Born in Switzerland, Armand Borel did his undergraduate work at the Federal School of Technology (ETH) in Zürich. He obtained his doctorate degree at the University of Paris in 1952 and then spent two years at the Institute for Advanced Study in Princeton. He has been professor there since 1957.
In the late twenties, Abraham Flexner, a prominent figure in higher education, had made an extensive study of universities in the U.S. and Europe and was extremely critical of many features of American universities. In particular, he deplored the lack of favorable conditions for carrying out research. In January 1930, while preparing for publication an expanded version of three lectures he had given in 1928 at Oxford on universities, he saw in the New York Times an article on a meeting of the American Mathematical Society (AMS), in which Oswald Veblen, professor at Princeton University, was quoted as having stated that America still lacks a genuine seat of teaming and that American academic work is inferior in quality to the best abroad. He immediately wrote to Veblen, saying there was not the slightest doubt in his mind that both statements were true and hoping that Veblen had been correctly quoted. In his answer, Veblen confirmed these views, described the context of his remarks and wrote in conclusion:
Here in Princeton the scientific fund which we owe largely to you and your colleagues on the General Education Board, is having an influence in the right direction, and I think our new mathematical building which is going to be devoted entirely to research and advanced instruction will also help considerably. I think my mathematical institute which has not yet found favor may turn out to be one of the next steps. Anyhow it seems to me to fit in with the concept of a seat of learning.
Here Veblen was alluding first to the efforts, initiated by Fine and pursued with the help of Eisenhart and Veblen, to improve research conditions in his department and to the construction of what became Fine Hall; second to a plan for an "Institute for Mathematical Research" he had outlined and presented (without success) around 1925 to the National Research Council and to the General Education Board of the Rockefeller Foundation. It was to consist of four or five senior mathematicians who would devote themselves entirely to research, their own and that of some younger men, and of some younger mathematicians. Members would be free to give occasional courses for advanced students. It could operate within a university or be entirely independent of any institution.1
Shortly before, Flexner had been approached by two gentlemen who were surveying medical education on behalf of two persons who wanted to use part of their fortune to establish and endow a medical college in Newark. Since Flexner was an authority on medical education in the U.S., it was only natural to seek his counsel. He advised against it, explaining why in his opinion there was no real need for a new institution of the type they had in mind. Instead, he showed them the proofs of his book on universities and outlined his plan for an institution of higher learning, where scholars would pursue their researches and interests freely and independently. They were so fascinated by it that they swayed the potential donors, namely Louis Bamberger and his sister, Mrs. Felix Fuld, born Caroline Bamberger, convinced them to look into this possibility and soon introduced them to Flexner. This initiated a series of discussions and a correspondence extending over several months, at the end of which the Bambergers agreed enthusiastically to back up Flexner's plan, on condition that he would be the first director. A certificate of incorporation for a corporation to be known by law as the "Institute for Advanced Study – Louis Bamberger and Mrs. Felix Fuld Foundation" was filed with the state of New Jersey in May 1930 and the New York Times announced in June the creation of an Institute for Advanced Study, to be located in or near Newark, on a gift of $5 million from Louis Bamberger and his sister, Mrs. Felix Fuld. Veblen learned about it for the first time through that press release, although there had been a little further correspondence between the two about the idea of an Institute, but carried out in abstracto, at any rate on Veblen's side. He wrote immediately to Flexner that he was greatly pleased and he expressed the wish that this Institute would be located in the Borough or Township of Princeton "so that you could use some of the facilities of the University and we could have the benefit of your presence." This heralded an increasing involvement of Veblen with this project, first as a consultant, then as a professor having the primary responsibility for the building up of the School of Mathematics.
The Institute was eventually to consist of a few schools, but Flexner decided early on to start first with one in mathematics, because "mathematics is fundamental, requires the least investment in plant or books and he could secure greater agreement upon personnel than in any other field".2 He began to make extensive inquiries in the U.S. and in Europe as to who would be the best choices for a faculty in mathematics. Among American mathematicians, the two most prominent names were those of George D. Birkhoff and Veblen. Flexner started with the former, on the theory that Veblen was already in Princeton anyhow. An offer was made, at an extremely high salary and accepted in March 1932, but Birkhoff asked to be released eight days later. After further inquiries, Flexner came to the conclusion that: "If the Princeton authorities agreed willingly and unreservedly, we could not do better than to select Veblen." They did so quickly, and Eisenhart telegraphed to Veblen in June:
Have talked with those concerned and they approve. Congratulate you heartily. Look forward to big things.
1932 was marked by extensive traveling, wide ranging consultations, and discussions, correspondence and negotiations with Veblen, Einstein and Weyl. (Of course, no outside advice was needed in the case of Einstein, and Flexner forged ahead as soon as he understood that he might be interested.) In October two faculty nominations were announced, that of Veblen, already effective October 1st, 1932 and that of Einstein, effective October 1st, 1933 (as well as the nomination of Walther M. Mayer, the then collaborator of Einstein, as an "associate"). It was also announced that the new Institute would be located in or near Princeton (a shift formally proposed in April 1932) and would be housed temporarily at Fine Hall. The school would officially begin its activities in Fall 1933, but in fact, during the academic year 1932-1933, Veblen already conducted a seminar in "Modern Differential Geometry."
It is well-known that Einstein was enthusiastic from the beginning ("Ich bin Feuer und Flarnme dafür," he had stated to Flexner) and excessively modest in his financial requirements, but the negotiations were not all that smooth. In 1933 Flexner learned that Einstein had also accepted a professorship in Madrid and one at the Collège de France. Since their residence requirements were minimal (in the former case, nonexistent in the latter), while those of the Institute were for him only from October to April 15, Einstein did not see any incompatibility; on the other hand, if Flexner felt otherwise, he would agree to terminate the arrangement with the Institute.... The Madrid offer also included the right to name a professor and Einstein tried to use it as a leverage to secure a professorship at the Institute for W. Mayer (without success). In summer of 1933, Flexner had asked whether Einstein could arrive soon enough to participate in a general organizational meeting of the members of the school on October 2nd. Einstein felt he could not because this would entail spending one month away from W. Mayer, which would be too detrimental to his work. He arrived on October 17. He was reminded of that when he complained later that he had not been consulted about invitations and stipends. The collaboration with Mayer was over within a few months.
In Europe, the two names of mathematicians mentioned to Flexner above all others were those of G. H. Hardy and H. Weyl. While in Cambridge, Flexner got readily convinced that there was no way to lure Hardy away from Cambridge and he turned his attention to H. Weyl. (Hardy and Einstein, as well as J. Hadamard, had singled out Weyl as the most important appointment to be made from Europe.) Both he and Veblen, who had received an offer in June and was in Europe at the time, began discussing the matter with Weyl. He was interested from the start, in spite of strong misgivings about leaving Germany, and immediately expressed some desiderata about the school. First he thought it was absolutely necessary to add to Einstein, Veblen and himself a younger mathematician, preferably an algebraist. Weyl commented (in a letter to A. Flexner, dated July 30, 1932):
The reason lies with the plans for filling the three main positions. By his personality, Veblen is certainly the most qualified American one can wish as the guiding spirit in an institution such as the one you have founded. But he is not a mathematician of as much depth and strength as say Hardy. The participation of Einstein is of course invaluable. But he pursues tong-range speculative ideas, the success of which no one can vouch for. He comes less under consideration as a guide for young people to problems which have necessarily to be of shorter range. I am of a similar nature, at any rate I am also one who prefers to think by himself rather than with a group and who communicates with others only for general ideas or for a final well-rounded presentation. Therefore I put so much value on having a man of the type of Artin or v. Neumann. 3
In fact, this was important enough to Weyl that Flexner included in his official proposal to him: "the understanding that when the right person has been found, an algebraist of high promise and capacity will be appointed". Later Weyl also pointed out the necessity for him to be allowed to give now and then regular courses. He was of course assured he would be welcome to do so, and he accepted in principle the offer in December 1932. But then, in three successive telegrams on January 3, 4, and 12, 1933 he withdrew, then accepted "irrevocably" ("unwiderruflich") and withdrew again. Later on he apologized profusely, explaining he had not realized he was suffering from nervous exhaustion. In his last telegram, he had given as his reason that he felt his effectiveness was tied to the possibility of operating in his mother tongue (a worry still faintly echoed in the foreword to his Classical Groups). But the deterioration of the conditions in Germany, in particular the passing of laws not only against Jews, but also against Aryans married to Jews (his case) made his leaving Germany all but unavoidable and in the course of the year he accepted a renewed Institute offer and began his activities at the Institute in January 1934.
The year 1933 also saw the addition to the school faculty of James Alexander and John von Neumann. It had been agreed between Eisenhart, Flexner, and Veblen that an offer would be made to either Lefschetz or Alexander, who both wanted the appointment. The choice fell on the latter, for reasons I have not seen stated anywhere. I have heard indirectly that Eisenhart had said he could more easily spare Alexander than Lefschetz. In view of the much greater involvement of the latter in all the activities of the department, this seems rather plausible. It is also well-known that later Lefschetz was not stingy with critical remarks about Veblen or the Institute. (In 1931, Flexner had asked his views first on the desirability, nature and location of an Institute and second on whom he would choose in mathematics, were he asked to do so. His answer to the second question was Veblen, Alexander and himself from Princeton, Morse and Birkhoff from Harvard; from Europe, he would add above all Weyl, but, since he was holding the most prestigious chair in mathematics in the world, there was no chance to attract him.) J. von Neumann had been half-time professor at the University for some time and the University was trying to make other arrangements. Veblen had suggested to offer him a position at the Institute but at first Flexner was reluctant to take a third mathematician from Fine Hall. However, after Weyl redeclined and after a further conference between von Neumann, Eisenhart, Veblen, and Flexner, an offer was made and quickly accepted. It was also agreed that the two institutions would, henceforth, jointly publish (and share the financial responsibility for) the Annals of Mathematics, with managing editors Lefschetz (who had been one since 1928) and von Neumann.
The appointment of Marston Morse in 1934, effective January 1st, 1935, brought to six the school faculty, which was to remain unchanged for the next ten years. To have assembled within three years such an outstanding faculty was an extraordinary success by any standard. In a report to the trustees of the Institute in January 1938, Flexner credited for this achievement Veblen and the help received from the University, in particular from L. P. Eisenhart, then dean of the faculty.
It was, of course, a tremendous boost for the development of the school that it could function in the framework of an outstanding department, strongly committed to research, and make full use of its facilities, vastly superior to those of any other mathematics department in the country. President Hibben and Eisenhart felt that the development of the Institute would be mutually beneficial, although the Institute was offering unique conditions for work, superior salaries, and therefore might again be successful in attracting faculty members besides Veblen. But others in the university community apparently had different opinions, so that, after the third appointment from the university faculty, Flexner and some trustees, in particular L. Bamberger, felt they had to assure the university authorities they would not in the future offer positions to Princeton University professors. As far as I can gather from the record available to me, they did so early in 1933 in one conversation with Acting President Duffield, (Hibben was retired by then). Whether this was meant for a limited time or forever, I do not know. I also have no knowledge of an official written statement by the Institute to that effect, nor of one by the University taking cognizance of such a commitment. On the contrary, the only university document of an official character on this matter I know of (prior to 1963, see below) takes a completely different position. To be more precise, L. P. Eisenhart had written to A. Flexner on November 26, 1932:
I agree with you that the relationship of the Institute and our Department of Mathematics must be thought of as a matter of policy extending over the years. Accordingly I am of the opinion that any of its members should be considered for appointment to the Institute on his merits alone and not with reference to whether for the time being his possible withdrawal from the Department would give the impression that such withdrawal would weaken the Department. For, if this were not the policy, we should be at a disadvantage in recruiting our personnel from time to time. If our Trustees and alumni were disturbed by such a withdrawal, as you suggest, they should meet it by giving us at least as full opportunity to make replacements intended to maintain our distinction. The only disadvantage to us of such withdrawals would arise, if we were hampered in any way in continuing the policy which has brought us to the position which we now occupy. This policy has been to watch the field carefully and try out men of promise at every possible opportunity. If it is to be the policy of the Institute to have young men here on temporary appointment, this would enable us to be in much better position to watch the field.
In my opinion the ideas set forth are so important for the future of our Department that it is my intention to present them to the Curriculum Committee of our Board of Trustees at its meeting next month, after I have had an opportunity to discuss them further with you next week.
Accordingly, Eisenhart presented on December 17 to the Curriculum Committee of the Board of Trustees a statement "on certain matters of policy in connection with the relation of Princeton University to the Institute", a copy of which was kindly given to me by A. W. Tucker. One paragraph reproduces in substance, even partly in wording, the first one quoted above. In conclusion, Eisenhart states that he is presenting this statement "With the expectation that you will approve of the position which I have taken...". It was indeed "approved in principle" by the committee. Obviously the latter was empowered to do so and to speak in the name of the Board of Trustees. Had it been solely advisory, Eisenhart could only have asked the committee to recommend to the board that it approve of his position. I am not aware of any other statement by university authorities addressing this question, again prior to 1963.
As already mentioned, Eisenhart was at the time dean of the faculty. Tucker pointed out to me that, in the organization of the University, this position was next in line to the presidency and that there was in fact no president in charge at that time: Hibben had retired in June 1932 and Dodds would be nominated and become president in late spring 1933. During the academic year 1932-1933, there was only an acting president, namely the Chairman of the Board of Trustees, E. D. Duffield, living in Newark, who mainly took care of off-campus, external affairs. Under those circumstances, Eisenhart was in fact addressing the Curriculum Committee as the chief academic officer of the University.
Although Flexner had not mentioned it in his formal report, he was of course acutely aware of another powerful factor for the rapid growth of the Institute, namely the anti-Semitic policies of the Nazi regime, without which the Institute could hardly have attracted Einstein, Weyl, and von Neumann. This was in fact only the beginning of the Institute's involvement with the migration of European scholars to the U.S. It is a well-known fact that Veblen played a prominent role in helping European mathematicians who had to leave Europe to relocate in the United States.4 He, Einstein, and Weyl, through a network of informants, were well aware of many such cases and often aided in a crucial way by offering first a membership, sometimes with a grant from the Rockefeller Foundation.
At the official Institute opening on October 1, 1933, the school already had over twenty visitors. The level of activities was high from the beginning. While emphasizing the importance of the freedom to carry on one's own research, and the opportunity of making informal contacts and arrangements, the early yearly Bulletins issued by the IAS list an impressive collection of lectures, courses and seminars. Among those given in the first four years, let me mention: A two-year joint seminar on topology by Alexander and Lefschetz, followed by a two-year joint course on topology, a joint seminar (extended over several years) by Veblen and von Neumann on various topics in quantum theory and geometry, a course and a seminar by H. Weyl on continuous groups (the subject matter of the famous Lecture Notes written by N. Jacobson and R. Brauer), followed by a course on invariant theory, courses and seminars by M. Morse in analysis in the large, a two-year course by von Neumann on operator theory, lectures on quantum theory of electrodynamics by Dirac, on class field by E. Noether, on quadratic forms by C. L. Siegel, and on the theory of the positron by Pauli. In 1935 H. Weyl started and for a number of years led a seminar on current literature. There was also of course a weekly joint mathematical club. The membership steadily increased and Veblen could state around 1937 that in Fine Hall there were altogether approximately seventy research mathematicians and an intense activity. This figure included the members and visitors of the University, too. There was no physical separation in Fine Hall between the two groups, which intermingled freely.5 Many faced the familiar dilemma of having to choose between attending lectures or minding one's own work. There were also some grumblings, that all this was too distracting for the graduate students. The trustees, mindful of the financial aspect, were asking for some limitation and even a reduction of the number of members; Veblen apparently was not too receptive. Almost from the start, Princeton had become a world center for mathematics, the place to go to after the demise of G6ttingen.
That the Institute had in this way a considerable impact on mathematical research in Europe and in the United States needs hardly any elaboration. Less evident, and maybe less easy to imagine nowadays, is its role in the improvement of the conditions in American universities by the sheer force of the example of an institution providing such exceptional conditions and opportunities to faculty and visitors. In 1939 Flexner was pleased to quote to the trustees from a letter written to him on another matter by the secretary of the AMS, Dean R. G. D. Richardson of Brown University: "... The Institute has had a very considerable share in the building up of the mathematics to its present level. ... Not only has the Institute given ideal conditions for work to a large number of men, but it has influenced profoundly the attitude of other universities. "
The School of Mathematics developed along lines certainly consonant with the vision of the founders, as outlined in the first documents, but not identical with it. Underlying the original concept was a somewhat romantic vision of a few truly outstanding scholars, surrounded by a few carefully selected associates and students, pursuing their research free from all outside disturbances, and pouring out one deep thought after another. Einstein, Weyl, and Veblen soon decided they were not quite up to that lofty ideal and that the justification for the Institute would not be just their own work but, even to a much greater extent, to exert an impact on mathematics, in particular mathematics in the United States, chiefly through a vigorous visitors program. The visitors (called "workers" initially, "members" from 1936 on) were to be mathematicians having. carried out independent research at least to the level of a Ph.D. and to be considered on the strength of their research and promise, regardless of whether or not they were assured of a position after their stay at the Institute. Furthermore, their interests did not have to be closely connected to those of one of the faculty members. Originally it was intended that the Institute would also have a few graduate students (but no undergraduates) and would grant degrees. It was officially accredited to do so in 1934. But already then, Flexner stated that it had been done because this seemed a wise thing to do, but it would not be a policy of the Institute to grant degrees, earned or honorary. Indeed, it has so far never done so. This view was confirmed in the 1938 issue of the yearly Bulletin, which stated that the Institute had discarded undergraduate and graduate departments on the ground that these already existed in abundance.
In short, the School of Mathematics had very early taken in many ways the shape it still has now, albeit on a different scale, at any rate for the visitors program. It was called School of Mathematics, although its most famous member was not a mathematician. In fact, when asked which title he would want to have, Einstein chose Professor of Theoretical Physics. However, it had been understood from the start that the school would also include theoretical physics. Internally, it was sometimes referred to as School of Mathematics and Theoretical Physics and there were always some visitors specifically in theoretical physics. The faculty had contemplated early on the addition of theoretical physicists; in particular Schr6dinger was suggested by Weyl in 1934 and then also by Einstein. Dirac was also mentioned. But the director felt that he could not increase the faculty in the school: He was at the time starting two other schools, in economics and politics and in humanistic studies. Moreover, the financial situation caused some worry and he and the trustees felt some caution was called for. Still, Dirac was a visiting professor in 1934-1935 and Pauli the following year. Later, Pauli spent the war years at the Institute and was offered a professorship in 1945. He was interested but felt he could not commit himself before he had gone back at least for a while to Zürich, where his position had been kept open for him. He stayed at the Institute for one more year with the official title of Visiting Professor, but functioning as a professor and chose later to go back definitely to Zürich. The first real expansion in theoretical physics took place under the first half of Oppenheimer's directorship. As theoretical physics grew at the Institute, the two groups operated more and more independently from one another until it was decided, in 1965, to separate them officially by setting up a School of Natural Sciences. In the sequel, "School of Mathematics" will be meant in the narrow sense it has today.
The Institute developed first very informally. As already stated, Flexner relied for mathematics largely on outside advice, mainly that of Veblen. He had to: "Mathematicians, like cows in the dark all look alike to me", he had said to the trustees at the January 1938 meeting. But this was to be an exception. He had already much more input in the setting up of the School in Economics and Politics and he expected fully it would be so in most aspects of the governance of the Institute. The correspondence with Veblen had shown already some differences of opinion on the eventual shape and running of the Institute, but they were not urgent matters at the time and could be overlooked while dealing with the tasks at hand, on which Flexner and Veblen were usually fully and warmly in agreement. However, as the Institute grew, differences of opinion between the director and some trustees on one hand, and the faculty on the other, became more apparent and relevant. The former liked to view the Institute as consisting of three essentially autonomous schools. They were willing to let each one run its own academic affairs; but there was a rather widespread feeling that professors were often conservative, parochial, not really able to see the Institute globally. Besides it was wrong for them to get involved in administrative matters (after all, Flexner had so often heard professors complain about those duties, which take so much precious time away from research and there he was offering them the possibility of having none... ). On the other hand, the faculties of the three schools, which had been chosen quite independently and did not know one another, began to meet, to discuss matters of common interest, to compare views and problems and as a consequence to develop some feeling of being parts of one larger body. Understandably, they wanted to have at least a strong consultative voice in important academic matters. This came to a head when Flexner appointed two professors in economics without any faculty consultation. Added to earlier grievances, it led to such an uproar that Flexner had to resign. But, at a more basic level, there was no attempt to reconcile these two rather antagonistic attitudes in order to arrive at a modus vivendi offering a better framework to resolve any conflict that might arise again. None did arise under the next director, Frank Aydelotte (1939-1947), who earned the confidence of the faculty by his way of handling Institute matters (but, as a counterpart, less than unanimous approval from the trustees). Some conflict did surface, not to say erupt, under the next two directors, J. Robert Oppenheimer (1947-1966) and Carl Kaysen (1966-1976). Fortunately, except in one case to which I shall have to come back, these disputes had comparatively little visible impact on the workings of the School of Mathematics, as unpleasant and distracting as they were to its faculty, so that with relief I may pronounce these matters as outside the scope of this account and ignore them altogether. To conclude this long digression, let me add that a prolonged, in my opinion largely successful, effort was made over several years and concluded in 1974 to set up some Rules of Governance for handling in an orderly way between trustees, faculty and the director all aspects of the academic business of the Institute. There has been no such crisis under the present director, Marvin L. Goldberger (1986- ), nor under the previous one, Harry Woolf (1976-1986).
In the fall 1939, a new chapter in the life of the Institute began with the moving of the Institute into the newly built Fuld Hall, on its own grounds. In preparation for this change, the school had begun to build up a library, aided in this first of all by Alfred Brauer, whom Weyl had taken as his assistant for this purpose. (Brauer did the same later on, on a bigger scale, for the Mathematics Department of the University of North Carolina at Chapel Hill.) In spite of the war, the Institute operated normally, although some professors were engaged in war work, albeit on a somewhat reduced scale. The influx from Europe increased and, again this had a direct bearing on the school: Siegel was given permanent membership, converted to a professorship in 1945. Kurt Gödel, after having been a member for about ten years, became a permanent member in 1946 and a professor in 1953. Why it took so long for Gödel is a matter of some puzzlement. There was of course unanimous admiration for his achievements and some faculty members had long favored giving him a professorship. The reluctance of others reflected doubts not on his scientific eminence, but rather on his effectiveness as a colleague in dealing with school or faculty matters (Siegel has been quoted to me as having said that one crazy man (namely himself) in the school faculty was enough) or on whether they would not be too much of an imposition on him. As a colleague of his in later years, I would say I found that, his remoteness not withstanding, he would acquit himself well of some of the school business hence that those fears were not all well founded. On the other hand, I have to confess that I found the logic of Aristotle's successor in more difficult affairs sometimes quite baffling.
After the war, the activities of the school and its membership increased gradually. There was a conscious effort to have members from Eastern Europe or East Asia, in particular Poland, China, India. 1946 was also the beginning of the first (and so far only) venture of the Institute outside the realm of purely theoretical work, namely the construction of a computer under von Neumann's leadership. This has been described in considerable detail by H. Goldstine in his book,6 to which I refer for details. The computer was used for a few years by a group working on meteorology and von Neumann wanted this to become a permanent feature at the Institute. But the faculty did not follow him. Even the faculty members who had a high regard for this endeavour in itself felt that it was out of place at the Institute, especially in view of the fact that there was no related work done at the University. The computer was given to the University in the late fifties.
Of the first faculty, Alexander resigned in 1947, remaining for some time as a member, Einstein became Professor Emeritus in 1946, Veblen in 1950 and Weyl in 1951. Siegel resigned in 1951 to return to Germany. Added to the faculty in 1951 were Deane Montgomery and Atle Selberg, who had been permanent members since 1948 and 1949 respectively, followed in 1952 by Hassler Whitney.
I came to the Institute in the fall of 1952, not knowing really what to expect. The only recommendation I can remember having received was to appear now and then at tea. This may have been prompted by memories of more formal days, but I soon realized that they were not counting heads. Instead, I found a most stimulating atmosphere, many people to talk to, and suggestions came from many sides. Let me indulge in some reminiscences of those good old days, with the tenuous justification that it is not out of order to describe in this paper some of the experiences and impressions of one visiting member.
F. Hirzebruch, whom I had known in 1948 when he spent some time in Zürich, came once to my office to describe the Chern polynomial of the tangent bundle for a complex Grassmannian. It was a product of linear factors and the roots. were formally written as differences of certain indeterminates; Hirzebruch proceeded to tell me how to interpret them but he could not finish: they looked to me like roots in the sense of Lie algebra theory and this was just too intriguing for me to listen to any explanation. An extension to generalized flag manifolds suggested itself, but it was not clear at the moment whether this was more than a coincidence and wishful thinking. A few days later however, it became clear it was not and that marked the start of our joint work on characteristic classes of homogeneous spaces, to which we came back off and on over several years. Conversations with D. Montgomery and H. Samelson led to a paper on the ends of homogeneous spaces. A Chinese member, the topologist S. D. Liao, lectured on a theorem on periodic homeomorphisms of homology spheres he had proved using Smith theory. Having the tools of "French topology" at my finger tips, I tried to establish it in that framework, succeeded and then, by continuation, obtained new proofs of the Smith theorems themselves. This was the beginning of an involvement with the homology of transformation groups. Of much interest to me also was the seminar on groups, let by D. Montgomery, including his lectures on the fifth Hilbert problem, solved shortly before by him, L. Zippin and A. Gleason, and the contacts with H. Yamabe, his assistant that year.
At the University, Kodaira was lecturing on harmonic forms ("a silent movie" as someone had put it. The lectures were perfectly well organized, with everything beautifully written on the blackboard, but given with a very soft, low-pitched voice which was not so easy to understand.) Tate was lecturing on his thesis in Artin's seminar. The topology at the University gravitated around N. Steenrod, and his seminar was the meeting ground of all topologists. Among those was J. C. Moore, whom I had looked for immediately after my arrival with a message from Serre. This was the beginning of extensive discussions, and a friendship which even moved him to put his life and car at stake by volunteering to teach me how to drive.
My discussions with Hirzebruch went beyond our joint project. He was at the time developing the formalism of multiplicative sequences or functors, genera and experimenting with reduced powers, the Todd genus and the signature. In the latter case, this was soon brought to a first completion after Thom's results on cobordism were announced. Sheaf theory, in particular cohomology with respect to coherent sheaves, had been spectacularly applied to Stein manifolds by H. Cartan and J.-P. Serre; Kodaira, Spencer, Hirzebruch were naturally looking for ways to apply such techniques to algebraic geometry. So was Serre, of course. Being in steady correspondence with him, I was in a privileged position to watch the developments on both sides, as well as to serve as an occasional channel of communication. The breakthroughs came at about the same time in spring 1953 (1 shall not attempt an exact chronology) and overlapped in part. Serre's first results were outlined in a letter to me, to be found in his Collected Papers (1, 243-250, Springer-Verlag, Berlin and New York, 1986); included were the analytic duality and a first general formulation of a Riemann-Roch theorem for n-dimensional algebraic manifolds. It was soon followed by the analogue for projective manifolds of the Theorems A and B on Stein manifolds. Spencer and Kodaira gave in particular a new proof of the Lefschetz theorem characterizing the cohomology classes of divisors. Soon came a vanishing theorem, established by Kodaira via differential geometric methods and by Cartan and Serre via functional analysis. Attention focused more and more on the Riemann-Roch theorem, whose formulation became more precise, still with no proof. During the summer, we parted, I to go to the first AMS Summer Institute, devoted to Lie algebras and Lie groups (6 weeks, about thirty participants, roughly two lectures a day, a leisurely pace unthinkable nowadays) and then to Mexico (where I lectured sometimes in front of an audience of one, but not less than one, as Siegel is rumored to have done once in Göttingen, a rumor which unfortunately I could not have confirmed).7
Back at the Institute for a second year, I found again Hirzebruch, whose membership had also been renewed. The relationship between roots in the Lie algebra sense and characteristic classes had been made secure, but this whole project had been left in abeyance, there being so much else to do. Now we began to make more systematic computations, using or proving facts of Lie algebra theory and translating them into geometric properties of homogeneous spaces. Quite striking was the equality of the dimension of the linear system on a flag variety associated to a line bundle defined by a dominant weight and of the dimension of the irreducible representation with that given weight as highest weight. Shortly after, I went to Chicago, described this "coincidence" to André Weil, and out of this came shortly what nowadays goes by the name of the Borel-Weil theorem. After I came back, Hirzebruch was not to be seen much for a while, until he emerged with the great news that he thought he had a proof of the Riernann-Roch theorem. This was first scrutinized in private seminars and found convincing. I also provided a spectral sequence to prove a lemma useful to extend the theorem from line bundles, the case treated by Hirzebruch, to vector bundles. A bit later, Kodaira proved that Hodge varieties are projective. All this, and the work of Atiyah and Hodge giving a new treatment of integrals on algebraic curves, completed a sweeping transformation of complex algebraic geometry. Until then, it had been rather foreign to me, with its special techniques and language (generic points and the like). It was quite an experience to see all of a sudden its main concepts, theorems and their proofs all expressed in a more general and much more familiar framework and to witness these dramatic advances. This led me more and more to think about linear algebraic groups globally, in terms of algebraic geometry rather than Lie algebras, an approach on which I would work intensively the following year in Chicago, bcnefitting also from the presence of A. Weil.
During that second year, I also gave a systematic exposition of Cartan's theory of Riemannian symmetric spaces and got personally acquainted with 0. Veblen, on the occasion of a seminar on holonorny groups he was holding in his office. I had of course no idea of his role in the development of the Institute, nor did I know about Flexner and his avowed ambition to create a "paradise for scholars". But I surely had felt it was one, or a very close approximation, so when I was offered a professorship in 1956, I was strongly inclined to accept it. It raised serious questions of course. I realized that, viewed from the inside, with the responsibilities of a faculty member, paradise might not always feel so heavenly. I had also to weigh a very good university position (at the ETH in Zürich) with the usual mix of teaching and research against one entailing a "total, almost monastic, commitment to research", (as someone wrote to us much later, while declining a professorship). In fact, the offer had hit me (not too strong a word) while I was visiting Oxford and in a conversation the day before, J. H. C. Whitehead had made some rather desultory remarks about this "mausoleum". To him it was obviously essential to be surrounded by collaborators and students at various levels. I also had to gauge the impact on my family of such a move. But, after some deliberation and discussions with my wife, who left the decision entirely to me, I felt I just could not miss this opportunity.
My professorship started officially on July Ist, 1957, but I was already here in the spring. I found Raoul Bott, with whom I had many common interests. Sometime before, Hirzebruch and I had made some computations on low-dimensional homotopy groups of some Lie groups and, to our surprise, some of our results were contradicting a few of those contained in a table published by H. Toda. There ensued a spirited controversy, in which the homotopists felt at first quite safe. Bott was very interested; he and Arnold Shapiro, also at the Institute at the time, thought first they had another proof of Toda's result on π10(G2), one of the bones of contention, but a bull session disposed of that. Later, Bott and Samelson confirmed our result. Eventually, the homotopists conceded. At the time, I had not understood why Raoul was so interested in those very special results, but I did a few months later when he announced the periodicity to which his name is now attached: Our corrections to Toda's table had removed a few impurities which stood in the way of even conjecturing the periodicity.
There was also a very active group on transformation groups around D. Montgomery who, with the Hilbert fifth problem behind him, had gone back fully to his major interest. My involvement with this topic increased, culminating in a seminar held in 1958-1959.
But I was now a faculty member in mathematics (together with K Gödel, D. Montgomery, M. Morse, A. Selberg, H. Whitney, as already mentioned, Arne Beurling, who had joined in 1954, and A. Weil from fall 1958 on) and had to have some concerns going beyond my immediate research interests. Foremost were two, the membership and the seminars. As regards the former, it was not just to sit and wait for applicants and select among them, but also of course to seek them out. Weil and I felt that in the fields somewhat familiar to us, a number of interesting people had not come here and I remember that for a few years, in the fall we would make lists at the blackboard of potential nominees and plan various proposals to the group. In this way, in particular, we contributed not insignificantly to the growth of the Japanese contingent of visitors, which soon reached such a size that the housing project was sometimes referred to as "Little Tokyo" and that a teacher at the nursery school found it handy to learn a few (mostly disciplinary) Japanese words. After a few years however, there was no significant "backlog" anymore and no need to be so systematic. As to the seminars, there were first some standard ones, like the members' seminar and the seminar in groups and topology, led by D. Montgomery. Others arose spontaneously, reflecting the interests of the members or faculty. We felt that the Princeton community owed it to itself also to supply information about recent developments and that beyond the graduate courses offered by the University and the research seminars, there should be now and then some systematic presentations of recent or even not-so-recent developments. In that respect, J.-P. Serre, a frequent fall term visitor during those years, and I organized in fall 1957 two presentations, one on complex multiplication and a much more informal one where we wrestled with Grothendieck's version of the Riemann-Roch theorem. As soon as he arrived, Weil set up a joint University-Institute seminar on current literature, thus reviving the tradition of the H. Weyl seminar, which he had known while visiting the Institute in the late thirties, and had also kept up in Chicago. The rule was that X was supposed to report on the work of Y, Z, with X \neq Y, Z. Later on, the responsibility for this seminar was shared with others. It was quite successful for a number of years, but was eventually dropped for apparent lack of interest. As I remember it, it became more and more difficult to find people willing to make a serious effort to report on someone else's work to a relatively broad non-specialized audience. Maybe the increase in the overall number of seminars at the University and the Institute, at times somewhat overwhelming, was responsible for that, I don't know.
During those years, algebraic and differential topology were in high gear in Princeton. In 1957-1958 J. F. Adams was here, at the time he had proved the nonexistence of maps of Hopf invariant one (except in the three known cases). Also Kervaire, while here, proved the non-parallelizability of the n-sphere (n \neq 1, 3, 7) and began his joint work with J. Milnor. In fall 1959 Atiyah and Hirzebruch developed here (topological) K-theory as an extraordinary homology theory, after having established the differentiable Riemann-Roch theorem; Serre organized a seminar on the first four chapters of Grothendieck's EGA. During that year, Kervaire, then at NYU came once to me to outline, as a first check, the construction of a ten-dimensional manifold not admitting any differentiable structure! M. Hirsch and S. Smale were spending the years 1958-1960 here, except that Smale went to Brazil in 1960. Soon Hirsch was receiving letters announcing marvelous results, so wonderful that we were mildly wondering to what extent they were due to the exhilarating atmosphere of the Copacabana beach, but they held out. (At the Bonn Tagung in June, as the program was being set up from suggestions from the floor, as usual, the first three topics proposed were the proofs of the Poincar6 conjecture in high dimension by Smale and by Stallings and the construction of a nondifferentiable manifold by Kervaire; Bott, freshly arrived and apparently totally unaware of these developments, asked whether this was a joke!)
During these first years at the Institute, my active research interests shifted gradually from algebraic topology and transformation groups to algebraic and arithmetic groups, as well as automorphic forms. That last topic was already strongly represented here by Selberg, and had been before by Siegel. This general area was also one of active interest for Weil, and it soon became a major feature in the school's activities. Without any attempt at a precise history, let me mention a few items, just to give an idea of the rather exciting atmosphere. I first started with two projects on algebraic groups, one with an eye towards reduction theory, on the structure of their rational points over non-algebraically closed fields, the other on the nature of their automorphisms as abstract groups. Some years later, I realized that Tits had proceeded along rather similar lines and we decided to make two joint endeavours out of that. But I was more and more drawn to discrete subgroups, especially arithmetic ones. Rigidity theorems for compact hermitian symmetric spaces, hyperbolic spaces and discrete subgroups were proved by Calabi, Vesentini, while here, Selberg and then Weil. It is also at that time that I proved the Zariski density of discrete subgroups of finite covolume of semisimple groups. Weil was developing the study of classical groups over adeles and of what he christened Tamagawa numbers. I. Satake, while here, constructed compactifications of symmetric or locally symmetric spaces. It became more and more imperative to set up a reduction theory for general arithmetic groups. The Godement conjecture and the construction of some fundamental domain of finite area became prime targets. The first breakthroughs came from Harish-Chandra. I then proved some results of my own; he suggested that we join forces and we soon concluded the work published later in our joint Annals paper. This was in summer 1960. The next year and a half I tried alternatively to prove or disprove a conjecture describing a more precise fundamental domain and finally succeeded in establishing it. Combined with the other activities here and at the University, this all made up for a decidedly upbeat atmosphere. But in 1962 rumors began to spread that it was not matched by equally fruitful and harmonious dealings within the faculty. Harish-Chandra, who was spending the year 1961-1962 here, asked me one day, What about those rumors of tremors shaking the Institute to its very foundations? We were indeed embroiled in a bitter controversy, sparked by the school's proposal to offer a professorship to John Milnor, then on the Princeton faculty.
Before we presented this nomination officially, the director had indeed warned us, without being very precise, that there might be some difficulty due to the fact that Milnor was at the University, and we could hardly an ticipate the uproar that was to follow. The general principle of offers from one institution to the other and the special case under consideration were heatedly debated in (and outside) two very long meetings (for which I had to produce minutes, being by bad luck the faculty secretary that year). A number of colleagues in physics and historical studies stated that it had always been their understanding that there was some agreement prohibiting the Institute to offer a professorship to a Princeton University colleague. In fact, the historians extended this principle even to temporary memberships. Fear was expressed that such a move would strain our relations with the University, which some already viewed as far from optimal. In between the two meetings, the director produced a letter from the chairman of the Board of Trustees, S. Leidesdorf, referring to a conversation he had participated in between Flexner and the president of the University, in which it had been promised not to make such offers. He viewed it as a pledge, which could be abrogated only by the University.
Those views were diametrically opposite to those of the mathematicians here and at the University, which were in fact quite similar to those of Eisenhart in the letter quoted earlier or in his statement to the curriculum committee, both naturally and unfortunately not known to us at the time. He really had said it all. First of all, the school used to give sometimes temporary memberships to Princeton faculty. This was on a case-by-case basis, not automatic, and it had never occurred to us to rule it out a priori. We also felt that our relations with Fine Hall were excellent and would not be impaired by our proposal. In fact D. Spencer had told us right away we should feel free to act. D. Montgomery stated that Veblen had repeatedly told him, in conversations between 1948 and 1960, that there had never been such an agreement. J. Alexander, asked for his opinion, wrote to Montgomery that he had never known of such an agreement (whether gentlemanly or ungentlemanly). He also remembered certain conversations in which an offer to a university professor was contemplated, or feared by some university colleague, conversations which would have been inconceivable, had such an agreement been known. Finally he had "no knowledge of deals that may have been consummated in 'smoke-filled rooms' or of 'secret covenants secretly arrived at.' All this sort of stuff is over my depth." A. W. Tucker, chairman of the University Mathematics Department, consulted his senior colleagues and wrote to A. Selberg, our executive officer, that in their opinion (unanimous, as he confirmed to me recently) the Institute should be free to extend an offer to Milnor. Of course, were he to accept it, this would be a great loss, but any such "restraint of trade" was distasteful to them and could well prove damaging in the long run. It would be much better, they felt, if the University would answer with a counteroffer attractive enough to keep Milnor. The point was repeatedly made that, when two institutions want the services of a given scholar, it is up to the individual to choose, not up to administrators or colleagues to tell him what to do; also, as Eisenhart had already pointed out, that such a blanket prohibition might be damaging to the recruiting efforts of the University.
In the course of the second faculty meeting a colleague in the School of Historical Studies, the art historian Millard Meiss, stated it had indeed been his understanding there was such an agreement; he noted that the mathematicians and his school acted differently with regard to temporary memberships he felt the rule had been a wise one in the earlier days of the Institute, but was very doubtful it had the same usefulness today. Accordingly, he proposed a motion, to the effect that the faculty should be free to extend professorial appointments to faculty members of Princeton University, with due regards to the interests of science and scholarship, and to the welfare of both institutions. He also insisted that this should occur only rarely. This motion was viewed as so important ("the most important motion I have voted on in the history of the Institute", commented M. Morse) that it was agreed to have the votes recorded by name, with added comments if desired. It was passed
After this, it would have seemed most logical to take up the matter with the president of the University, R. Goheen, but nothing of the kind was done at the time and the tension just mounted until the trustees meeting in April. There, as we were told shortly afterwards by the director, the Milnor nomination did not even come to the board: The trustees had first reviewed the matter of invitations to Princeton University faculty, with regard to the Meiss motion, and had votes a resolution to the effect that the agreement with Princeton University to refrain from such a practice was still binding.
In this affair we had worked under a further handicap: In those days, it was viewed as improper to talk about a possible appointment with the nominee before he had received the official offer (nowadays, the other way around is the generally accepted custom). Consequently, none of us had ever even hinted at this in conversations with Milnor. But he had heard about it from other sources and it became known that he would have been seriously interested in considering such an offer. The director and the trustees may not have felt so fully comfortable with their ruling after all. At any rate, they soon proposed to offer some long-term arrangement to Milnor, whereby he could spend a term or a year at the Institute during any of the next ten years. This was of course very pleasant for Milnor, and we gave this proposal our blessing, but it fell short of what we had asked for. Finally, eighteen months later, in October 1963, we were informed that, following instructions from the trustees, the director had taken up the matter of general policy with President Goheen in January 1963 and we received a copy of a letter written on January 21, 1963 by President Goheen to the director, outlining one. Although cautious in tone, it allowed one institution to extend an offer to a faculty member of the other, after close consultation "to the end of matching the interests of the individual with the common interests of the two institutions to the fullest extent possible." In conclusion, he urged that "this agreement supplant any specific or absolute prohibition that we may have inherited from our predecessors."
Right after the next trustees meeting the director wrote to Goheen on April 22, in part: "The Trustees asked me to tell you that they welcome your letter, and that they have asked me to let it be a guide to future policy of the Institute." As far as I know, the matter was never reconsidered and this agreement is still in force. At the time we were apprised of this (October 1963), it would have therefore been "legally" possible for us to present again our proposal, although Milnor was still a Princeton faculty member.
But we could not! During 1962-1963, we had asked for two additions to our group; they had been granted and no chair was available to us anymore. How had this come about?
This experience had left strong marks. It was not just the decision of the trustees, but the way the matter had been handled and the breakdown in relations within the faculty (also contributed to by conflicting views on some nominations in the School of Historical Studies), the ruling from on high by the board, without bothering to have a meaningful discussion with us, bluntly disregarding our wishes, as well as those of the faculty as expressed by the Meiss motion, all this chiefly on the basis of a rather flimsy recollection of the chairman of the board, promoted to the status of an irrevocable pledge. Some of us were wondering whether to withdraw entirely into one's own work or to resign, and were sounded out as to their availability. One Chairman, who had for some time wanted to set up a mathematics institute within his own institution, toyed with the idea of making an offer to all of us. We still had the option of making another nomination and there were indeed two or three names foremost on our minds. But just choosing one and presenting it would not suffice to restore our morale. Something more was needed to help us rebound. It was Weif who suggested that we present two nominations instead of just one, as was expected from us. After some discussions, we agreed to do so and nominated Lars H6rmander and Harish-Chandra.
This took the rest of the faculty and the director completely by surprise. The latter did not raise any objection on budgetary grounds. He also made it clear at some point that if granted, this request would have no bearing on faculty size for the other groups. Since our nominations were readily agreed to be scientifically unassailable, it would seem that our proposal would go through reasonably smoothly, but not at all. Our request had been addressed by A. Selberg, still our executive officer, directly to the director and the trustees, bypassing several steps of the standard procedure for faculty nominations, which seemed unpracticable in the climate at the time, and also not fulfilling one requirement in the by-laws. And it is indeed on grounds of procedure that the director and some colleagues raised various objections. There was overwhelming agreement on the necessity of major changes in our procedure for faculty appointments. The question was whether this review should precede or follow the handling of our two nominations. Again, this grew into a full-size debate and we did not know how our proposal would fare at the April trustees meeting. There, as we were told at the time, the director recommended to postpone the whole matter, but the trustees, after having heard Selberg present our case, voted to grant our request under one condition, namely that a faculty meeting be held to discuss our nominations. This was really only to restore some semblance of formal compliance with the by-laws, and they were anxious that this matter be brought with utmost dispatch to a happy conclusion, so that the Institute would soon regain its strength and some measure of serenity. This meeting was held within a week and the offers were soon extended.
Harish-Chandra accepted quickly, Hörmander after a few months. Finally, this sad episode was behind us. We felt and were stronger than before and could devote ourselves again fully to the business of the school. In fall 1963 there were the usual seminars on members and faculty research interests. Harish-Chandra started a series of lectures, which became an almost yearly feature: every week two hours in a row, most of the time on his own work, i.e., harmonic analysis on reductive groups (real, later also p-adic), documenting in particular his march towards the Plancherel formula. He was not inclined to lecture on other people's work. One year however he did so, he "took off", as he said, viewing it as some sort of sabbatical, and lectured on the first six chapters of Langlands' work on Eisenstein series (then only in preprint form). There were also some seminars on research carried out outside Princeton: I launched one on the Atiyah-Singer index theorem, for non-analysts familiar with all the background in topology. Eventually, R. Palais took the greater load and wrote the bulk of the Notes (published in the Annals of Math. Studies under his editorship). The following year, there was similarly a "mutual instruction" seminar on Smale's proof of the Poincaré conjecture in dimensions >= 5. Still, we felt some imbalance in the composition of the membership and the activities of the school. Of course, there is no statutory obligation for the school membership to represent all the main active fields of mathematics. In any case, in view of the growth of mathematics and of the number of mathematicians, as compared to the practically constant size of the school (the membership size hovering around 50-60 and that of the faculty around 7-8), such a goal was not attainable anymore. Nevertheless, it has always been (and still is) our conviction that the school will fulfill the various needs of its membership best if it offers a wide variety of research interests, and that this is a goal always to keep in mind and worth striving for, even if not fully reachable. For this and other reasons we decided in 1965 to have more direct input in part of the work and composition of the school by setting up a special program now and then. This idea was of course not to have the school fully organized all of a sudden, rather to add a new feature to the mathematical life here, without supplanting any of the others. Such a program was to involve as a rule about a quarter, at most a third, of the membership, with a ix of invited experts and of younger people. It would often be centered on an area not well represented on the faculty, but not obligatorily so. We did not want to refrain from organizing a program in one of our fields of expertise, if it seemed timely to gather a group of people working in it to spend a year here. It was of course expected that such a program would include a number of seminars for experts to foster further progress, but we also hoped it would feature some surveys and introductory lectures aimed at people with peripheral interests, and would also facilitate to newcomers access to the current research and problems. Pushing this "instructional" aspect a bit further, we also decided to have occasionally two related topics, hoping this would increase contacts between them.
The first such program took place in 1966-1967 and was devoted to analysis, with emphasis on harmonic analysis and differential equations. In agreement with the last guideline stated above, the second one (1968-1969) involved two related topics, namely algebraic groups and finite groups. As a focus of interaction, we had in mind first of all the finite Chevalley groups and their variants (Ree and Suzuki groups). They played that role indeed, but so did the Weyl groups and their representations, as can be seen from the Notes which arose from this. The third program (1970-1971) centered on analytic number theory.
In 1971, again with an eye to increasing breadth and exposure to recent developments, another activity was initiated here, namely an ongoing series of survey lectures. In the sixties and before, the dearth of expository or survey papers had often been lamented. The AMS Bulletin was a natural outlet for such, first of all because the invited speakers for one-hour addresses are all asked to write one. But this did not seem to elicit as many as one could wish and various incentives were tried, with limited success. It had always seemed to me that most of us are cold to the idea of just sitting down to write an expository paper, unless there is an oral presentation first. But the example just mentioned showed that this condition was not always sufficient. Already in my graduate student days, I had been struck by some beautiful surveys in the Abhandlungen des Math. Sem. Hamburg. They were usually the outgrowth of a few lectures given there. This suggested to me that one might have a better chance of getting a paper if the prospective author were invited to give some comprehensive exposition in a few lectures, not just one. However I had done nothing to implement such a scheme, just talking about it occasionally, until the 1970 International Congress in Nice. There K Chandrasekharan, then president-elect of the IMU, told me he wanted to set up a framework for an ongoing series of lectures sponsored by the IMU, to be given at various locations, with the express purpose to engender survey papers. Would I help to organize it? Our ideas were so similar that we quickly agreed on the general format: A broad survey, for non-specialists, given in four to six one-hour lectures, within a week or two. Expenses would be covered, but the real fee would be paid only upon receipt of a manuscript suitable for inclusion in this series. A bit later, I suggested as an outlet for publication the Enseignement Mathématique, mainly for two reasons: First, it is in some way affiliated to the IMU, being the official organ of the International Commission for Mathematical Education. Second, it has the rare, if not unique, capacity to publish as a separate monograph, sold independently, any article or collection of articles published in that journal.
The first two such sets of lectures were given at the Institute in the first quarter of 1971, by Wolfgang Schmidt and Lars Hörmander (who was a visitor, too, having resigned from the faculty in 1968), both soon written up and indeed published in the Enseignement Mathématique. But a difficulty arose with our third proposal, namely to invite Jürgen Moser, then at NYU, to give a survey on some topics in celestial mechanics. From the point of view of the IMU, these lectures were meant to promote international cooperation. Accordingly, the lecturer was to be from a geographically distant institution, so that the invitation would also foster personal contacts. They felt that we did not need an IMU sponsorship to bring Moser from NYU to the Institute. They certainly had a point. On the other hand, it was also a sensible idea to have such a set of lectures from Moser. In the school, we were really after timely surveys, whether or not they were contributing to international cooperation, while this latter aspect was essential for the IMU. Also, they wanted of course to have such lecture series be given at various places ana their budget was limited. Since we planned to have about one or two per year, our requests might well exceed it, so that some difficulties might be foreseen also on that scare. We therefore decided to start a series of similar lectures of our own, and to call them the Hermann Weyl Lectures, an ideal label, in view of Weyl's universality: It was a nice touch to be able on many occasions to trace so much of the work described in those lectures to some of his. We planned to publish them as a rule, though not obligatorily, in the Annals of Mathematics Studies. Otherwise, the conditions and format of the lectures were to be the same. Our series started indeed with J. Moser's lectures, resulting in an impressive two-hundred page monograph. For a number of years, the H. Weyl lectures were a regular feature here, at the rate of one to two sets per year. As to their original purpose, namely to bring out survey papers, I must regretfully acknowledge that our record is a mixed one, and that the list of speakers who did not contribute any is about as distinguished as that of those who did. Maybe Moser's contribution was a bit daunting, although F. Adams and D. Vogan rose to the challenge, even topping its number of pages (slightly in the former case, largely in the latter). Overall, the high quality of the monographs growing out of the H. Weyl lectures has made the series very worthwhile. Their frequency has declined in recent years. Since we started this, "distinguished" lecture series have sprung up at many places. Also, symposia, conferences and workshops on specific topics have proliferated, often leading to publications containing many surveys or introductory papers. There is indeed nowadays quite a steady flow of papers of this type so maybe the need for our particular series has decreased. One of the nice features of the Institute is that we need not pursue a given activity if we do not feel it fulfills a useful function in the mathematical community. So we may weft leave this one in abeyance and revive it whenever we see a good opportunity.
In 1966 C. Kaysen had taken up the directorship and found the school faculty in good shape. He thought that, at least with our group, he would not face requests for new appointments. But we pointed out to him that our age distribution was a bit unfortunate and would later create some problems, with retirements expected in 1975, 1976, 1977, and 1979. Therefore it might be desirable to consider some advance replacements; also that some minimal expansion might be to the good. He agreed. In 1969 Michael Atiyah joined the faculty. Originally, this appointment had been meant to be an expansion, but it was not anymore, after H6rmander had resigned in 1968. Later, we made offers successively to John Milnor and Robert P. Langlands, who came to the faculty in 1970 and 1972 respectively.
In the sixties, considerable progress was made in the general area I had already singled out as a very strong one here: Algebraic groups, arithmetic groups and automorphic forms, number theory, harmonic analysis on reductive groups. Much of it was done here, but also at the University by G. Shimura, and by R. P. Langlands who was there for three years. It continued unabated, or even at an increased pace, after Langlands joined us. This whole general field had become such an active and important part of "core mathematics" that it was all to the good. However, that was not matched by activities of similar scope in other areas and created some imbalance, accentuated by Atiyah's resignation in 1972. For reasons already explained, in our view it was not in the best interest of the school in the long run and to correct it by increasing activities in other areas became a concern. There were two obvious means to try to remedy this: the special programs and new faculty appointments. But they were not available to us during the energy crisis and the immediately following years. The financial situation of the Institute was worrisome and we had not even been authorized to replace Atiyah. Also, we had not been able to take care completely within our ordinary budget of the special programs, which entailed invitations to well-established people. We always had had to get some outside support, besides our standing NSF contract, and that was hard to come by in those years. But we resumed both as soon as it became possible: Enrico Bombieri came to the faculty in 1977 and Shing-Tung Yau in 1980, broadening greatly its coverage. We had also to wait until 1977 for the programs but have had one almost every year from then on.
In 1977-1978, our program was devoted to Fourier integral operators and microlocal analysis with the participation in particular of L. H6rrnander and M. Kashiwara. This was again an attempt to increase contacts between two rather different points of view, in this case the classical approach and the more recent developments of the Japanese school around M. Sato. It led to a collection of papers providing a mix of both. The next one was on finite simple groups and brought here a number of the main participants to the collective enterprise to classify the finite simple groups. 1979-1980 was the year of the biggest program to date, on differential geometry and analysis, in particular nonlinear PDE. The number of seminars was somewhat overwhelming. Several were concentrated at the end of the week, so as to make it easier for people in neighboring (in a rather wide sense including New York and Philadelphia) institutions to participate. Roughly speaking, the main activities were subdivided in three parts: differential geometry, minimal submanifolds, and mathematical physics, with seminar coordinators L Simon for the second one, S. T. Yau for the other two. A remarkable feature of the third one (devoted to relativity, the positive mass conjecture, gauge theories, quantum gravity) was the cooperation between mathematicians and physicists, probably a first here since the early days. Two volumes of Notes resulted from this program.
There was none the following year but then, in 1981-1982, we had one on algebraic geometry, at least as big as the previous one. Again, seminars were also attended by visitors from outside, two even coming from Cambridge, Massachusetts: D. Mumford and P. Griffiths would visit every second or third week for two to three days, each to lead one of the main seminars. We had decided to concenttate on the more geometric (as opposed to arithmetic) aspects of algebraic geometry, since we intended to have in 1983-1984 a program on automorphic forms and L-functions. But even with that limitation, it was of considerable scope (Hodge theory, moduli spaces, K-theory, crystalline cohomology, low-dimensional varieties, etc.). Griffiths' seminar also led to a set of Notes. This was again very successful but the evolution of these seminars betrayed a natural tendency, namely to try each time to improve upon the previous one, leading not unnaturally to bigger and bigger programs. As already stated, our original intention had been to add an activity, not to suppress any, and we began to wonder whether these programs, carried out at such a scale, might not hamper somewhat other important aspects of the mathematical life here, such as variety, informality, the opportunity for spontaneous activities and unplanned contacts, quiet work, etc. So we decided to scale them down a bit. Again, this was not meant as a straightjacket; rather, that the initial planning would usually be on a more modest scale. But, if outside interest would lead to a growth beyond our original expectations (as is the case with the present program on dynamical systems), we would of course do our best to accommodate it. We were aided in fact in our general resolve by the emergence of the Mathematical Sciences Research Institute at Berkeley: Big programs are an essential feature there and they have more financial means than we to carry them out. There is no need to compete for size.
S. T. Yau had resigned in 1984 and was soon replaced by Pierre Deligne. The retirements we had warned C. Kaysen about had caught up with us for some time and our group was reduced to six, two fewer than the size we were entitled to at the time, so that we had the possibility of making two appointments. We were anxious to seize this opportunity to catch up with some new major trends in mathematics. There had been some very interesting shifts in the overall balance of research interests, partly influenced by the development of computers, notably towards nonlinear PDE and their applications (with which we had lost first-hand contact after Yau's resignation), dynamical systems, mathematical physics, as well as an enormous increase of the interaction with physicists, the latter visible notably around string theory and conformal field theory (CFT). These last two topics were very strong at the University, but underrepresented here (not only in the faculty, but also in the membership). As a first attempt to improve this situation, I suggested in fall 1985 to E. Witten to give at the Institute a few lectures on string theory aimed at mathematicians. They were very well attended, so that the next logical move was to think about organizing a program in string theory and to ask Witten whether this seemed to him worth pursuing and, if so, whether he would agree to help, first as a consultant and then as a participant. That same year, we made two successful offers to Luis Caflarelli and Thomas C. Spencer, thus increasing considerably our range of expertise in some of the "most wanted" directions.
The first question put to Witten was not entirely rhetorical, given the abundance at the time of conferences and workshops on these topics. But it was agreed after some thought that a year-long program here would have enough features of its own to make it worth trying. A bit later, an expert to whom I had written about it warned that, in view of the usually rather frantic pace of research in physics, this might be all over and passé at the time of the program (1987–1988); but it seemed to us there was enough new mathematics to chew on for slower witted mathematicians to justify such a program on those grounds (later, that expert volunteered to eat his words). Anyway, we went ahead. The program had originated within the School of Mathematics, but the School of Natural Sciences became gradually more involved and eventually contributed to the invitations. In fact, the borderline between the two schools became somewhat blurred, the physicists D. Friedan, P. Goddard and D. Olive being members in mathematics, while the mathematicians G. Segal and D. Kazhdan were invited by the School of Natural Sciences. A primary goal of this program was to increase the contacts between mathematicans and physicists and to help surmount some of the difficulties in communication due to differences in background, techniques, language and goals. Accordingly, we had invited several mathematically minded physicists and some mathematicians with a strong interest in physics, all rather keen to contribute to the dialogue. The program was very intense, too, with an impressive array of seminars, notably many lectures on various versions of CFT, and many discussions in and outside the lecture rooms.
Our last two appointments, succeeded by that of E. Witten in the School of Natural Sciences, have quickly made the Institute a major center of interaction between physics and mathematics and also increased significantly the membership in analysis. Altogether, the school faculty seems to me to be about as broad as can be expected from seven people. I hope it is not just wishful thinking on my part to believe that by its concern for the school and its own work, it is well on its way to maintain a tradition worthy of the vision of the first faculty.
The reader will have noticed that, from the time I came to the Institute, this account is largely based on personal recollections and falls partly under the label of "oral history", with, as a corollary, an emphasis or maybe even an overemphasis on the events or activities I have been involved with or witnessed from close quarters. Even with those, I have not been even-handed at all and this paper makes no claim to offer a balanced and complete record of the school history and of all the work done there.8 Such an undertaking would have brought this essay to a length neither the editors nor the author would have liked to contemplate. Also absent is any effort to evaluate the impact of the school on mathematics in the U.S. and beyond: How much benefit did visitors gain? How influential has their stay here been on their short-range and long-range activities? What mathematical research was carried out or has originated here? How important has been the presence and work of the faculty? These are some of the questions which come to mind. To try to answer them would again have had an unfortunate effect on the length of this paper. Besides, an evaluation of this sort is more credible if it emanates from the outside, at any rate not solely from an interested party of one. Moreover, as a further inducement for me to refrain from attempting one, two evaluations of relatively recent vintage do exist. First, a report by a 1976 trustee-faculty committee, whose charges were to review the past, evaluate the Institute and provide some guidelines for the future. Its assessment was based in part on the letters of a number of scholars and on the answers (over five hundred from mathematicians) to a questionnaire sent to all past and present members on behalf of that committee. Second, one by a 1986 visiting committee, chaired by G. D. Mostow. Both, though not exempt from criticisms, conclude that the School of Mathematics has been successful in many ways. As a brief justification for this claim and without further elaboration, let me finish by quoting from a letter written in 1976 by I. M. Singer to the chairman of the review committee, Martin Segal, who was happy to share it with the committee:
Their [the members'] stay at the Institute under the guidance of the permanent staff affects their mathematical careers enormously. Their contacts with their peers continue for decades. They leave the Institute, disperse to their universities, and carry with them a deeper understanding of mathematics, higher standards for research, and a sophistication hard to attain elsewhere.
Such was the case when I was here twenty years ago. Last fall when I signed the Visitors' Book I turned the pages to see who was here in 1955–1956. Many are world famous and they are all close professional friends. I notice the same thing happening now with the younger group. Before I came in 1955, the Institute was described to me as I am describing it to you. It remains true now as it has been for the last thirty years.
In preparing this article I benefitted from the use of some archival material. I thank E. Shore and M. Darby at the Institute for their help in dealing with the Institute archives and R. Coleman at the University for having kindly sent me copies of some documents in the University archives. I am also grateful to A. Selberg and A. W. Tucker for having shared with me some of their recollections, and especially to D. Montgomery for having done so in the course of many years of close friendship.
[1] For this and the development of mathematics in Princeton until WW II, see William Aspray's article in A Century of Mathematics in America, Part II (editor, P. Duren, with assistance of R.A. Askey and U.C. Merzbach), Amer. Math. Soc., Providence, R.I., 1989, pp. 195–215.
[2] Flexner, I remember, Simon and Schuster, New York, 1940, pp. 359–360.
[3] Der Grund liegt mit in der Art der in Aussicht genommenen Besetzung der drei Hauptstellen. Veblen ist zufolge seiner menschlichen Qualitäten sicher der geeignetste Amerikaner, den man sich als führenden Geist in einer solchen Institution wie der von Ihnen gegründeten wünschen kann. Aber er ist doch nicht ein Mathematiker von ähnlicher Tiefe und Stärke wie etwa Hardy. Einsteins Mitwirkung ist nattürlich unbezahlbar. Aber er verfolgt spekulative Ideen auf lange Sicht, deren Erfolg niemand verbürgen kann. Als Führer junger Leute zu eigenen, notwendig auf näher gesteckte Ziele gerichteten Problemen kommt er weniger in Betracht. Ich bin von ähnlicher Natur, jedenfalls auch Einer, der lieber einsam als mit einer Gruppe gerneinsam denkt und mitteilsam nur in bezug auf die allgemeinen Ideen oder in der fertigen gerundeten Darstellung. Mit darum lege ich so viel Wert auf einen Mann vom Typus Artin oder v. Neumann.
[4] See in particular the articles by L. Bers, D. Montgomery and N. Reingold in A Century of Mathematics in America, Part I (editor, P. Duren, with assistance of R. A. Askey and U. C. Merzbach), Amer. Math. Soc., Providence, R.I., pp. 231-243, pp. 118-129, pp. 175-200, respectively.
[5] For many recollections about Fine Hall at this time, see The Princeton Mathematics Community in the 1930s. An Oral History Project, administered by C. C. Gillespie edited by F. Nebeker. 1985, Princeton University (unpublished, but available for consultation).
[6] H. H. Goldstine, The computer, Part III, Princeton University Press, Princeton, N.J., 1972.
[7] (Added in proof) B. Devine just drew my attention to the interview of Merrill Flood by A. Tucker in the collection referred to in footnote five above, according to which such an incident did indeed take place once in Fine Hall.
[8] In that connection, let me mention that A Community of Scholars. The Institute for Advanced Study, Faculty and Members 1930-1980, published by the Institute for Advanced Study on the occasion of its fiftieth year, contains in particular a list of faculty and members up to 1980 and, for most, of work related to IAS residence.
