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Sigurdur Helgason Autobiography
Sigurdur Helgason gave an interview on 4 August 2008 in which he spoke about his career. We have taken Helgason's words, removed the questions that he was asked, and created what is effectively an autobiography.
We also give the autobiographical information given by Helgason in the Introduction to his Selected Works at THIS LINK.
We also give the autobiographical information given by Helgason in the Introduction to his Selected Works at THIS LINK.
1. Sigurdur Helgason Autobiography from 2008 Interview
I am born in north Iceland, as a matter of fact. The town is called Akureyri. It was - as Iceland was at the time - rather primitive. It had around three thousand people, and yet it was the second largest town in Iceland: Iceland even now has only 300,000 people, total. And Akureyri was very isolated because there was in 1930 no passable road from Reykjavík, the capital, and the only communication was by boat or by horse. The farmers in that area brought their goods to Akureyri - milk and so on - and this was all by horse-drawn carts. So I remember very vividly horses in the town. At that time there were probably five or seven private cars in town. The owners were primarily medical doctors who needed them for their practice. There was a taxi station, too. So there were some decent roads in the town. The roads in the northern part of the country were mostly one-lane gravel roads.
Life in Akureyri was simple: no TV, of course, and the radio reception was rather poor. Sports and social life at all ages was quite active. The large fjord would be frozen over much of the winter, providing infinite space for skating. As children we would skate for hours until we were exhausted, then lie down on the ice, look up in the black evening sky, and admire the dazzling display of northern lights in various colours.
My father was born in 1892 and my mother in 1900. Both of them lived in Reykjavík in the early part of their years. My mother lived in Reykjavík because her father was postmaster general there. My parents. families go way back. You see, in Iceland it is very easy to trace families, partly because there are so few people but in addition because there is available recent software called Íslendingabók, which can trace how an arbitrary person is related to you. My mother happened to be of a very large family called Briem, which went way back to 1750 or so. Iceland does not have family names, except from old times. Now you can use an existing family name but you cannot introduce a new one. You have to use the patronymic system. That is, your last name comes from your father's first name: Helgason means son of Helgi, and so on.
My father Helgi Skúlason became a medical doctor in Reykjavík, and he practiced for a couple of years in Iceland before he went abroad for specialisation in ophthalmology in Germany, in Freiburg and Berlin. So he became an eye doctor and was offered the position in Akureyri by the surgeon general. At that time the number of eye doctors in Iceland was maybe two or three. He settled in Akureyri in 1926 and served until 1970 as the only eye doctor in north Iceland. This meant that in the summer he travelled to the various villages in the north and set up consultations, two or three days in each place, depending on the size. There was very often a medical doctor in these areas where he would be housed. Thus people didn't have to travel to Akureyri, but they could see him in these villages. This took him, I guess, a little over a month every year.
This travel was rather primitive: first a little by car, but then by boat and finally by horse. So he had his equipment on two horses, and he set up consultations in schools or little buildings where he could house his equipment. I was with him on one of these trips, I remember. It was to a village called Raufarhöfn: there was a herring factory in the village, and I and several gymnasium students got jobs there for the summer, mid-June through August. This was convenient because the herring season coincided with the school vacation. My father happened to be going to that place on the boat with me at the time, so I saw his setup.
He actually produced the glasses too; he ground the lenses. After the summer he would be preparing dozens and dozens of glasses for the people he had seen during the summer, and the whole family was involved in packing these things. We would send them by mail to people, by boat.
My father found glaucoma to be abnormally frequent in North Iceland and gave several talks on the radio about it. With no TV and only one radio station, this reached many people and may have made some difference.
The work at the herring factory in Raufarhöfn consisted of two six-hour shifts each day. It was a hard and messy job. I remember several hours of rowing a boat inside a huge herring oil tank while my partner stood in the boat spreading protective paint on the inside walls of the tank. We alternated positions occasionally. The smell was overpowering and falling out of the boat because of the slippery bottom meant certain death, since nobody could swim in that oil - he would simply sink. However, the pay was great, at least when the fishing was good. The last summer I worked there, the herring season lasted until mid-September. The bus that took us back to Akureyri got stuck in a snow bank during a blizzard, the motor died, and we spent the night in the bus. Neither cell phone nor radio communication there. Help came the following morning from the village of Húsavík. I was sixteen or seventeen at the time.
In earlier summers I had done the usual thing for boys of my age, did some light work on a farm, haymaking, attending to animals, sheep, goats, cows and horses. It was quite enjoyable.
I went to gymnasium in Akureyri, which started in 1930 as the second gymnasium in the country. A gymnasium is about equivalent to high school, although it goes normally until the age of nineteen or twenty. So the last classes are a little bit more advanced than an American high school. At that time, there were just two: one in Reykjavík and one in Akureyri. Since then there are several others. There was a lot of time spent on languages, because Icelandic is only used by Icelanders: it resembles Norwegian but has more complicated grammar. So we had classes in Danish, English, German, and French. If you chose the "language line," as my brother Skúli Helgason (1926-1973) did, you would also have Latin. Now this has changed, of course, because Latin has lost its usefulness. But the programme was actually quite varied. This was a six-year school, so a bit longer than high school. I had chemistry, astronomy, physics, and mathematics, in addition to several other subjects.
I think the chemistry was somewhat inadequate because the laboratory facility was rather limited. Physics was pretty good. It was modelled after the Danish system: we used Danish text books. The mathematics programme was fine. Our final exam included finding the radius of a sphere inscribed in a regular dodecahedron. So this was a pretty good school, I would say.
The teacher that I benefited most from was actually an astronomer, but he had certain training in mathematics. He had a doctorate from Göttingen in Germany, which was a very prominent place, perhaps a centre of mathematics in the world at the time. This would have been around 1930 or so. His name was Trausti Einarsson (1907-1984). He is no longer alive, but he was a highly respected astronomer and geologist. And he had this infectious respect for mathematics, which he conveyed to the students. He became a professor at the University of Iceland in Reykjavík.
I was in the gymnasium during the years of World War II. The war affected Iceland in several ways. Iceland of course did not have any army. But after Germany invaded Denmark, England believed that Iceland might be next, so England decided to occupy Iceland. That was in 1940. So they moved in there with a certain amount of army and their equipment. We were very conscious of the army there in Akureyri, because they had a fairly large group there.
Now, that meant that Germany no longer considered Iceland neutral, and that was very costly to Iceland because Iceland had this unlimited trade of fishing with Britain. So there were constant shipments of fish to England from Iceland, and this was a very significant part of Iceland's economy. But by the end of the war, Germany had sunk just about all that fleet. The war had this effect in Iceland that there was considerable loss of life, mainly because of the sinking of the boats. This included many boats in normal travel from one town to another, as well as ordinary fishing boats. These were completely unprotected, you see. My mother's sister and her husband were returning from the United States at the end of the war, in 1945. Two weeks before they left, he had defended his PhD thesis at Harvard in Medicine, and she was a paediatrician. The boat was in a convoy coming from New York, and it was quite close to Iceland when it was shot down by a U-boat. The whole family was wiped out: both parents and three children. It was very tragic. As a matter of fact, in percent of population, Iceland lost more lives during the war than did the United States.
Iceland benefitted from the war in some ways though. After America entered the war, the British were essentially replaced by Americans. Various construction projects, including the airport in Keflavík, caused considerable boost to the economy. At that time, Iceland became able economically to start on the geothermal energy project, first in Reykjavík but gradually it spread over the whole country. This energy source, combined with plentiful hydroelectric power, has eliminated the use of coal and oil for heating.
When the war ended in 1945, I went to University in Iceland school of engineering. That was a temporary move, because I intended to go to Denmark. Copenhagen was the traditional place. Of course prior to that, around 1930, Göttingen was also popular. But in those days it was out of the question to go to Germany. Göttingen actually did not suffer bombing during the war, so the university escaped more than many others. But the Hitler regime had decimated the mathematics faculty.
But I was told that the University of Copenhagen hadn't quite gotten started during that first year. The teaching had become a bit fragmentary during the last years of the war. In mathematics, a couple of the professors had to settle in Sweden. I think Harald Bohr had to settle there, as did his brother Niels Bohr. Werner Fenchel was another of Jewish descent; he had also to go to Sweden. So I was told it would be more practical for me simply to study at the engineering school in Reykjavík, because, as I said, the science education in Iceland was very much modelled after the Danish one, including the use of the same textbooks. So this was a practical move, just to stay in Reykjavík one year.
During that time, my fellow engineering students spent a lot of time on technical drawing. Actually the professor was the father of Vigdís Finnbogadóttir, who later became president. He had very high standards. Nowadays that drawing has been computerised, but at that time, much of the students' time went to that. But I could skip it. The courses I took were in mathematics, physics and chemistry, and a small one in philosophy.
In 1951, I won the University of Copenhagen's Gold Medal for an answer to a Mathematical Prize problem, to establish a Nevanlinna-type value distribution theory for analytic almost-periodic functions. The University of Copenhagen had a tradition that goes far back and they still have this system: every year the university posts certain prize questions, not just in mathematics, but in all fields. In literature it may be a literature study about some specific author. There is one in theology and one in physics and so on. So it would generally be an essay about a specific topic. In mathematics it is usually a real research problem.
Now, in Copenhagen, I had plenty of time. You see, the degree there was the so-called Candidatus magisterii. This meant you studied mathematics, physics, chemistry and astronomy to a certain level so that you could with ease teach these subjects in gymnasium throughout Denmark. Then after the two or three first years, you specialised - and I specialised in mathematics - and that would be like a beginning of a graduate school. But the number of courses available was rather limited, which meant that I had lots of time on my own. I could have finished my degree much earlier than I did. But then I noticed one of these prize problems, because it was posted there in the university. So I decided to try that, mostly for the fun of it, also because there was plenty of time for me.
The system is that you hand in your response after a year: you have the calendar year for this. The only condition is that you had to be under the age of thirty, so technically anybody in the world is allowed to try. And you do it anonymously: the author's name is not on the paper, but rather a certain symbol. I used the letter "Aleph". Along with the paper is a closed envelope where the author's name and the symbol is included. If there is no reward for the paper, the envelope is not opened. So you had nothing to lose if you didn't win! I worked on it for a whole year. Yes, it was a hard problem, at least for me, because I didn't really have the background needed, and since it was supposed to be handed in anonymously, I could not consult with anybody: I felt duty bound not to tell anyone. So I was on my own trying to get the background. It wasn't even clear what background I needed! So I think the first four months went on just finding out what the problem was about. But an indirect benefit, perhaps, was that I got used to working on my own without any guidance.
There were two subjects involved. One was a subject founded by a Finnish mathematician, Rolf Nevanlinna. There was a school of Finns that all went into that line, including one who later became a professor at Harvard, Lars Ahlfors, and who became a good friend later, after I got to MIT. Then there was the other side, the theory of Harald Bohr, who was a very prominent Danish mathematician, and he had started this almost-periodic function theory. The problem was to somehow combine these two theories. That is, to show that the Nevanlinna theory could be modified so that it fits the theory of almost-periodic functions and then prove the analogues of Nevanlinna's two main theorems as well as the defect relations. So this was what I did.
The problem isn't completely solved yet because I had to make a certain restriction on the functions considered. I did actually revise the paper later with the intention of pushing further, getting rid of this assumption. This was after I got to the States. But then I somewhat lost interest. I have published an excerpt of the paper in the proceedings of a memorial conference to Harald Bohr. I have been approached lately by two Russians about what remains of the problem, namely certain subtleties concerning the so-called defect relations. But almost-periodic functions is a rather limited subject, although it was intensively studied in Denmark during the 1930s and 1940s even. Harald Bohr developed the theory in the 1920s, and it was very prominent for a while. But it has dried up a bit.
The theory was a kind of a natural extension of the theory of Fourier series, which is a very prominent subject in mathematics. Later on, one can say it has been to some extent absorbed in the theory of Fourier series on compact groups, except for the part that I was involved with, which was more related to Dirichlet series. I would say that this part is very much alive. In fact, analytic almost-periodic functions do constitute a very interesting generalisation of Dirichlet series, in that the frequencies are quite arbitrary.
Rolf Nevanlinna had a strong connection to Switzerland. There used to be a Nevanlinna colloquium in Switzerland regularly, and he enters a little bit in the story of André Weil. Weil was in Finland during the second half of 1939, and he was arrested by the Finns during the Russian-Finnish war and was in danger of being executed as a spy. But Nevanlinna managed to have him deported to Sweden instead. André Weil wrote his memoirs published by Birkhäuser - not very large, but very interesting. There he relates the story. He had to go back to France and serve in the army, and he got into some difficulties, having been away when the war started.
Somehow at that time I wasn't thinking much about the future: I didn't have any particular sights on doing research in mathematics. I suppose I assumed that I would go back to Iceland and take a position as a gymnasium teacher, because that seemed a very pleasant job at the time. I think the teaching load was quite high, although nobody thought about that. But, the teachers in the gymnasium in Akureyri, they were extremely respected people. In town, these were the people who were looked up to and they seemed to enjoy their job. So I felt this was just an agreeable position to have.
At that time I discounted the possibility of having some kind of a position or a job at the University in Denmark. It just didn't seem possible: there were very few positions and relatively little research activity. As I pointed out to you, the amount of mathematics courses that were given was rather limited, and Denmark, in mathematics, took a while to recover after the war. Mathematics was only present at one university: the University of Copenhagen. There was another university, Aarhus, but there was no mathematics department there at the time. The mathematics department in Aarhus came later through the visionary initiative of Svend Bundgaard. That meant that there were two mathematics departments in Denmark, and that turned out to be very beneficial to both places. There was not a real competition, but somehow they both got revitalised in the process. But it wasn't until the 1960s when that really took off. So now Danish mathematics is very versatile, very dynamic, but in those days it seems in retrospect to have been somewhat dormant, although the teaching was first rate.
My parents told me about this possibility of a fellowship to go to the United States. So I applied to something called the Institute for International Education, kind of a Fulbright. So I filled out an application there, and I had the choice to go to either Harvard or Princeton, but my professor, Börge Jessen, recommended Princeton, and it was a very wise choice because Princeton did not have such a structured graduate programme. There were no exams in the graduate courses. No homework. Complete academic freedom. And I didn't take many courses there! As a matter of fact, the courses in Princeton were, at that time, somewhat specialised, in the sense that it was nothing like the spectrum we have now at MIT - and maybe now in Princeton too. But the compensation for that was that there were many seminars. In particular, students would often organise seminars on topics they wanted to study. Then there was the Institute for Advanced Study, which had regular lectures. I went to some of them, particularly to some given by Arne Beurling. So I was on my own as a graduate student, which suited me fine. It was a great time! But at one time there was a serious oral exam, called the "Generals". On my examination committee I had Artin, Lefschetz and Spencer.
I was only two years in Princeton, so when I finished my thesis there I really knew very little, quite ignorant about several fields in mathematics. At the time I didn't feel that way, but later I realised that two years was a little short. So I could have stayed a little longer perhaps, but when you finish your thesis you are supposed to leave!
Salomon Bochner was fairly natural choice for my thesis advisor because he was one of the major names in the theory of almost-periodic functions. He went to other things after he came to the States, but while he was in Germany, he was one of the founders of the theory. So I went to his lectures and somehow got acquainted with him. During the term when he was on leave I came across a problem in almost-periodic function theory. After Bochner came back I mentioned this to him. He was at first sceptical: "this problem has already been considered by many people." But he quickly agreed that it was far from being solved. Actually, the solution was quite easy, but it required methods which were only known after 1940. It was not directly related to what I had done in Denmark, but it was still within that field. I connected it to another theory, which had become very fashionable at the time, the so-called theory of Banach algebras. So my thesis was on Banach algebras and almost-periodic functions and contained some theorems on each of these two topics. Bochner was very accessible and supportive.
I spoke to Salomon in English - it seemed easier. I could read German, but I couldn't speak it. I had English for four years, German for just two. But neither language was comfortable for me. I hadn't spoken a word of English for five years, so it took some effort. But I had no problem understanding lectures: Mathematics is easy so far as vocabulary is concerned.
Bochner was Jewish and Orthodox, so of course he had to leave Germany in 1933, and he came to Princeton later that year. That was good, because many other German mathematicians had very great difficulty finding positions here. He was a big star, no question about that. There has not been much written about his life. There is an article by a colleague of his at Rice, where Bochner went after he retired from Princeton. That colleague was William Veech, who was a former student of Bochner's, and he wrote a little article about his life. He sent it to me, but I am not sure it has been published. Anthony Knapp has an article about him in the biographical memoirs of the National Academy of Sciences, 2004.
Bochner was inspiring to talk to, and he spoke freely about many things. I also liked the fact that although he talked to me about certain problems that he wanted me to get interested in, the fact that I was not interested in those problems but started on some other problems instead - he reacted positively to that and was very supportive. So I got into this habit of just following my own taste in the choice of topics to work on. This suited me better.
I was good friends with Arthur Mattuck at Princeton; Frank Peterson also. We were in classes together, as a matter of fact. Frank and I played tennis together, too. Arthur was of course in a different field: he was in algebraic geometry. Frank Peterson was again in another field, in topology. So mathematically, we were separate. But we all had dinner together in Procter Hall in Princeton. We wore black gowns, and at dinner time we talked about mathematics - it was largely shop talk throughout. This was a complete novelty to me, because I lived in a fifty-student dorm in Copenhagen, and there were no mathematicians there except me. They were mostly engineers, but somehow talking shop over meals was something that you just didn't do. In Princeton that was quite common among the mathematicians, but frowned upon by others, who even complained to the dean.
Frank and I were in one course together the very first term. That course was given by Ralph Fox. I happened to have met Fox in Copenhagen. He came there for a visit. So he knew that I was going to Princeton, and I contacted him when I got there. Anyway, this was a course in topology, in which I knew next to nothing. It was done partly by the so-called Moore method, where there is no text book; he simply handed out some notes on theorems that we, the students, were supposed to prove. Moore was a guy from Texas who is famous for that method of teaching. Moore went further in that he forbade the students to look at any sources. No books are allowed; you had to do it all on your own. Fox didn't take that attitude: he welcomed that we consulted the literature. And it was a kind of a seminar. Fox would say, "Is there anyone that has solved one of these problems here?" And then somebody would say "Yeah." So we would be usually occupied with listening to somebody giving a talk on a problem. But, as I say, there were no exams: there was complete academic freedom. So if you were not interested, or if you were lazy, you didn't have to do anything. But Frank was in the course, and Frank knew the subject already pretty well, so I would talk to him sometimes about topology.
Another person in that course was Gian-Carlo Rota. It was apparent to me that this guy would become quite a scholar; whenever I took a book or journal out of the library, his name was on the list of earlier borrowers. He was a senior, and he had to take this course for credit. He was kind of humble, because he was a senior in a class with mostly graduate students, who by custom did not talk much to undergraduates. So I never talked to him at that time, and I didn't get to know Gian-Carlo until I came to MIT and we became close friends. However, he did an enormous amount of the work in that course, and he was very often the one who was called upon to talk: I would say that he probably gave more problem solutions in that class than anybody else. Later I asked him, "Why did you do that?" He said, "Because unlike you I had to take it for credit!"
After finishing my PhD I came to MIT for the first time in 1954. There were these that you could apply to, so I applied to MIT and I applied to Harvard, both of them. And then a friend of mine, Walter Baily, was driving up to Boston - it was probably early spring 1954 - and he said, "Why don't you come along? You can then take a look at MIT, where I studied." So I did. I liked it, so when I was offered the Moore Instructorship, I accepted right away.
Ted Martin was the chairman then, so I'm sure that Bochner wrote to Martin. I think the procedure was simpler in those days, because I don't remember asking anyone for a recommendation letter except Bochner. Nowadays the custom is to have several different letters.
Of course, the number of graduate students during those years in the 1950s at MIT was very small, and the offering of courses was relatively limited. But there was a big jump at the end of the 1950s - the Sputnik atmosphere caused a lot of expansion. In earlier days, undergraduate seniors were I think usually encouraged to go elsewhere for their graduate school, because there wouldn't be enough new to offer them. But that changed in the 1960s, and as a result, we have now some graduate students who were undergraduates at MIT.
Ted Martin was extremely successful in building up the department. Later chairmanships have been rotating, usually for just a three-year period, so people tried to do their research at the same time. Whereas in Ted's time, he was a full-time, permanent chairman. Through his dedication he accomplished a lot for the department. Later he became Chairman of the Faculty for the whole Institute, which he was for several years. He was a talented administrator, and extremely tactful in dealing with people. I later served on several committees within the department, for example for hiring Moore Instructors, where he was a member. I remember that very well. He would always be willing to accept another point of view.
He had a monthly get-together at his house. He sent a card, "We will be home every Sunday during this month, and feel free to drop in," and people did. Nothing very formal, but it was a positive effort to make people feel comfortable. I admired him very much as a chairman. I think he really deserves tremendous credit for what he did for the mathematics department at MIT. He showed up with his family at a celebration of fifty years of the Moore Instructorship, which he started.
Martin died just a couple of years ago on 30 My 2004. I remember there was an Institute-wide gathering at the Faculty Club, when he was retiring, and he gave a little speech there. He was telling a joke about another guy who was retiring, and who was asked, "What are you going to do now that you're retired?"
"Oh I think I'll finish my book."
"Oh, I didn't know you were writing a book!"
"Oh, no, I'm not writing one, I'm reading one."
That's the kind of joke that Ted Martin would tell.
When I was in Princeton I did somehow find out about Harish-Chandra's work. Bochner never talked about it - it was very far from his work. I saw it in the literature. It was unrelated to my thesis, but I was led to it because I got interested in Banach algebras and their relation to abstract harmonic analysis, so it was natural to get involved in Harish-Chandra's work. But that was on a very high level, so the first thing I had to study on my own was Lie group theory. Once in the Spring 1954 Harish-Chandra was invited to give a lecture to the physicists. On the faculty were the two physicists Valentine Bargmann and Eugene Wigner, who were among the pioneers of representation theory. I was very happy to go but there were, I think, no other mathematicians in the audience.
I understood it, because he really wanted to be understood by the physicists. And Wigner, he wasn't afraid of asking simple-minded questions. He was a modest man, and might be asking questions to which he knew the answer very well but pretended not to, just so that the students would then benefit from the answers. It is not very usual that a professor would play that role. But I think he did! Bargmann was there too; he wrote a very important paper in the field, and so did Wigner.
I studied Lie theory on my own in Princeton and I continued after I got to MIT.
Lie group theory has nowadays become a lot easier. In one year's course you can go much further than you could in the 1950s, because the whole thing has become more streamlined. At the same time, the subject has grown enormously, so reaching the frontier is still quite difficult. Lie groups and representation theory, which is an outgrowth of Lie theory, is a strong subject at MIT now. But as I said, this was not in a course given during those two years I was here, when the department and the variety of courses was much smaller. So I would say that even after those two years at MIT, I didn't really have any serious insight in it, partly because nobody else in the area was involved in the subject. Nowadays students have a much easier time of it.
I started systematically reading Harish-Chandra's papers. But it did not come so easily to master Harish-Chandra's work, because although I had gotten some facility with Lie group theory, Harish-Chandra's work was way beyond that, too. The background for reading his papers was really rather inaccessible. There were some notes from Hermann Weyl at the Institute, which Harish-Chandra always quoted and probably had mastered: Harish-Chandra had been at the Institute, so he had absorbed some of this from Hermann Weyl. While Weyl was a spirited writer, I did not always find him all that clear.
Then there was Élie Cartan's work. But his work was, for one thing, relatively little understood, in spite of its great importance. He was one of the great mathematicians of the period, but his papers were quite a challenge. Hermann Weyl, in reviewing a book by Cartan from 1937 writes: "Cartan is undoubtedly the greatest living master in differential geometry. I must admit that I found the book like most of Cartan's papers, hard reading."
Geometry has always appealed to me very much. It's just that global differential geometry, at the time I was interested in it, was rather inaccessible. It is an old subject, but it was rather quiet until the mid-1940s or so. It has now become a very popular subject, but it took a while.
Chevalley's book Theory of Lie Groups came out in 1946. He was a professor at Princeton University at the time. His book was very useful, in that it clarified a lot of material that was obscure. It's a book on Lie groups, but it had considerable effect in differential geometry also. Chevalley's book and Seminaire Sophus Lie in France, 1955, were the principal background sources for me. Later, in 1956, appeared a nice little book by Katsumi Nomizu, Lie Groups and Differential Geometry. It did not go very far - eighty pages or so - but it was very useful to me. However I did not really get seriously involved until I went to Chicago for two years, 1957 to 1959.
In some ways I was still in graduate school at MIT. I only taught courses that did not take much time. Ted Martin did allow me to give one graduate course, and that required much more preparation. It was called Functional Analysis. That is a very large subject, so I just simply picked out subjects that I liked to talk about.
I picked up a lot of things from communication with other mathematicians. I got acquainted with John Nash right away when I came here, and I was, of course, very impressed with him. I knew of his great work in Riemannian geometry, and he also would often talk about what fields in mathematics were worthwhile and what fields were not. He had plenty of opinions about that, which I took seriously, though with some grain of salt. We were both bachelors, and neither of us liked to cook, so we went to restaurants all the time. Arthur Mattuck was often with us. He was an instructor at Harvard the first year I was at MIT and moved to MIT the following year.
Nash actually started the MIT colloquium. There was a Harvard colloquium, which met once a week on Thursdays. Then Nash felt MIT should have one too, so it was set up on Wednesdays. After every colloquium, at Harvard and MIT, there would be a party, and there you could talk mathematics with everybody - and you did! And you picked up a lot of information that way. I didn't have a car, so I usually relied on Nash or Arthur for transport. So it was very different from what things are now. I mean the community was smaller, but there would be, as I say, a mathematics party twice a week! So it was kind of a graduate school for me.
My period knowing Nash was 1954 to 1956. After that I saw him very little. Nash had become sick, back in 1959. I saw him later when I was at the Institute in Princeton in 1965. He was not involved in mathematics in any way, and he was under care at some clinic in Princeton. I remember running into him at the library once, we got into a conversation, and I said, "Well, why don't you come for afternoon coffee one day." So he did, but it was very sad. Somehow one didn't know what to do. He was not happy at that clinic, although he talked about sessions with his doctor being productive in some ways. But I didn't follow that up. He was still quite ill at the time.
I didn't see him again until 1998. Then he was completely different, and he had recovered a lot. He was a little subdued compared to his older days and didn't quite have the same sense of humour as I remembered. But otherwise okay.
One of the people I remember from the first years was Norbert Wiener. I visited his course once in while, out of curiosity. It was interesting. We did not talk mathematics, but he did talk to me about his stay in Denmark. He told me that he learned to speak Danish and said a few sentences to me in Danish, but he said, "Oh, but I never could master Icelandic."
He came to the common room quite often, where all these graduate students were milling around. He wanted to mix with them a little bit. I played chess with him a few times. I was no good as a chess player, but I usually won over him. Chess is usually a rather competitive game. But for Wiener, no! He did it strictly for the fun of it. So we would play a game, and after I won he would say, "Well, you win, let's try another - let's change sides." I found it refreshing: here is this great mathematician, who probably had a considerable competitive streak in him; yet he took chess as strictly a game, nothing more. Philip Franklin however was a clever chess player. I was no match for him. He even looked a bit like Lasker, the chess champion.
After two years at MIT, I got a kind of a lectureship in Princeton, and there I worked on applications of Lie theory to differential equations. For example I generalised Ásgeirsson's mean value theorem to Riemannian homogeneous spaces.
Princeton was not set to be a one-year job, but I assumed it was just that. Then I got this offer from Chicago for two years, so I jumped at that.
The chairman was Saunders Mac Lane, so he essentially hired me. He was wonderful, a great guy. Mathematically we had nothing in common, really: he was in abstract algebra. But he was a very enterprising chairman. He had actually invited me to a summer session in 1955: they had a special summer on functional analysis, of which I learned a lot from there. He was visiting MIT, and I think he was looking for younger people to invite to that summer in Chicago. He had some funding for it, so I went there. George Mackey was there. Also Kaplansky and Halmos. And I met Irving Segal there. At the time he was giving a course on group theory in quantum mechanics. So this was a very productive summer.
I met Grothendieck there. He was becoming a star at the time. He had not yet entered algebraic geometry with full steam, but he just delighted in talking about mathematics. I remember we both stayed at the International House at the University of Chicago, and we had our meals there. Once after dinner he said, "Would you like to take a little walk and talk about mathematics?" Sure! Delighted. And I could pick up whatever topic I wanted to talk about, because he knew everything. We discussed a paper by Godement that I had been reading and he knew very well. I was also writing a paper, and I was telling him what was in it. And he gave some good suggestions. He was not preaching to me at all: he just enjoyed talking about mathematics.
This was before he entered algebraic geometry and changed that subject. He had written his thesis, a very strong one, in topological vector spaces. At the time he worked with Laurent Schwartz and Jean Dieudonné. But then he changed fields completely.
I was one year at Columbia in between my two years at Chicago and returning to MIT. I shared an office with Harish-Chandra. At that time I was teaching a course on Lie groups, and I began my book on the subject. He himself gave a course on Carl Ludwig Siegel's work on quadratic forms, but I followed that too. He was very pleasant and enjoyed discussing mathematics.
At that time he had never taught Lie groups at Columbia; he had been there quite a few years, but somehow never lectured there about his own work or even the background to it. That would have involved quite a bit of effort for him, because this material was available mainly from Weyl's seminar notes from the Institute, but he probably realised that it would be so much work to make this comprehensible to graduate students that he had never done it. In fact, Weyl had been lecturing on material where lots of stuff was taken for granted because Cartan had done it.
I published Differential Geometry and Symmetric Spaces in 1962 - quite an early stage in my academic career to take time off to write a research level textbook. It was a project that I became committed to in Chicago: I already started planning it there in detail. Chicago was on the quarter system, so I gave a quarter course on topics in Lie theory. It was very small, very few people were there, but it was okay. I remember discussing the project with Chern, who was a major mover in the field of differential geometry. He was very encouraging.
We had a seminar in Chicago that went on probably a whole quarter. I was getting interested in not just Lie groups but symmetric spaces, an outgrowth of Lie group theory, which was the title of the seminar. Joe Wolf, who later became a big contributor in the subject, was a graduate student, but he was the one that instigated that seminar. Participants were Shiing-shen Chern, Edwin Spanier, Richard Palais, Bruce Rinehart, Richard Lashof and me, and a couple of graduate students. So that was quite stimulating. But it was unsystematic. People picked out what they could talk about, and some notes materialised. It was not something where one lecture was a continuation of the previous one. You jumped around at will.
I gave a more systematic course in Lie groups for a whole year at Columbia. That was on a lower level, but still at graduate level, and at the same time I was working on this book: I would say one-third of the book, the differential geometry part, was done that year. Then I continued after I came to MIT: the first year, 1960, I gave an undergraduate course on differential geometry. I had the notes already written out, and I simply tested out the first chapter of my book on the students.
The book was in two parts. Differential geometry was the first half of it and some of the second part, but the main topic was symmetric spaces, which was a natural continuation. The first part dealt with things that were classical, but where the proofs were rather hard to dig out. So in some ways the first part was more difficult and a little frustrating, because the main results were known, but the proofs were often either not rigorous or were in Élie Cartan's difficult style.
Now, Warren Ambrose was a serious expert at MIT in differential geometry at the time. He had also studied Cartan's work very much, and a book by Richard Bishop and Richard Crittenden was in part based on his lectures and his collaboration with Isadore Singer. But his taste was a little different. His differential geometry was more in the direction of fibre bundles, which included lots of machinery that I did not want to include, because it wasn't quite needed for what I had in mind. Fibre bundles came about in topology, but they fit into differential geometry in a very natural way. But my main aim was an exposition of these so-called symmetric spaces, and there this machinery of fibre bundles wasn't really necessary.
Writing an advanced text book at that age was a little risky, because that doesn't weigh very highly in the job market. I wrote some short papers in the fifties and a long one in 1959, that is true. The last one was related to the book without overlapping it. Of course, it was an advanced book, containing some new results, and I hoped it would be a useful one. But still I thought, better get it done quickly! So I was working very hard on it. I finished it in 1961 just before my son was born. My wife Artie typed it all on ditto masters, and our hands were always blue from the ink when making corrections. A later book of mine, in 1984, just went to the printer handwritten. That was a useful shortcut.
During that year 1960-1961, I was a kind of a hermit: I didn't go anywhere, in spite of the place being full of old friends! Arthur Mattuck and Frank Peterson and Gian-Carlo Rota, they were all colleagues. It was very nice. I don't think there were parties twice a week at that time - at least, maybe there were, but I didn't go to them twice a week. I was busy. But it was worth it, because the book became reasonably useful. I revised it thoroughly back in 1978, with title Differential Geometry, Lie Groups, and Symmetric Spaces, for example taking into account very important contributions by my colleague Victor Kac. He had not published this material in full before and had not been able to take it out of Russia so he had to reconstruct it from memory.
In August 1988 both Gian-Carlo Rota and I won the Steele Prize. There was a centennial meeting of the American Mathematical Society, so as a result there was a lot of historical talks, and it attracted a lot of people. It's in Providence, only an hour drive from Cambridge, so I just drove down there, and I gave Gian-Carlo a ride back, because he was also at the meeting. So I asked him, when I was about to take him home, "Are you going back there tomorrow?"
"Yah, I am going back there tomorrow, I - I have to go, because I am getting the Steele Prize."
I said, "That's curious, because I am, too!."
They tell you in writing, but they keep it confidential, so nobody knows until the time. Neither of us knew that the other had received it. So I said, "Well, look, we are near to a very nice restaurant, Le Bocage on Huron Avenue in Cambridge, why don't we celebrate?" So we had a fancy dinner together, right there.
I did not do many administrative things in the department. I have just served on various Ad Hoc and standing committees and was chairman of the Graduate Committee for seven years, sometime in the eighties. Phyllis Ruby was a great help to me at that time. She was the first graduate administrator and with George Thomas set up the Graduate Office in the form it is today. She served in the department for forty-one years, and was a helpful friend of the graduate students.
Arthur Mattuck has truly done remarkable service for the department, both as department head but also with his effective organisation of the undergraduate courses and setting up the Undergraduate Office.
I retain contact with people in Iceland. I go back there most summers, I just was there in June. It's not very far - same distance as to California, because you fly directly from Boston now. I have, of course, several relatives that I keep in touch with. There are reunions where we go and see old gymnasium classmates, every ten years certainly, even now. What is left of the class is actually only about fifty percent, but they get together quite regularly. But then I have good contact with the mathematicians, too, although there is nobody there that works directly in my field.
They had the 'International Conference on Integral Geometry, Harmonic Analysis and Representation Theory' there a year ago in my honour, which attracted people in my area. David Vogan and a couple of Icelandic mathematicians did a great job in organising it. It dealt with three topics: harmonic analysis, Radon transforms, and representation theory. So it was all Lie group-oriented. And it was not too large for Iceland to organise, maybe at most ninety participants, so very easy to manage. The unpredictable Icelandic weather cooperated nicely, so country sightseeing during half a day was quite successful. The Icelandic landscape is unusual in its absence of trees, providing a view towards infinity in all directions.
2. Introduction to Helgason's Selected Works
I am deeply grateful to the American Mathematical Society and to Gestur Ólafsson and Henrik Schlichtkrull for their interest and their proposal to produce this volume with selections of my mathematical papers. The editors and publisher kindly asked me to write some explanatory comments on these papers.
My interest in mathematics I trace to my mathematics teacher in the Gymnasium in Akureyri in North Iceland, Trausti Einarsson. Although an astronomer, his deep respect for mathematics was highly infectious. The geometry textbooks by the remarkable mathematician Ólafur Danielsson, the pioneering founder of mathematics education in Iceland, were written by a man with a real mission. The program was on a respectable level - for example the final exam, for students at age 17, included finding the radius of a sphere inscribed in a regular dodecahedron of edge length 1. Danielsson, who wrote several beautiful papers in geometry along with his demanding Gymnasium teaching, was profoundly influenced by the rich geometry tradition in Denmark, which started around 1870 with J Petersen and H Zeuthen.
When I entered the Gymnasium in Akureyri around 1940, the Icelandic population totalled about 125,000. Thus the Gymnasium program put considerable emphasis on the teaching of several foreign languages. Our French teacher, Thórarinn Björnsson, was particularly memorable because of his outspoken admiration of French authors like Maupassant, Daudet and Rolland, in addition to his fondness for the French language in which he left us with a decent reading ability. Thus my later difficulties in understanding Élie Cartan's papers were not of linguistic nature.
Later, at the University of Copenhagen, I benefited from splendid lectures by Harald Bohr, Børge Jessen and Werner Fenchel. The first two gave substantial courses in complex function theory and real analysis, respectively. With Fenchel I had also a kind of reading course in projective geometry. At that time I obtained a copy of Julius Petersen's remarkable book, "Methods and Theories for Solutions of Geometric Construction Problems" (1866). I worked on many of the 300 problems there. Example of such problems: Construct a triangle from and . This was consistent with my expectation later to teach at a Gymnasium in Iceland.
One day I happened to notice a posting of the collection of prize questions which were posed by the university each year. The mathematics problem on the list called for a generalisation to analytic almost periodic functions of the value-distribution theory of Ahlfors and Nevanlinna. Everyone under 30 was free to try for a year. A solution was to be submitted and judged anonymously so I could not consult anybody. Thus four months went by before I understood what the problem was about. My paper solving the problem became my masters thesis. A summary is given in a paper in this volume.
After completing my thesis in Princeton enjoying supportive and stimulating supervision by Bochner, my interest shifted to Lie group theory with the intention of studying Harish-Chandra's work. A related interest was geometric analysis, having received from my friend Leifur Ásgeirsson the page proofs of Fritz John's famous book Plane Waves and Spherical Means. Although the term 'group' does not appear in this book I sensed quickly that some of its themes were ripe for group-theoretic variations.
At an early stage it was a source of inspiration to me how at the hands of Sophus Lie, the study of differential equations led to the concept of Lie groups which in turn went on to penetrate numerous other fields of mathematics. Since Lie group theory had in the mid-fifties become so well developed I became interested in a kind of converse to Lie's program, namely to investigate differential operators and differential equations invariant under known groups. Examples were readily at hand: the Laplace-Beltrami operator on Riemannian or pseudo-Riemannian manifolds, the Dirac operator, the wave equation and its variations.
In the account that follows I describe some of my own work connected to analysis on homogeneous spaces. This is accompanied by brief mention of some prior and subsequent work which is closely related. These personal viewpoints do not represent any attempt at a historical account, the feature of relevance and of value being highly subjective. As Peter Lax aptly observed: "For a mathematician the central problem is the one he happens to be working on".
Inspired by Harish-Chandra's brilliant work on the representation theory of semisimple Lie groups G, my taste for geometry led me to a related topic, namely a study of the symmetric spaces of É Cartan. Here enter several differential geometric features: geodesics, curvature, K-orbits, totally geodesic submanifolds and horocycles; for the compact dual spaces we have also closed geodesics, especially those of minimal length, equators, antipodal manifolds, and totally geodesic spheres. Geometric analysis means, in this context, analysis formulated in terms of these geometric objects.
I have had the fortune of teaching for 50 years at M.I.T. but have also enjoyed the privilege of extended, stimulating visits at the Institute for Advanced Study in Princeton and at the Mittag-Leffler Institute in Djursholm, Sweden.
In conclusion I want to emphasise my wife's role in my life and work. Her support during these 50 years has been precious, and her tolerance, at times when nothing but mathematics mattered, has been admirable. She even typed my first two books in pre-TEX times, giving stylistic suggestions along the way.
Last Updated June 2024