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Isadore M. Singer (1924–2021)

In Memoriam Part 2: Personal Recollections

Robert Bryant
Jeff Cheeger
Phillip Griffiths

Communicated by Notices Associate Editor Chikako Mese

Figure 1.

Singer giving the Killian lecture at MIT, 2006.

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Lenore Blum

At Is’ 90th birthday celebration, Mike Sipser, then head of the MIT Math Department, read a letter Is wrote in the spring of 1962 suggesting how he might contribute to the department on returning from sabbatical in the fall. “Am willing to serve as Faculty Counselor or Freshman advisor, whichever you need most. …Happy to lecture in 18.01–18.02 or any of the special calculus sections. In fact, am willing to teach any elementary course so long as it is not advanced calculus for engineers.” He also added, “would be happy to teach some form of Modern Algebra…whatever the department desires.”

I was stunned. That letter, I realized, had changed my life.

Since I was 10 years old, math was my favorite subject, though I also loved art. When I told my high school math teacher that I wanted to major in math in college, he said, “Why would you want to do that? The best math was done 2000 years ago.” Not so strange in retrospect. After all, this was a missionary high school, and my math teacher was a missionary who taught Euclidean geometry. Not wanting to go into a dead field, I decided that architecture would combine my love for art and math, that is, until I took advanced calculus for architects. During my second year in college (Carnegie Tech), I was able to switch to math by taking an experimental course in computer science (the first such academic course ever given on the planet). Although fascinating, this was not the math I was looking for. I married that summer, moved to Boston, and transferred to Simmons College. The head of the Math Department, Marion Walter, a wonderful teacher said, “we don’t have anything more for you here” and arranged for me to be a special student at MIT. So, in the fall of 1962, I enrolled in Modern Algebra at MIT. The teacher was Is Singer. His course was just so beautiful. It was what I had always been looking for. The first semester was abstract algebra, the second semester linear algebra, but really terrific abstract linear algebra with Grassmannians. It was beautiful underlying theory and pretty advanced. I loved it and did really well. So that spring, I got up my courage and applied to grad school at MIT.

I remember going for an interview with the head of the Math Department in his office. As I walked in, he handed me a list of schools and said, “if I had a daughter who was going to graduate school, these are the places I’d tell her to go. MIT is not a place for women.” I was devastated. But then, I got a ‘lucky’ break. The next Saturday the Math Department had a faculty party. Somehow, at the party they were joking about this girl who was applying to the PhD program in mathematics. Is was there and asked, “who are you talking about?” They gave my name and he said, “Oh! She’s the best student in my class.” I got accepted the next day.

Lenore Blum is a career professor of computer science at Carnegie Mellon University. Her email address is lblum@cs.edu.

Robert Bryant

My first meeting with the legendary Is Singer was when I was a graduate student at the University of North Carolina at Chapel Hill. He, Atiyah, and Hitchin had just announced their beautiful work classifying instantons on using an artful combination of index theory and ideas of Penrose, and he came to our department to deliver one of those inspiring colloquium talks that live in my memory as highlights of my introduction to the world of research mathematics. I remember the contagious delight that he radiated as he told his story. He managed to convey the importance and beauty of the techniques and results to us graduate students without hauling out all the machinery that underlay the proofs. For example, when he needed to pass from the case of instantons with finite action on to instantons defined on , he referred to Karen Uhlenbeck’s “beautiful removable singularities theorem” and urged us to read about her work. (I believe that this was the first that I heard of Karen’s work.) I understand now, as I did not then, how important inspiring colloquium talks are to young mathematicians. That Is took the task seriously and didn’t just give a brilliant technical seminar talk to the experts in the audience is an indication of how much he cared about the life of our mathematical community.

Figure 2.

Singer in the Army Signals Corps during WWII, ca. 1945.

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I got to experience Is’ generosity in another way that evening. Back then, it was customary for a faculty member to host a reception/party in their home for the colloquium visitor, and graduate students were invited. After such a great talk, I couldn’t pass up the opportunity. I had an ulterior motive, though. In my study of Élie Cartan’s works on Lie transformation groups, which was just beginning, I had been referred to a paper by Singer and Sternberg 4, Part I, for a modern treatment. I wanted to ask Professor Singer, “What happened to Part II?” In Part I, they had thoroughly explained Cartan’s method of classifying the primitive, transitive Lie transformation groups and even pointed out places where Cartan’s arguments were incomplete (and how to complete them). Moreover, whereas Cartan worked in what was essentially a holomorphic category, they were able, with some extra hypotheses, to extend his results to the smooth category. However, this was just in the transitive case, and it was clear that there were many interesting intransitive simple Lie transformation groups; for example, the gauge group of a principal -bundle where is a simple (finite dimensional) Lie group. When I got up the courage to ask my question at the party, Is gave me his full attention, found a quiet corner where we could sit and talk, and told me the story of how he and Sternberg had become interested in Cartan’s work, what had motivated them, and what still intrigued him about Cartan’s old papers on the subject. The intransitive case had turned out to have several new and unexpected features, and they had intended to continue to work on it, but, after Part I was finished, Singer had gone on leave to the UK to visit Atiyah, and Atiyah immediately engaged him in the work that ultimately led to the their famous Index Theorem. Singer had never had time since then to return to work on Part II and, as far as I can tell, he never did, but he encouraged me to keep working on developing Cartan’s ideas. The enormous amount of attention that Professor Singer paid to a random graduate student far from the major centers in geometry is just one example of the way that he fostered and inspired a generation of mathematicians.

Of course, over the years, as I became aware of the enormous service that he did for the mathematics and physics communities and the National Academies, his example has remained an inspiration to me in my own career.

Perhaps the most direct way that he influenced my career, though, was that he, Shiing-shen Chern, and Calvin Moore joined forces in the early 1980s to propose a new NSF-sponsored research institute, the Mathematical Sciences Research Institute, in Berkeley, CA. They conceived of it as a permanent, independent entity from UC-Berkeley, supported by a consortium of academic institutions and with national reach.

Dan Burns

“Very dashing, Dr. Singer,” Ronald Reagan said in reaction to Is’ brightly colored cravat as the President rushed into his meeting with the White House Science Council. Is relished recounting this moment which had gone the way he felt all one’s life should go: vivaciously full-throttle. Earlier, however, during the Vietnam War era, he marched in protest at MIT in a suit, cravat, and sunglasses to increase the ‘credibility’ of the largely hippie student throng. He remarked, shocked, that his old friend Dirk Struik had not recognized him in his more ‘establishment’ outfit!

Many know that Is grew up poor during the Depression in Detroit. He was wistful recalling the letter from the University of Michigan awarding him a tuition scholarship of $50 per semester, making it possible for him to attend. Many years later he returned with his three youngest children to show them the campus and recount his days in the student coops living on less than $2.50 per week, taking turns preparing the meals for the residents and other duties the members shared. He had a deep sense of the challenges overcome in that part of his life and wanted to pass on some of that resilience to his daughters.

Is is famous for his endless series of very influential seminar courses on a broad array of the latest active areas in mathematics, and he would push his stable of students to pick up expertise in these areas to bring back to the group. The topics would range from algebraic geometry to PDE’s and later to mathematical physics. His advising of grad students could be similarly ‘bird’s eye view,’ students just being assigned a very general area to explore. I remember his oracular pronouncement long ago that the Poincaré Conjecture would be solved by a PDE method, his confidence based on the pattern of the uniformization theorem in dimension two. He was right, of course, but it was unreachable until much later, by methods he certainly did not foresee in any degree of detail.

In the time I worked with him, he was often away on leave and would return about once every month or two and hold a marathon session all in one day to catch up on theses, projects of post-docs, and so on. I found it was always a good idea to be last in line since his energy even late at night seemed boundless, and when it was getting late, with no dinner in the meantime, the last student could get dragged along to a continuation of the scientific discussion over a midnight sandwich or more at Ken’s in Kenmore Square. Discussions often drifted over to Boston’s need for more jazz and Chicago blues clubs, some of his musical loves.

Is went to college to study either literature or physics. He said much later that he had thought of trying to have another go at writing, perhaps buoyed by being a relative of Isaac Bashevis Singer, though my memory is not confident I have that claim correct now. But even with his enthusiastic embrace of intellectual breadth, Is admitted that he thought that people really didn’t excel at such cross-sectional careers, that it was already a rare enough gift to be good at even one pursuit, even if taken in the broad sense he showed in mathematics and physics. Is continued actively to a very ripe age, retiring in 2010 at the age of 86. He was always keen to be active, on the move. He had very little use for dwelling on past accomplishments. In 1984, Jeff Cheeger and I and others discussed having a 60 Birthday conference for him, an idea he shot down unceremoniously. Five years later he was awarded an honorary degree from the University of Michigan, and we had a small ‘birthday-like’ cake in the department common room for him, which he admitted he was grateful for, but we shouldn’t think he didn’t notice the unmentioned coincidence with his 65th birthday. He said to come back in twenty years, maybe then he would feel like 65 and be ready for a party. That was exactly what he got with the famous Gang of Four Conference for him and his friends Atiyah, Bott, and Hirzebruch right on time for his 85th.

I think of Abraham Pais’ exquisite biography of Einstein, Subtle Is the Lord, where he says that for such a committed scientist, his science was of the essence in any biography since that was an integral part of his view of life. This is certainly true of Is, and hopefully some day somebody will write such a history. These few recollections are admittedly personal and barely scientific, but they are a part of our placeholders until this scientific biography is written.

Dan Burns is a professor of mathematics. His email address is dburns@umich.edu.

Alain Connes

Is Singer was a great analyst who discovered and developed, completely independently of his work on index theory, many new fundamental notions of analysis with great impact in geometry such as analytic torsion. He was also a rare example of a mathematician with true influence in physics with, for instance, the use of zeta function regularization in renormalization (he told me about his lecture on that topic in front of Richard Feynman). Of course, the same independence holds for his contributions to operator algebras, such as the problem he formulated with Dick Kadison, whose resolution took more than fifty years.

Figure 3.

Singer as a young man with his father, 1940.

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I first met him in Rome in September 1975, when I had just finished my work on the classification of factors, and Is Singer was lecturing on the Extension theory of -algebras by Brown–Douglas and Fillmore. I remember vividly our discussions together while walking to the small church San Pietro in Vincoli, because he wanted to pay a visit to the Moses of Michelangelo. The impact of his ideas on my own trajectory was simple; I realized at that point the limitations of working on operator algebras per se while the potential of the Hilbert space operator interpretation of -homology was offering a bridge to reach geometric notions in the realm of the formalism of quantum mechanics.

Is Singer’s paper 3 already contains the key ideas that would give the operator theoretic paradigm for geometry, which continues to make perfect sense in the noncommutative framework. I knew from my work on factors that, in the measure-theoretic context, with key examples provided by spaces of leaves of foliations, noncommutative spaces possess an intrinsic time evolution while ordinary spaces are static, but meeting and discussing with Singer gave me the ferment of the intrinsic formulation of their geometry in a spectral manner, in particular by the role of the Atiyah–Singer Dirac operator as a generator of -homology. This paper of Is Singer, and the ideas he explains there, have a rare generating power that gives them everlasting value.

Is Singer was a mathematician with great analytic power and unique conceptual vision. In many ways the mental picture I keep of him in my mind is very close to the fatherly figure of the Moses of Michelangelo.

Alain Connes is a professor emeritus at the Collège de France. His email address is alain@connes.org.

Harold Donnelly

Although I wrote only one joint paper with Atiyah and Singer 1, these two mathematicians were crucial in my early career. In 1973, S.S. Chern, my thesis advisor at Berkeley, suggested that I visit MIT for a semester, to meet some of the mathematicians in Cambridge. Professor Singer kindly agreed to be my host. Patodi was also visiting MIT at that time, and we had many joint discussions about about spectral geometry. A few years later, Patodi and I wrote a joint paper. Patodi was then at the Tata Institute, and I was a Moore instructor at MIT. This long-distance collaboration might never have taken place without our previous acquaintance. Jeff Cheeger was visiting Harvard, and we had a discussion concerning Chern–Simons invariants. These two important contacts were only possible because of Singer’s generosity in acting as host.

I completed my doctoral program at Berkeley in 1974 and accepted a two-year Moore instructorship at MIT. Singer was the likely sponsor, because he was the faculty member most cognizant of my thesis work and some other results. During the years 1974 to 1976, there were some discussions with Singer about mathematics. For example, he liked my paper about the heat equation and volume of tubes, which solved a conjecture by Seeley. It was somewhat disappointing that Singer was on leave for much of the years 1974–76. However, I talked extensively with Bott at Harvard and Ray at MIT. Bott informed me of an old conjecture of his, which I solved. My interaction with other faculty at MIT was also very enlightening. So, again Professor Singer was indirectly supportive, although the personal contact was quite limited.

After my time at MIT, I spent two years at Johns Hopkins, 1976–78. For the years 1978–79, a Sloan Fellowship supported a return to Berkeley. Professor Singer was one of the letter writers in my application for the Sloan. Professor Singer had moved from MIT to Berkeley by that time. During the year, I was working on spectral theory for complete Riemannian manifolds, partly with Peter Li. Again, Singer attended my lectures and made favorable comments. Purdue was looking for a mathematician specializing in partial differential equations and differential geometry. The department head contacted Singer who suggested my name. I have been a tenured faculty member at Purdue since 1979.

My collaboration with Atiyah and Singer began in 1980. Several years earlier, they had provided an outline for the solution of a conjecture by Hirzebruch. The outline suggested that only one crucial step was missing. I worked on that step and proposed a joint paper to Singer. However, the problem was more difficult than anticipated. The work was only completed and published in 1983. An independent proof was given by Werner Muller; 2.

Harold Donnelly is a professor of mathematics at Purdue University. His email address is hdg@purdue.edu.

David Ebin

My contact with Singer was limited to being his student—I don’t recall talking to him except about mathematics. Thus, I am limited to describing him in his role as an advisor. He was everything a student would want. He would both ask and answer questions. Then he would ask me to write out pieces of my work and then make comments and suggestions.

Figure 4.

With Natasha and Emily Singer, 2010.

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Singer was my thesis advisor from 1965 to 1967. This was shortly after the Atiyah–Singer index theorem came out, so Singer was very much in demand. However he always made time for students. For a thesis problem, Singer first suggested that I work on the conjugate locus of a Riemannian manifold. Two previous students, Frank Warner and Nathan dos Santos, had written theses in this area previously, and John Mather had recently achieved new results on singularities of differentiable mappings. Singer thought I should try to apply Mather’s results to the conjugate locus. Unfortunately I was unable to make any headway in this area. As far as I know, this still has not been done.

Instead, I started working on the space of all Riemannian metrics on a given manifold. The group of diffeomorphisms naturally acts on this space, and my thesis came out as a construction of a subspace transversal to its orbits. It involved a lot of technicalities using Sobolev spaces and finally encompassed about a hundred and fifty pages. This is where I believe Singer showed his value as a fine advisor. He exhibited infinite patience while I worked it out, again sharing the asking and answering of questions. I say much patience because the project went well beyond the end of the semester and into the summer. This became a bit complicated because Singer was in California for the summer. However, he accepted my hand-written work and gave it to two other readers who were willing to sign off on the project.

Once during this time I went to the Singer house and met Mrs. Singer and their daughter Natasha, who was then about six years old. I saw Natasha again recently at a social function in New York, and she confirmed that she is the Natasha Singer who writes for the New York Times. I found that rather curious because Singer had several times told me that he was the world’s worst writer. For many of us in mathematics, the strenuous task of writing out the details does not compare to the satisfaction of working out the ideas.

One might ask how a graduate student in mathematics manages to write a thesis? It is usually the first time that one faces the daunting task of solving an original problem. The task is double for there are two a priori unknowns: ‘Can this problem be solved?’ and ‘Can I solve it?’ Singer’s advice and patience provided the help necessary to surmount the difficulties.

David Ebin is a professor of mathematics at Stonybrook University. His email address is ebin@math.stonybrook.edu.

Phillip Griffiths

Although over the years Is and I were together at conferences and other gatherings, my main contacts with him were through his mathematical works. With his special and, in many ways, unique perspectives from physics, Is was a central pioneer in the confluence in mathematics between topology and analysis that has taken place.

Among the interactions, direct and indirect, that I had with Is, three stand out. In the mid-1960s, we were together at a small conference in Mexico City. I had recently finished my degree at Princeton under Don Spencer and, not surprisingly, was interested in deformations of complex structures, a subject that had then been recently originated by Kodaira and Spencer. Is was also interested in this, especially how the Maurer–Cartan equation

enters and the subsequent properties of its solutions. Here is in a graded Lie algebra and is a derivation. In application to geometry, is usually a section of a bundle over a manifold and is in a first-order linear differential operator. In deformations of the complex structure on a compact manifold , in a tour de force Kuranishi had just completed his work using properties of 1 to show the existence of a unique versal deformation space, the Kuranishi space, . Although by no means an expert, Is asked me to explain as much as I could of the technical aspects of Kuranishi’s argument. It was later that I came to understand his interest as a variant of the inhomogeneous version of 1 appears in deforming connections in gauge theory as, e.g., in his work with Atiyah and Hitchin on instantons.

A second instance was many years later during the period when I was Director of the Institute for Advance Study. The Institute had organized a special event commemorating the establishment of the School of Mathematics. For one part of the program, Is, Raoul Bott, and Michael Atiyah were invited to come and give a general discussion of the year in the mid 1950s they had been together at the IAS. This was the beginning of one of those golden periods in an era of mathematics where there is a confluence of areas, in this case topology and analysis, that together solve major outstanding problems and create a whole new area in the subject. The three of them were among the principal founders of this new area, and their reminiscences and reflections, at once mathematical and personal, were the highlight of the event.

The third is indirect and reflects Is’ great breadth of interests. In the 1970s, I was working in Nevanlinna theory, which is the general subject of the geometric properties of holomorphic mappings between complex manifolds. Two milestones in the classical theory were due to Lars Ahlfors. The classical defect relations of R. Nevanlinna concerned the solutions to the equation

where is an entire meromorphic function and is a point on the Riemann sphere . One may view 2 as a holomorphic mapping , and Ahlfors extended the theory to the case of a holomorphic mapping

The second was Ahlfors’ interpretation of the classical Schwarz lemma, formulated as saying that a holomorphic mapping , ( = unit disc) is distance decreasing in the hyperbolic metric. Ahlfors showed that the same result is true if one uses any metric on the image whose Gauss curvature is . This ‘method of negative curvatures’ can be used to give a differential-geometric proof of the defect relation. Reflecting his broad interests in almost anything that combines geometry and analysis, Is was interested in Nevanlinna theory and asked one of his students, Michael Cowen, if the method of negative curvature could be used to prove Ahlfors’ defect relation for 3. Michael talked with me and together we were able to do this. Although far removed from Is’ main interests involving PDEs, operator theory, and topology, it was the combination of differential geometry and analysis that he appreciated. And this example is only one of the many instances that, through his own work and through his personal interactions with his students and colleagues, Is had such a central and unique role in our field.

Victor Guillemin

As a devotee of Is Singer, I sat in on many of the graduate-level courses that he taught at MIT during the years that we were colleagues in the math department. His lucidity and insights into the mathematics of spectral theory, index theory, and the geometry and topology of differential manifolds made me an ardent Singer fan. Moreover, I had an opportunity several years back to teach a course dealing with the Atiyah–Singer index theorem, and, as a result, an opportunity to appreciate the depth and beauty of one of his greatest achievements. I am extremely pleased that Is is being remembered in this volume of the Notices.

Victor Guillemin is a professor of mathematics at MIT. His email address is vwg@math.mit.edu.

Richard Palais

The way I am most closely connected with Singer is through volume 57 of the Annals of Math Studies book series, published in 1967 with the title Seminar on The Atiyah–Singer Index Theorem (SASIT). This lists me as its ‘Author,’ though I was in fact the sole writer of only nine of its twenty-one chapters, and there were six other participants (Atiyah, Borel, Floyd, Seeley, Shih, and Solovay) who wrote the remaining chapters or otherwise made major contributions to the volume, so I should better be referred to as its ‘Editor.’

This volume has an interesting history. In Bonn, in the Spring of 1962, Michael Atiyah gave a fascinating Arbeitstagung talk outlining his recent joint work with Singer: a remarkable new connection they had discovered between analysis and topology. This was what has come to be known as their Index Theorem. Armand Borel and I were in the audience and, like everyone else present, we both were excited by what we heard. Borel knew that I would be at IAS the next year and asked if I would be willing to cooperate with him on a seminar working out the details. (Borel, Deane Montgomery, and I had run a successful Seminar on Transformation Groups when I had been at IAS in 1959). I of course readily agreed, but then in midsummer, after our seminar had been announced, Borel wrote to me that something had come up and he would have to spend the better part of the academic year in Paris, so would I be willing to take over the direction of the Seminar. I felt I had to agree, and then spent the next two and a half years, first at IAS, with other Institute members, working through the details that had been roughly outlined in the Atiyah–Singer announcement, and then back at Brandeis, carefully writing up those details for publication in SASIT. It was a very rewarding but difficult experience, and it would have been a great help if Singer had been at MIT where I could have easily consulted with him. But he was in England, working with Atiyah on the details of an even more powerful result, using a different approach, and since this was long before the period of instant email communication, I had to get along without that resource. SASIT was published a couple of years prior to when Atiyah and Singer felt satisfied enough with their revised version to publish it, so for a while SASIT was the ‘go-to’ source for the Index Theorem, and even now it is sometimes still suggested as a source for less-experienced mathematicians to get started learning the prerequisites for its understanding.

Richard Palais is an adjunct professor of mathematics at the University of California, Irvine, and a professor emeritus of mathematics at Brandeis University. His email address is palais@uci.edu.

Hugo Rossi

Once I had passed my orals, I was officially qualified to do research with my advisor, Professor Singer. My job became that of formulating problems, looking at examples to understand them, and ultimately solving them. Throughout this period I had weekly meetings with my advisor. Sometimes these were group meetings with all of his advisees, but mostly they were one-on-one. Rarely did I have some progress to report—almost all of the meetings were spent trying to understand why the latest method of attack failed.

At one of those meetings in early March of my third year, my advisor stopped me in the middle of a tortuous computation of what we thought was a signal example, and said, “Good. Write it up.” I asked, “Write what up?” and he said, “Your thesis. What you’ve been telling me for the past months. That’s your thesis. Congratulations. And, by the way, I am nominating you for a postdoc at Princeton, and an abstract of your thesis is due there by the end of the week. Get something to me by Thursday and I’ll add it to my letter.”

Figure 5.

With Annabelle, 2005.

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I was shocked. What did I do? Where was the fabulous new result? The remarkable insight? The surprising twist? What was there in anything I’d reported over the past year that could impress anyone at Princeton? He could see what I was thinking. “Just the facts. No embellishment. Just ‘here’s the problem I’ve been working on, here’s where I’ve gotten, here’s how, and here’s what I want to do next.’ Just as if you were talking to me, here in this room.”

So, I went home that evening and started work on an abstract just as he had advised. Indeed, I had solved some problems, but not those originally proposed. I realized that I had to invert history: start by stating that the problems I had solved was the object of the study and end with the original problems as prospects for future work. Over the next few days I followed that course and went to Singer’s office with a draft of my abstract. To me, it was a catalogue of failures—a confession of incompetence.

When my advisor looked at it, he took a different view. “This will work just fine.” He noticed the look of despondency in my eyes. “OK, it’s a work in progress, but there’s no doubt you’re on the right track, and there’s lots of promise in this approach. They’ll appreciate that in Princeton. Don’t worry, you’ll do just fine.” I didn’t feel just fine. I knew in my bones that—on the day I arrived in Princeton—they’d uncover the hoax we were perpetrating on them, and expose me as a fraud.

“But, Dr. Singer…I’ve applied for a lectureship at Dartmouth College, and I’ve been told that I’ll probably get it. I love to teach. I think I prefer…”

“That’s fine. It’s good to have options. If they don’t take you at Princeton, we can talk about options. Maybe Dartmouth is one. But I think you’ll get the Princeton postdoc, so let’s proceed on that premise.”

I must have looked like I was being led to the gallows. “Look, Hugo—you’ve learned a lot of things here at MIT. You have the tools to learn what Spencer is doing and to work with him. While we’ve been scratching around in the desert looking for gold, he’s been out in the galaxy finding diamonds. You go there for a year and next summer come back here and tell me all about it. I really want to learn that stuff. So, you better get busy writing your thesis in detail, because it’s due in the office by the second week of April—about a month. And it has to be read by your committee before you can submit it.”

Hugo Rossi is a professor emeritus of mathematics at the University of Utah. His email address is rossi@math.utah.edu.

James Simons

While I was an undergraduate at MIT I first saw Is around midnight at Jack and Marion’s restaurant in Brookline where he was sitting with Warren Ambrose (whom I already knew) doing mathematics. I thought that was very cool—being a mathematician was certainly a great job! Thereafter I saw them frequently there but didn’t talk with either of them.

I graduated MIT after three years but stayed on as a graduate student working under Is. He taught me about Lie groups and Lie algebras as well as some differential geometry, which I very much enjoyed. Later in the year he told me that Chern was about to leave U. Chicago and go to Berkeley. He urged me to transfer to Berkeley so I could work under him. That sounded good. I got a very nice fellowship and hightailed it to the West Coast. Much to my disappointment, Chern was spending his first year at Berkeley on sabbatical leave!

Soon I met Bert Kostant, whom I liked, and he took me on as his student. I proved a few things that seemed interesting and showed them to Bert. He also thought they were interesting and said they might be used to solve an open problem: Why are all holonomy groups on undecomposable Riemannian manifolds transitive on the unit sphere? I said I would like to work on that, but he said don’t even try. He said Borel and Singer had each tried and failed. That just got me excited!

I made some progress with the problem and by mail shared my progress with Singer. He was encouraging. I kept making progress but then got stuck. It was then that my newly married wife and I were going to Boston over Christmas, and I made an appointment to meet Is at his office. That day there was a huge blizzard, but somehow he and I reached his office. I showed Is where I got stuck, and he immediately pointed out that I hadn’t used the undecomposable assumption, and, in a second, I became unstuck! After a few months back in Berkeley, I finished the proof.

In the next several years, we socialized occasionally but did not work together. Then I became Chair of Math at Stony Brook and became friendly with C.N. Yang, the famous physicist. One day Yang told me about the Bohm–Aharonov effect. It showed that if one constructed a magnetic field completely confined to the -axis in -space and sent electrons in a circle around that axis but very far away, then as they met, they made a phase change that would vary with the strength of the magnetic field. It is quite amazing! Mathematically, it could be described as a flat vector bundle with varying holonomy.

That year Singer was in New York. We met for dinner, and I told him about Bohm–Aharonov. He got very excited, and I believe that inspired him to start doing physics with Atiyah.

In subsequent years, we met socially from time to time. He and his wife Rosemarie accompanied us on our new boat, Archimedes, thirteen years ago. We occasionally visited them at their home, and once or twice they visited us at our home. When we started the Center for Geometry and Physics at Stony Brook, Is became co-chair of the Board. He gradually began to fail. The last time I visited him, he recognized me and was glad to see me, but it seemed clear that the end was nigh.

I wish I had seen more of him in these past sixty-plus years, but each occasion was a great pleasure. I miss him very much.

James Simons is co-chair of the Simons Foundation. His email address is jim@Euclidecap.com.

Elliot Singer

Is Singer, my father, was the quintessential Boston driver. He was the only person who could get to Logan Airport without taking the bridge or the tunnel. He never demeaned himself with paid parking for Red Sox games—free side-street spots miraculously opened on yet another go-around. He would find circuitous routes to avoid traffic jams on Memorial Drive. When, in my middle-age, I pointed out this saved no time, he acknowledged the fact, but had no patience for the inelegant solution.

Figure 6.

With Elliot.

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As a child, there were always mathematicians at the house speaking a foreign language, which fascinated me—words like ‘fiber-bundle,’ ‘isomorphic,’ and ‘manifold.’ Before Dad’s 50-plus years as a professor at MIT, he did the typical pre-tenure itinerant faculty gigs—UCLA, Columbia, Institute for Advanced Study in Princeton. The year in New York, when I was 4, Dad and Dick Kadison would visit jazz clubs, late into the night, and Dad got to sip coffee with Billie Holiday, one of his most precious memories. (Stale black coffee and two packs a day of Chesterfields were his staples.)

The next year, in Princeton, I knocked over J. Robert Oppenheimer while learning to ride a bike, with Dad running alongside, holding the seat. Arnold Shapiro was another of the mathematicians there—Arnold’s wife, Janine, was a second mother to me—and Dad always remained close to Janine and their son Gregor. The Shapiros had a copy of the newly released Harry Belafonte calypso album, and “Jamaica Farewell” became our family song. I still know it by heart.

We used to go to Celtics games with Ambrose. Dad had been in the Signal Corps with Johnny Most, the famous Boston sports announcer, and we would visit the broadcast booth, where I was allowed to announce. I never knew Ambrose had a first name, and I never called anyone ‘Mr.’ or ‘Mrs.’ Dad referred to some mathematicians, like Dick and Arnold, only by their first names. Others like (Warren) Ambrose, (Raoul) Bott, and (Michael) Atiyah, had only last names.

I was 11 when we spent the year at Oxford, where the Atiyah–Singer Index Theorem was born. I remember putting shillings into the meter for heat, how bad the food was, and learning algebra and Latin, and scuffing my dress shoes playing football (the English kids brought boot-black to retouch) at Magdalen College School for Boys. Before retuning to Boston, we visited Paris, where Dad taught me quadratic equations on the banks of the Seine.

Memories are funny, and I don’t really have many of Dad from my interminable years in junior and senior high. We moved to a more suburban-style house in Newton, and he hated the need to keep up appearances, like weeding and mowing (often assigned to me, contributing to my own lifelong contempt for suburbia). We went to see Goldfinger together, after which I devoured, in a week, his collection of Bond novels, before graduating to Hammett and Chandler (I still have his copies, in tatters). Then Kafka, The Brothers K., and browsing together at the Paperback Booksmith in Harvard Square—Dad, absolutely not school, instilled in me a love of literature. Of course, there was Casablanca, and the Bogart festival at the Brattle Theatre. We had no television.

By the late 1960s, when I was an undergraduate at MIT, I was a budding anti-war and civil rights protestor, which Dad never discouraged, though himself not an activist. October 1969 was the first huge Boston anti-war march, and Dad and his colleagues dressed in suits, with the hope this would make a difference. A few months later, when we were occupying Jerry Wiesner’s office to protest the war, Dad was among those outside the door (along with Chomsky). In later years, Dad took great pride in the photo of me on the front page of the Boston Globe from my graduation, handing a red armband to Wiesner, though I don’t think he was so sanguine at the time.

Dad always said, “science is just one of the humanities.” I sometimes try to reconcile my own research, as a folklorist, with his, as an abstract mathematician who considered String Theory, ‘applied.’ We both hated the term, ‘social science,’ and he loved my dismissal, stolen from Tom Lehrer, of anti-logocentric philosophers who incessantly bemoan the failings of analytical scholarship: “the very least you can do is shut up!” Somehow his search for simple, elegant, proofs and explanations must be isomorphic with my exhaustive (and exhausting) search for interconnectedness of versions and variants.

I was fortunate enough to attend the 1979 Joys of Research Einstein centennial conference at the Smithsonian Institution in Washington, where Dad was one of the presenters—another was Linus Pauling, whom we were both thrilled to meet. This was probably the occasion on which he best shared his views, in light of his personal experiences, and those interested in his intellectual biography should be familiar with this easily overlooked source (Walter Shropshire, Jr., ed., The Joys of Research, Washington, D. C., Smithsonian Institution Press, 1981), from which I quote:

My own motivations for doing research are: first, the private joy in the exercise of one’s talents. When I was a youngster I envied many around me. They had much talent. There were those who could play musical instruments well and those who were good at sports. (A great tragedy of my youth was I couldn’t hit a curve ball!) I found, however, that I seemed to be able to think more abstractly than most. When I learned about science and mathematics as a teenager, I discovered that the manipulation of abstract objects, their construction, and their rearrangement, were things I could do very well. Exercising this talent has always been a joy…

I find great satisfaction in creativity, for I then feel a kinship with artists and scientists the world over. A Matisse exhibit thrills and inspires me. I rush home and attack my own research problems with zest, feeling I am part of the world of Matisse. A good ballet affects me the same way. I love it and am inspired to go home and try to do my little bit—like a juggler before the gates of heaven.

Isadore Singer (On His Early Years)

(As told to Hugo Rossi)

I was in Luzon when the war ended [in 1945], and nine months later I was shipped home by boat and train—that is to Chicago, where I was discharged. Before returning home, I thought it would be a good idea to stop by the University of Chicago to see what it would take for me to become a graduate student in mathematics or physics. After talking with some people in the Math department, I was told, “You are now a graduate student in Mathematics—show up here in September.”

Three years later I, received my PhD—my advisor was Irving Segal, and the subject was mathematical physics. I was offered a visiting assistant professorship at MIT. Paul Halmos suggested that I get in to see Warren Ambrose when I got there late that summer. I didn’t know it at the time, but he had written to Ambrose to tell him, “look out for this kid—you’ll like him.”

When I arrived at MIT late in early September, I looked for Ambrose. When I walked in his office, he looked at me and said, “You’re Singer. I’m Ambrose. Sit down and let’s talk.” And we started talking, but about 15 minutes into that I told him that I had better go to the Math office to meet the chair and associate chair. “Nonsense,” Ambrose said, “There’s plenty of time for that. What’s going on here is what is important.” So we talked in his office for an hour and a half, and then Ambrose said, “I’ve heard that Chern is doing some fantastic stuff—completely changing geometry. Tell me about it.” I said that I had little knowledge of what Chern was doing; I was a student of Segal—a mathematical physicist. Ambrose retorted, “You’re from Chicago, that’s good enough for me. You’ll teach me Chern’s differential geometry this Fall.”

Ambrose, Halmos, and I were abstract analysts of one stripe or another, that’s what connected us. But here was Ambrose wanting to be in on what was going on that was really new, and I was his link to that. So, Ambrose and I were abstract analysts, but he saw, and I accepted, that the action at the time was in geometry. I told him that I’d prepare weekly meetings on Chern’s geometry. He said, “Where have you been since you got here?” I answered, “North Station and the train here.” He said, “Let me show you around.” I protested that I didn’t want to take up his time, and he said, “My time and your time are not at stake—when you get up tomorrow morning, it will all be there. But Boston is here now and you have to get to know Boston, if you want to do mathematics at MIT.” I had no choice.

We drove around and talked. Ambrose was infatuated with what was new and profound. His gut feeling was that Chern’s geometry was right at the forefront and will finally make geometry understandable to mathematicians. As it turns out, we had a couple of great years and came to understand and further develop this new technique in geometry. Some time later, Ambrose was asked: “how would you define geometry?” His response was: “Geometry is the study of things that are invariant under a change of notation.” Chern and a few others knew what he meant. I was one of them.

Ambrose was a taciturn fellow, uninterested in conventions. If you came up to him and said, “Hello, Warren, I’m…,” He would just walk away. He was Ambrose, not Warren Ambrose, not Dr. Ambrose, and not even ‘hey, you.’ Just Ambrose. He once wrote a review of a paper, saying that “it filled a much needed gap in the literature.” That was his whole review.

In any event, we drove all around Boston, up and down the coast and stopped to go see something that would have been just a tilted post, until Ambrose explained what it was. We ended up late in the evening, at a coffee shop—Jack and Marion’s. Ambrose explained that this was a hangout for the wives of mathematicians, whose husbands were in their studies proving theorems. He said that we could do our mathematics there, benefiting from the energy propelled by the women.

Indeed, when we got there, he took me over to a table and introduced me to Mrs. X, Y, Z, and so forth. I was in awe to meet the wives of these famous men—who I had not yet met! Was Ambrose trying to tell me something?

After two years at MIT as a visiting Assistant Professor, I accepted a position at UCLA as a tenured Associate Professor. I went back to abstract algebra and happily and excitedly worked with [a colleague] at UCLA. We did great stuff, but it didn’t have significant impact on mathematics. The work of “Ambrose–Singer” did. In 1957, I returned to MIT—not just to work with Ambrose, but to develop my own approach to geometry—especially as a tool to understand physics, both macro and micro.

In 1974, the Soviet Union opened up enough to hold a meeting of differential geometers in Novosibirsk. The main theme of the conference was the impact of Atiyah–Singer and Chern–Simons. I had not been invited, so I contacted the organizers (friends I had developed over the years) to remind them that I was the Singer of Atiyah–Singer. The response I received was that there were political issues, but they’d get me an invitation.

As they did. It was wonderful—I gave a half-hour talk, and the following discussion was another half-hour (that conference was advertised as taking place from 2pm to 6pm, but, every day, went on until midnight). It turned out that the issue was that I am a cousin of Isaac Bashevis Singer—and therefore a threat in their eyes.

Natasha Singer

My father Isadore, known to his colleagues and friends as ‘Is,’ passed away in February, and with him went the first person who taught me how to tell a story.

During his 50-year career as a mathematician at M.I.T., he produced and collaborated on major research discoveries that helped catalyze developments in both math and physics.

Figure 7.

Singer with Rosemarie, ca. 1998.

Graphic without alt text

But, to his family and students, my dad was first and foremost a teacher.

For years, professors and students in the Boston area would gather for his weekly seminars at M.I.T. where he hosted mathematicians and physicists presenting their latest ideas on string theory and other fast-evolving science topics. The seminars always ran late, largely on purpose, so that students and faculty could go out together afterward for Chinese food and continue the discussions.

Even after my father became an Institute Professor at M.I.T., a status that freed him from formal teaching duties, he continued working with undergraduates. For several years toward the end of his career, he even volunteered to work as a teaching assistant for a freshman course: first-semester calculus. The gig appealed to him, he said, partly because T.A.s in the course were able to freely mentor students—without needing to test or grade them.

“The students understood that I was there to help them and not to judge,” he said in an interview in 2010, adding that he thought empathy was the key to being a good teacher.

My father often said that math was “just one of the humanities” and he had a gift for explaining the intricacies of science in simple terms to non-scientists. One time when I was in sixth grade, I remember having dinner with him at a restaurant in Harvard Square and asking him to explain how the telephone worked. Delighted, he quickly launched into a lesson that started off with an ode to Marconi’s discoveries on radio waves, covered Alexander Graham Bell’s work on electrical currents and ended with a caution on government wire-tapping.

Along the way, he grew so animated, using his arms to illustrate the undulating electrical currents, that people at neighboring tables began to stare at us. I was 11 years old at the time and recall feeling embarrassed by the attention.

When the science lesson ended, however, a stranger at the table next to us immediately leaned over to my dad. “That was amazing!” she said in a loud whisper. “What are you going to explain next?”

It wasn’t until much later that I absorbed the real lessons of that evening: that understanding how science and technology work can be a powerful tool—and that explaining the inner workings of tech power to others can be a public service.

Natasha Singer is a tech accountability reporter at The New York Times and a Knight Science Journalism Fellow at MIT

Nancy Stanton

I met Is Singer the first day of his Complex Manifolds course in the spring of 1970. He came in the first day and said, “For those of you enrolled in the course, I require no work. I would like volunteers to write up notes.” I volunteered—and worked very hard. By the middle of the semester, it was clear that I would be one of Is’ ‘complex manifolds’ generation of students.

Several things stand out about his course. He introduced numerous important examples in the first few classes, so we would not just learn theory without knowing what it applied to. When possible, he introduced multiple ways of thinking about concepts, usually with a theorem starting “The following are equivalent,” and communicated that which way was best depended on the question. The course continued for several semesters, bringing a group of students to the point that we could do research in the area.

One of my nonmathematical memories from the first semester of the course is chatting with Is on the march from MIT to Boston Common for an anti-Vietnam War rally in the spring of 1970. Is wore his banker suit to help make sure the protesters wouldn’t be described as just some radical students. Is’ son Elliot, who was an MIT undergrad at the time, refused to march next to Is because he would be embarrassed to be seen with a ‘banker,’ but he did give Is a red armband to wear on his suit.

From Is I learned the importance and beauty of bringing together different areas of mathematics. The interplay and relations between different areas of mathematics and also physics was a theme throughout his research. As an advisor, he made sure his students worked on problems which involved several areas of mathematics so we would not become narrowly focused, but also so we would appreciate the unity of mathematics. The interplay between complex geometry, several complex variables, and partial differential equations became a theme through much of my work.

In the years after I received my PhD, Is inspired me to work on serious, interesting hard problems. When I struggled with a difficult problem he always encouraged me to continue working on it and made me feel that it was tractable. His strong encouragement was crucial to me at many points of my career.

I have very fond memories of Is and his wife Rosemarie welcoming Is’ former students to their house for dinner during the 1999 conference in honor of Atiyah, Bott, Hirzebruch, and Is. I last saw Is when he and Rosemarie again warmly welcomed us, along with other attendees and colleagues, to a garden party at their house at the end of the 2009 conference in honor of Is’ 85th birthday.

I am very happy to have been Is’ student, thankful for his teaching me to bring together different areas of mathematics and to look at the big picture, and grateful for his inspiration throughout my career.

Nancy Stanton is a professor emerita of mathematics at Purdue University. Her email address is stanton@nd.edu.

Shlomo Sternberg

At the time of our collaboration, Singer was an established mathematician. See for example, the Ambrose–Singer theorem and others. I had proved then one big theorem. The Linearization Theorem. We started working together on Cartan’s theory of Infinite Lie Groups 4. We worked every evening in the living room of our first rented apartment in Brookline, Mass. These sessions lasted very late into the night.

Is was devoted to his son and brought him to our home for regular visits. I particularly recall our evening together with this young son at our Passover Seder. Is was a joyous man. This joyousness that he so clearly brought to his mathematics was always there. It was part of who he was.

Shlomo Sternberg is a professor emeritus of mathematics at Harvard University. His email address is shlomo@math.harvard.edu.

References

[1]
M. Atiyah, H. Donnelly, and I.M. Singer, Signature defects of cusps and values of -functions: the nonsplit case, Addendum to: Eta invariants, signature defects of cusps and values of -functions, Ann. of Math. (2) 119 (1984), 635–637. MR0744866
[2]
W. Müller, Signature defects of cusps of Hilbert modular varieties and values of -series at , J. Differential Geom. 20 (1984), 55–119. MR0772126
[3]
I.M. Singer, Future extensions of index theory and elliptic operators, Prospects in Mathematics (Proc. Sympos., Princeton Univ., Princeton, N.J., 1970), pp. 171–185. Ann. of Math. Studies, No. 70, Princeton Univ. Press, Princeton, N.J., 1971. MR0343319
[4]
I.M. Singer and S. Sternberg, The infinite groups of Lie and Cartan. I. The transitive groups, J. Analyse Math. 15 (1965), 1–114. MR0217822

Credits

Figure 1 is courtesy of MIT.

Figures 27 are courtesy of Natasha Singer.