 1957 NYU lecture notes by Friedrichs and Shapiro. The book also contains a careful discussion of scattering for boson fields under the influence of linear homogeneous forces. This book was especially important in that it isolated and gave precise definitions to many of the basic notions used by physicists in quantum field theory. It is even more remarkable when we recall the amount of confusion present in the physics literature of the time. Remember that the formal work on renormalized perturbation theory done by the physicists Fineman, Dyson, and Schwinger was done in 1949. Friedrichs’ first paper in the series appeared in 1950. He was also in his early 50’s when this work was published.
I feel that Friedrichs’ most important contribution to quantum physics was his book “Perturbation of Spectra in “Hilbert Space” published in 1965. It was an expanded version of lectures he gave in Boulder, Colorado in 1960. The first two chapters of this work of art contain Friedrichs’ description of his approach to perturbation and scattering problems. Chapter 3 contains his formal perturbation approach to. non-linear quantum field theory. He explains clearly what some of the main dif-ficulties are in constructing quantum fields that require renormalization. In particulars for a restricted class of perturbations, he outlines the construction of a renormalized Hamiltonian, intertwining operators, and Heisenberg scattering matrix. The methods used are generalizations of his methods for studying the perturbation of continuous spectra. Extensions of this work were given by Jack Schwartz, Raphael Hoegh Krohn and James Glimm. In particular the first con-struction of the non-linear boson quantum field theory, by James Glimm and Arthur Jaffe, made significant use of Friedrichs’ diagrams, formulas, and ideas. If Glimm and Jaffe are to be regarded as fathers of the new branch of mathe-matics, called constructed quantum field theory, then Friedrichs is certainly a grandfather.
Friedrichs’ most recent contribution to quantum mechanics was his 1978 paper “Remarks on the Notion of State in Quantum Mechanics.” This paper is a con-tribution to one of the most fascinating controversies of twentieth century science. I’m referring to the debate between Niels Bohr and Albert Einstein over the com-pleteness of quantum mechanics. Einstein felt that the laws of nature should be deterministic or causal, and for this reason he felt that the quantum mechanical notion of state of a particle was incomplete. Friedrichs introduced a new notion that he called the “intrinsic state of a particle” and he argues that for his intrinsic state, causality is valid but in accordance with quantum mechanics, not verifiable. We may all speculate on what Einstein and Bohr would have thought of this notion of Friedrichs.
However, there is no doubt in my mind that a slight rephrasing of Einstein’s words of praise for his friend Max Planck apply especially to Frieder. I would like to paraphrase what Einstein said. In the temple of science are many mansions and various indeed are they that dwell therein. Many take to science, out of a joyful sense of superior intellectual power. Science is their own special sport to which they look for vivid experience and the satisfaction of ambition. Many others are to be found in the temple who have offered the products of their brains on this altar for purely utilitarian purposes. Were an angel of the Lord to come and drive all the people belonging to these two categories out of the temple, the assemblage would be seriously depleted, but there would still be some men of both present and past times left inside. Our Friedrichs was one of them. That is why we loved him. Thank you.
There is both an advantage and a disadvantage in being late in the program. Some of the things I intended to say have been said already, even better. The disadvantage is that “the natives are getting restless.” Some of the things I have to say will seem to disagree with what has been said, especially about the like for elegance or the dislike for spoiling things. I think that is partly in the. eye of the beholder and also it’s automatically the case, since Friedrichs was not a simple person and therefore there can’t be a simple statement that applies to him. To compress an account of Friedrichs’ scientific work into one afternoon is a patent absurdity. I think that best that we can hope for, is to give you something of the flavor and more than that, some of the impact, not only on the field, but more important on the people.
I was asked to report on one particular field of the numerous that he worked in, and that is MHD and fusion energy. Of course it can’t be easily separated from the other fields that he worked in or from what other speakers have been talking about. His work is coupled. Indeed I would say that the key to Frieder’s great-ness as a scientist was the way he transferred concepts from one to another, providing cross-pollination and advancing both fields simultaneously. Incidentally, the extreme difficulty of doing this successfully, that is -- in working in two fields simultaneously, is not just the difference in language; it is the matter of differences in philosophy. Nevertheless, he was frequently very successful in carrying over different methods from one field of human knowledge to another.
Let me start with a very old story that is important to me personally: how I came to enter the field of mathematics. Friedrichs played a critical part in this and it was only a few years ago that I mentioned this to him, which he hadn’t known. I’d first come to NYU as a graduate student in 1943 or 1944. There was a war going on and I was working 60 hours a week, commuting 3 hours a day for 6 days a week and evening classes were very convenient, so that was the reason I chose New York University. I was undecided at that time whether I wanted to major in mathematics or physics, and I decided to test this by taking one graduate course in mathematics and one in physics -- sort of a sudden death playoff. I might add that I never regretted the results. There wasn’t a large number of courses to choose from at the time. I chose electromagnetic theory in the physics department and special functions in mathematics taught by Friedrichs, who was my first con-tact with graduate mathematics. At the present time, I know both of these fields much better than I did at that time. Electromagnetic theory, which I have taught frequently, is a beautiful, elegant structure as well as being useful, and special functions by comparison is rather dull.
There are many different kinds of inspired lectures, many different ways of giving inspired lectures. The type that is most commonly praised is one from which all human failings and frailty have been carefully removed. I have been fortunate a number of times to have been exposed to such elegance and beauty in mathematical lectures and I usually wonder, while listening to such a talk, how could the mind of a mortal man conceive of this beautiful edifice, its total perfection without warts. For a potential working mathematician, the beauty and elegance is inspira-tional but it’s incomplete because you also want to know how it grew. The crea-tion of such an edifice is usually one step forward and two steps back, just like anything else worthwhile. The question of how to discover them or how it might have been discovered is a very important part of the picture. In other words with some of the nuts and bolts visible -- and that’s how Friedrichs did it. This is the way Friedrichs usually taught. It’s not just the courses that he taught -- it’s typical of his work. It was in some sense intentionally inelegant, but he wanted to allow an ordinary human being to understand where it came from. His work was inseparable, considering the elegant and important generalizations that some of his work consists of, and the solution of very difficult, critical, important, particular problems. These particular problems weren’t just examples of the general theory, they were essential to it. They were essential to understand-ing.
I’ve just described the start of my mathematical learning experience from Friedrichs, and I can easily say that I learned more about mathematics from him than from any other person. In preparing this talk, I reread several of his papers on subjects that I thought I was well versed in, and that I thought I knew exactly what was in them, and I find out that I am still learning from Friedrichs. Strangely enough, I rarely had substantial discussions with him in his office. We talked sometimes at home and most often when we both found ourselves ‘away at the same trip.
Let me mention just two such occasions -- one of which was in Los Alamos in 1954, and the other was in Lille in 1959. The Los Alamos meeting was in fusion energy. We were trying to fathom the way that physicists’ minds worked and how to com-municate with them. The subject was classical and relativistic NIH D. After several years of lack of success, he finally was able to show that the relativistic, the covariant form of the equations, could be written in symmetric hyperbolic form and therefore all the elegant theorems could be proved. The alternative, up to then, was using some extremely complex, much more inelegant and much more difficult methods of Leray which were the only ones available. So this simplified it by just rearranging the equations, because symmetry is not an inherent property of a system, it’s a matter of introducing the right variables and combining them in a different way. He was very pleased with this result, because after having done it, there was almost no paper to write. You just wrote down the equations and you could see that everything followed from that.
With regard to symmetric hyperbolic systems, he held that if something was to arise from a physical problem it really should have come from some stage of its existence from a symmetric hyperbolic system or one related to it by some simplifications or complications. I differed with him and argued with him about this. In plasmaphysics you are presented with equations that are extraordinarily com-plicated. The importance is to try your best to find something related to the problem that you stumbled across that is more elegant or solvable. If you can identify it then you may feel that it is worth spending a couple of years of your life trying to find properties of the equation.
Let me return to specifically the Los Alamos meeting in 1954. I had previously come across a variational principle which I had used to prove stability of certain free-boundary configurations and I showed that subject to existence of a certain curvature criteria for the magnetic lines was necessary and sufficient for stability. Thus, was discovered the first stable magnetic model. There were many generalizations that followed from this in axial symmetries and free dimensions. These were even treated numerically in the 1950’s but no one was interested and they got lost; it was too early, and eventually it led to a patent. Friedrichs and I were joint patent owners together with the AEC of this confine-ment concept. It is my only patent and it was Frieder’s only patent also. Unfortunately, the patent has expired before this has been brought to fruition, and it looks as though it will be another 40 or 50 years before anything like this reaches something that will be patentable.
Not only did I spend most of my life learning from Frieder, I shall probably con-tinue to do so.
I would like to comment on a few aspects of Friedrichs’ life. I won’t try to describe his profound originality but none of my reflections here would be inter-esting, if not for that originality. Friedrichs always had a great desire to be honest and fair. That quality was an ingredient in the vital contribution he made to the development of the Courant Institute. Let me start by noting that Fried-richs hated to say or write anything that he didn’t believe. I remember one occasion when Frieder and Nellie invited me to come along with them to visit the Greenwich Village studio of a well-known New York artist. The studio was spacious with white walls on which were hung huge paintings, mostly bright stripes of color. Nellie was sincerely enthusiastic and said repeatedly “How wonderful.” Frieder appeared to be puzzled by the show and I wondered what he would say. Finally, he was put on the spot and after coughing for a minute, he said, “Well, I think I like them better when I shut one eye.”
His ability to withstand social pressures and maintain integrity was particularly important when it came to judging scientific worth. The Institute is its people and the most serious issues in administering the Institute concern appointments. Who should be appointed? Who promoted? Who given tenure? Disagreements on these questions naturally aroused passions. On the right decisions hangs the future success of the Institute. One could always depend on Frieder to make an impartial, scientific judgement. Frieder would undertake the labor of reading a candidate’s papers, sometimes in an area in which he himself didn’t work. He’d come back a week or so later with a detached, just and penetrating opinion.
Friedrichs always wanted to spend at least several hours every day to work on his mathematics and Nellie was a tiger in guarding him from interfering distractions. I think mathematics owes a real debt to Nellie for her understanding devotion to Frieder. I once mentioned to Frieder how marvelous it was to be paid a salary to do mathematics. He replied that he always felt he was paid his salary for the time he spent not doing mathematics.
Friedrichs was a complete mathematician, but he was committed to being in the world as well -- he was involved at all times in the affairs of the Institute. He was always concerned about his students and his teaching. He was careful to carve out a position that suited the balance of activities he sought for himself, and of course he was lucky that his partners, Courant and Jim Stoker, tolerated his work style and relied on his unique capabilities.
Once Frieder had an offer from Rockefeller University. He was invited up there and when he came back he told me how they had shown him his large office with a view of the river, introduced him to his secretary, and described all the advantages of the position. When he asked them what his duties would be, they said, “No duties.” He told me that was the moment he knew he wouldn’t accept the offer.
Frieder was not unworldly, but he didn’t suffer from the vanity or ambition that would have led him into positions that he wouldn’t have enjoyed. However, Fried-richs did serve one year as Director of the Courant Institute. He contributed that year’s service to provide a smooth transition from the leadership of his generation to the next. I was his Assistant Director and it was my great pleasure to work with him. He knew how to delegate the work but he understood that there were always some painful decisions that only the Director can make. He worried much more than I had imagined he would about people’s feelings. It also struck me then, how much he liked to have a philosophical principle for whatever he did -- even if the philosophy sometimes had to be invented after the actions.
Finally, I would like to say a few words about Friedrichs as a classroom teacher. His lectures were an experience that transformed many of his students. They were not at all like the remarkable performances of Artin or Bers. Friedrichs held a very independent view of what teaching a course should be. Since he was so original in the way he thought about mathematics, his courses were often quite hard, but he taught many people to do mathematics. He used to say that he believed the first part of a course should be confusion: that a student had to feel the problem, to learn what wouldn’t work, to go into the dead ends. The second part of the course should be deconfusion: clarification and illumination. “Of course,” he said, “I sometimes don’t have time for the second part.”
He would run out of alphabets: English, Greek, Gothic, German, and Hebrew were scarcely enough, even with the possibilities of multiple subscripts and super-scripts, on the left as well as on the right of the main symbol. As he lectured, he would draw a box about some important formula or phrase. The blackboard would gradually become covered with rectangular boxes, each containing a few equations and possibly a keyword or two. Soon there would emerge a patchwork quilt of boxes covering the board. As he proceeded he would erase the contents of boxes no longer needed refilling them with new equations. Sometimes he lost sight of a box he needed and there would be a panicky search for it, and some-times, heaven forbid, he had erased the box it turned out he needed later on. I always imagined that the boxes were the way he disassembled the subject, demystified it, and let us see how one could reassemble it all, in different ways.
Friedrichs liked to make schedules and plans. At any time he knew what he wanted to teach 3 years later. He always wanted to learn something he didn’t know and then teach it. There are not many professors whose courses range from quantum mechanics and fluid dynamics to algebra and topology.
Frieder had an unusual, profoundly original mind. His brilliant mathematical research came from his thinking very deeply about fundamental issues. His discoveries are those of a trail blazer. He got to the good problems first. He was a man driven all his life to carry out the possibilities he sensed in himself and he consciously arranged his career to nurture his great gifts.
Friedrichs, I am sure, would deeply resent any suggestion that he was happy or well-adjusted. He was always suffering from asthma or deafness or something else. But the fact is, he worked successfully and serenely into his old age with-out any of the bitterness that powerful men sometimes feel when they are retired. No one can try to imitate him, but his life, professional and private, creative and thriving, makes us all feel grateful for having been touched by it.
I am Martin Friedrichs, his youngest of five children. Whereas the others have talked about basically my father’s contribution to mathematics and the Institute, I wanted to talk a little bit about what mathematics and the Institute really con-tributed to my father’s life and to the lives of our entire family.
My father considered himself a mathematician from childhood. He always said he knew that’s what he wanted to do and as he has said in his version of his family’s history, he would have been desperate if his parents, at a time of financial prob-lems, had been forced to cut off funds from his studies. He would have been a mathematician anyway, even if he had had to have some other job to earn a living. He alway8 considered himself very fortunate and grateful to be able to earn a living at something he would have been doing anyway and that he would have been doing in his pastime.
Mathematics also gave him an opportunity to start a new life when he left Germany in 1937, without having to fundamentally change careers and without, in some cases, having to change working partners. It gave our entire family -- this career he had -- an opportunity to travel: all over this country, to Europe, to Colorado, to California. It was something that we all loved, and brought what was already a close family, even closer together. Of course, mathematics and the Institute gave my father and my mother and our entire family a very large community of good and close friends and sense of belonging to that community which was very valued by our entire family. It’s something not necessarily possible in many other fields or other kinds of endeavors.
Even beyond that, many of the characteristics that perhaps fit in with and helped my father in his mathematical work, also helped enrich our own family life. I wanted to mention a few of those characteristics as we saw them from the family’s point of view. One was his constant concern for clarity and orderliness, and I’m sure that others saw this in his mathematics. Everything had to be made very clear and so that it was clearly understood. We got it too, in the sense that if you ever wanted to talk to my father you had to think things out very well ahead, you had to explain it, you could never assume that he knew what you were going to talk about, even if it was on the headlines of the newspaper that he had just read - you would have to spell it out. You would always have to be prepared, because he stressed being prepared; when we took a 2-hour hike it was like taking an expedition, there was nothing we wouldn’t bring along. Orderliness was also important, although Louis Nirenberg saw it a little differently, to us it seemed very orderly. Dishes always had to be passed counter-clockwise around the table. Even if what you wanted was just to your right, you had to wait till it would go all the way around, but we learned to appreciate that. We would spend 2 hours packing our car for a trip so that at lunchtime you could get what you wanted without wasting any time. It taught us the sense of importance of that and we all appreciated it.
Another characteristic was the enormous commitment and dedication he’d give to whatever he considered important and we always realized his intense involvement with his work. He went up to his study often, he would close the door, put an “Occupied” sign on and we knew that we were not to disturb him. It was the cardinal rule of the household -- when my father was in his study he wasn’t to be disturbed unless the house was on fire and then only if it couldn’t be put out.
At the same time, that same sense of commitment and dedication he gave to his family in the time he spent with us, which was a lot. Sunday afternoons, at dinner, during the summers, he always spent a lot time with us and then he wasn’t distracted by other things. When he gave his time to his family, he gave it fully to his family. We always got the sense that whenever we needed help he was there; sometimes you had to make an appointment to talk to him, but when you got that appointment you knew he was devoted to you and giving you his full attention.
The intensity of whatever he studied was the same. He loved nature but he wouldn’t just admire a tree, he would want to know what species it was, why it was there, and what problems it had. We’re the only family in New Rochelle that had labeled trees, which happened on occasions when he wanted to take people on his nature walks.
Another characteristic was that he had an almost rigid sense of personal integrity but he was very open minded as well, enormously open minded, always willing to look at a different angle, a different viewpoint, understand how somebody else saw a problem. This certainly helped in his mathematics -- being able to constantly be willing to put his views out of his mind and look at a different approach. It was very important to all of us because each of the five children were very different. We were very different from him and very different from each other. He would be able to relate to each of us separately, look at us as individuals, and understand what our interests were and what our thoughts were -- those were very different from his -- but he always tried to do that.
My father had so many interests that even as we all went into different careers, real estate, languages, sociology, history, operations research -- whatever it was -- he was interested in it and he usually knew a great deal about it, so we always had something to relate to him with. He was never disappointed that none of us went into mathematics -- he never expected that, he never planned on that, it never bothered him, because whatever we went into was something that he was interested in. As long as we enjoyed it, and were interested in it and achieved what we could in it that was what was important.
Of course, the whole family was very proud of his mathematical accomplishments and the recognitions he achieved in that field. From the family’s point of view, the warm, really wonderful and loving relationship he had with our mother for almost 50 years was an equally important achievement as the foundations that he may have set for others in terms of mathematics for future generations. That relationship set the foundation for our family and for future generations of our own family. For this and because he was a very wonderful father, we are very grateful and we will all miss him. My brother Walter also has a few remarks.
I’m the oldest of the five children and I realize that we are running a little bit behind time here and I’m pleased that more than just my mother is here to still hear me. It doesn’t surprise me though; I can see tonight that my father was well appreciated here and I would really like to thank from the family all of the people who spoke tonight.
Since his death I have been thinking about him a lot. My earliest memories as a child, during the Second World War, was that at home there was constant reference made to “the Institute”, as the Courant Institute was then called. It seems that everything that both of my parents did, in some way involved the Institute. Most nice weekends were spent in the Courants’ backyard where there was always a large group of people from the Institute. Every summer the whole Institute would have a lovely picnic in Pound Ridge and in the winter it seemed that whole Institute went skiing at Lake Placid. As a small child it was very clear that our whole family life seemed to revolve around the Institute. I kept on asking “Can’t I see it?” At last, when I was about 6 years old, I was told that I would visit the Institute with my grandmother and meet my father there. At that time the Institute was located in the old Bible Building opposite Coopers Union. We found my father out in front and he gave us a tour around the building. While doing this we bumped into Professor Donald Flanders who invited us into his office. His office was total chaos: books and papers and magazines were all over the place lying around on the chairs and desk and so on. Then we saw Dr. Courant’s office which was more organized but also full of books and papers all over the place. At last we entered my father’s office which was extremely well organized with hardly any books and papers to be seen and I recall thinking at that time, “Gee, I guess daddy doesn’t do much work”. I learned differently tonight.
My father was with the Courant Institute for about 45 years from its beginning with Courant, whom my father was associated with from the early 1920’s. At the Courant Institute both my parents made a large number of their closest friendships which gave them much joy. Besides the family relationships, my father had many professional relationships here which were very satisfactory to him. As the years went by he took great pride in seeing how former students of his and associates did so well in mathematics and how the Institute grew from the Judson Church Building, to the Bible Building, then to 25 Waverly Place, and now this beautiful Warren Weaver Hall. I know from time to time he had opportunities to leave, but the Courant Institute meant so much to him that he never really had any inclina-tion to leave.
I. want to extend my family’s great appreciation and I want to also add that this applies to my brother Christopher and my sister Liska who are not here today -- they couldn’t make it -- but we all wish to thank those of you at the Courant Institute who made it possible for him to have such a rewarding and successful career and also for the condolences recently given to our family.
My father had a long and fruitful life but when he died, I thought, why couldn’t he live a few more years as he and my mother were enjoying their lives very much these last few years. But at least he lived to see that his work was appreciated and to receive many honors. One that gave him the greatest joy was the 80th birthday party held for him right here at the Courant Institute. It was nice for the rest of our family too -- to see with what esteem his colleagues held him. At that affair I think Nina Courant summed up his life at the Institute best when she said, “We had good times together here.”
While my father’s career was a very important part of his life, it was by no means his whole life, lie had many other interests which he was very knowledgeable about and which gave him much satisfaction, much too numerous for me to list. I must mention that he had a good and active, happy family life which my brother Martin also mentioned. For those people who did not know much about his family, it might be surprising to learn that he had five children but those who knew him well found, like everything else, that he had his family life well organized. Some-times when we would meet him -- Sunday afternoons for the whole family, which were well planned affairs, at our dinners together -- he often determined and controlled the subjects to discuss so that it was a good learning experience. We children were fortunate that his profession involved trips all over the world, some of which we took with him. On these trips and at all other times he was con-stantly passing on his vast knowledge to us at the same time showing us how to organize things, passing on his discipline to work and his integrity too.
One cannot talk about my father without mentioning his daily walks. When family members would join him on these walks or mountain hikes or walking around his beloved weekend property, we would get great pleasure in seeing how he enjoyed all aspects of nature and its beauty. While no one can question his own high and successful standards of what one should do with one’s life, he never imposed his will on us to the extent that we felt we couldn’t live up to it. He was also very open minded as to how we wanted to conduct our own lives and my father was always available, sometimes as Martin said with appointments, but he was available to us when we needed him. For this and many other reasons our family will miss him and I know you too here at the Courant Institute will also miss him.
CATELEEN S. MORAWETZ
I’m here to close this meeting but I would like to say a few last words about an aspect of Friedrichs that may, in the larger picture of his great scientific con-tributions, seem to some unimportant. I’d like to add that I owe practically my whole scientific life to him and so I would have liked to take a little longer to talk about that but I think we should come to an end.
He devoted, many times with the very direct help of Nellie, long hours to making the Institute function in special ways: to seeing that our visitors were comfort-able, that their offices were near people with whom they would make stimulating contact. He watched over the design of this building, caring how it would look, how it would function, what would make the library a good place to go to, what would improve a classroom. I could go on. He also, for example, back in 1948, when the Communications journal first came out, contributed four or five long papers to the first volume. They were all very important papers, difficult. They weren’t just something he rushed off in a hurry because it had to get into that new journal, it was a very important contribution to make that journal get started on a good footing. It was also typical of his very concrete devotion that he spent almost 5 years in such close touch with Constance Reid about the book on Courant that he read it and, I might add, every comma in it, at least six times.
I want to close on another very specific one of those contributions. Friedrichs believed very strongly that mathematicians lead lonely lives and need much more than other special contact. He envisioned a tea place to gather and exchange thoughts, some mathematical, some just everyday stuff. Our daily teas were begun back in the early 1950’s when the standard at most departments was a dry biscuit and tea in a paper cup once a week. Nowadays we gather every afternoon in our very pleasant lounge on the top floor. We hope you will join us there now, even at this belated hour, for a last tea party, I might even say with Friedrichs. I close by thanking you for coming to share with us this afternoon in his memory.