Remembering Sigurður Helgason (1927–2023)

Sigurður Helgason was born September 30, 1927, in Akureyri, Iceland. His parents were Helgi Skúlason (1892–1983) and Kara Sigurðardóttir Briem (1900–1982). He studied at the Gymnasium in Akureyri from 1939 to 1945 where he became interested in mathematics. At that time, the University of Iceland had no mathematics department, and to study mathematics one had to go abroad, mainly to Copenhagen or Göttingen. In 1945 the University of Copenhagen was still recovering from the war, so Sigurður spent one year in the engineering department at the University of Iceland. In 1946 he left Iceland for Copenhagen to study mathematics. His main topic was almost periodic functions, which was very fashionable there at that time. He won the Gold Medal in Mathematics at the University of Copenhagen in 1951 for his work on Nevanlinna-type value distribution theory for analytic almost periodic functions. Afterward he studied at Princeton from 1952 to 1954, writing a PhD thesis with S. Bochner on Banach algebras and almost periodic functions. From there, he took temporary positions at the Massachusetts Institute of Technology (MIT), Princeton, and Columbia until he became a professor at MIT in 1960 where he stayed until retirement, spending several periods at the Institute for Advanced Study in Princeton and other travel destinations. He was an invited lecturer at the International Congress of Mathematicians (ICM) in 1970. Another honor was the Leroy P. Steele Prize for Mathematical Exposition in 1988 for his books Hel78 and Hel84. He was elected a member of the Icelandic Academy of Sciences in 1960, of the American Academy of Arts and Sciences in 1970, and of the Danish Royal Academy of Sciences and Letters in 1972. He was awarded honorary doctorates by the University of Iceland in 1986, the University of Copenhagen in 1988, and Uppsala University in 1996. In 1997 he was made an honorary member of the Icelandic Mathematical Society, and in 2012 he became a fellow of the American Mathematical Society.

During his first time in Princeton after graduate school his focus started to shift towards harmonic analysis, in particular Radon and Fourier transforms on symmetric spaces resulting in his article Hel58.

Sigurður is best known for his series of books dealing with geometry and analysis on Riemannian symmetric spaces, starting with Hel62. At that time there was very limited literature on those subjects and his books became instant classics. Eight years later his opus Hel70 was published in the Advances. This long article included several of his results on Radon and Fourier transforms as well as new work and open problems. It was a motivational source for several younger mathematicians, a paper where one could learn basic ideas concerning harmonic analysis on homogeneous spaces. His first article was published in 1954 and his last article appeared in 2018, a span of 64 years.

His interest in harmonic analysis on Riemannian symmetric spaces was motivated by classical harmonic analysis as well as the work of Harish-Chandra and others on Lie groups. His main topics were (a) invariant differential operators, (b) the Radon transform, (c) Fourier analysis, and (d) joint eigenfunctions. He was among the first to systematically investigate the analysis of differential operators on reductive homogeneous spaces. His research on Radon transforms on homogeneous spaces presaged the resurgence of activity on this topic which continues to this day. This work also led to his idea of the double fibration framework for integral geometry on such spaces. Likewise his work offered a geometrically motivated approach to harmonic analysis on symmetric spaces. Of course there is much more but due to lack of space we refer the reader to the introduction to his selected works Hel09 and the synopsis of his work ÓS13.

Sigurður has had a large impact on both of our lives and mathematics, so we finish this short introduction with a few personal comments:

Fulton Gonzalez: I got to know Sigurður the canonical way. I took his class on analysis on Lie groups in 1979, became enthralled with the subject (and even more by the instructor), then asked him to be my adviser.

Siggi’s students appreciated the precision and clarity of his lectures, which can likewise be said of his books and papers. Interest in integral geometry was picking up at that time, no doubt popularized by his first book on Radon transforms Hel80. This book elicited numerous research problems, one of which became the subject of my thesis.

Siggi exuded an old world charm that was complemented by an expansive personality. He often invited students to the frequent get-togethers in his house, and this is how I got to know some prominent mathematicians I would otherwise not have had the courage to approach.

Siggi’s work was voluminous, and until the mid-aughts produced at a pace such that keeping abreast of it became almost a full-time job. It was my good fortune to have landed a position in Boston allowing me to hear about his mathematics firsthand. I was also lucky that he took a personal interest in my career, for which I am profoundly grateful.

Gestur Ólafsson: I knew about Iceland’s most famous mathematician Sigurður Helgason before I first met him during a summer school at Laugarvatn, Iceland, in 1977. Among others, Mogens-Flensted Jensen gave a series of lectures on harmonic analysis on Riemannian symmetric spaces. At that time my interest was in $C^*$-algebras and abstract topological groups. But long evening discussions with Sigurður changed that: I began by studying his 1970 paper and then wrote a master’s thesis on $\mathrm{SL} (2,\mathbb{R})$ and the upper half plane. A few years later when I was looking for topics for my PhD, I met Sigurður again. This time he suggested I look at the work of Flensted-Jensen and G. van Dijk. Since then analysis and representation theory related to symmetric spaces has been my subject and joy.

In addition to serving as my mentor and mathematical inspiration, Sigurður also extended to me his great warmth and kindness. We kept in regular contact by phone, email, or meeting in person. Often on my way to Europe or back, or to conferences close to Boston, I stayed with Sigurður and his wife Artie for a day or two, enjoying their company and hospitality.

By Jean-Philippe Anker

I grew up mathematically with the books of Sigurður Helgason on my desk. Actually, when I was a student only his first book Hel62 was available. I learned a large part of the content of his second book Hel84 from his lecture notes Hel72 and from the pioneering works of Harish-Chandra, complemented by his own contributions and those of Gindikin-Karpelevich, Gangolli, Kostant, Flensted-Jensen, and Koornwinder. Thus, when it was published, I welcomed Hel84, which was for me a relatively easy and quite clarifying read. I had a similar experience with the third book Hel94, which was published after I was already acquainted with its content. Thus, fortunately I didn’t have to face all three books at once, whose encyclopedic approach may nowadays impress newcomers and possibly prevent them from getting immersed in the subject. Obviously, Helgason’s trilogy had quite an important impact by providing access to geometric analysis on symmetric spaces in book form, and by presenting it in a rigorous style. By comparison, analysis on $p$-adic symmetric spaces, i.e., affine buildings, developed much more slowly. In my opinion, this is largely due to the fact that, for a long time, Macdonald’s lecture notes (at the Ramanujan Institute in Madras) was the only reference on the subject in book form (and moreover was hardly available).

I first met Sigurður Helgason in person during a workshop in Switzerland, and next during the Elie Cartan conference in 1984 in Lyon; “met” is actually a big word as I was too shy to approach him for a true discussion. I started interacting mathematically with him during my postdoc stay in the US. While working on Fourier multipliers on symmetric spaces, I realized that the argument I was using could simplify the study of the spherical Fourier transform on the $L^p$ Schwartz space ($0<p\le 2$) and I sent him the resulting preprint. He reacted quickly and enthusiastically, managing to invite me for a talk at a summer conference in 1989 at Bowdoin College; this was a great opportunity for me to meet several experts in harmonic analysis and representation theory on reductive groups. Later on, I took part in the conferences organized for his 65th birthday in Roskilde, his 70th birthday in Copenhagen, and his 80th birthday in Reykjavik. There I remember that he was quite talkative about his youth in Iceland and his ties with his native country; it was the last time I met him in person. Sigurður kept following the literature closely until very recently. For instance, a couple of years ago my last graduate student Hong-Wei Zhang was impressed to receive a request from him for preprints about the wave equation on (locally) symmetric spaces. Now that I have retired, I hope to stay as fit and accurate as he was during the latter years of my life.

Jean-Philippe Anker is an emeritus professor at the Université d’Orléans. His email address is anker@univ-orleans.fr.

By Erik van den Ban

As a PhD student, I first encountered Helgason’s work through his expository paper Hel73, which offered quick access to the beautiful Fourier theory of Riemannian symmetric spaces $G/K$. In particular it covered the Paley-Wiener theorem due to him. The very clear writing was of great help to a newcomer in the field.

A bit later I learned from Hel70Hel76 about the Poisson transform. In Hel76, Helgason suggested that possibly every joint eigenfunction of the invariant differential operators on $G/K$ could be represented as the Poisson transform of an analytic functional on the boundary component $K/M$. This became known as Helgason’s conjecture and was established in 1978 (see KKM$^{+}$78).

I met Siggi personally in the fall of 1982, when I spoke in the Lie group seminar at MIT. Siggi’s genuine interest and encouraging attitude meant a lot to me. As a member of the Institute for Advanced Study (IAS) in Princeton from 1982 to 1983, I learned about exciting developments in the theory of semisimple symmetric spaces $G/H$; those with $H$ compact are the Riemannian ones. Although Siggi did not contribute directly to this field, his work on Riemannian symmetric spaces would play a prominent role in the development of the extended theory.

A number of years before, partly while he was a guest of Siggi’s at MIT, Mogens Flensted-Jensen had constructed part of the series of unitary representations appearing discretely in $L^2(G/H)$. Surprisingly, his method exploited the Poisson transform of a dual Riemannian symmetric space $G^d/K^d$.

In the spring of 1983, Harish-Chandra showed me the announcement made by Toshio Oshima and Toshihiko Matsuki that they could use Mogens’s idea in combination with the proven validity of Helgason’s conjecture for $G^d/K^d$ to describe the discrete series for $G/H$.

These developments made me focus all my attention on harmonic analysis on $G/H$. A year later Henrik Schlichtkrull and I started a long-term collaboration, first on questions related to Helgason’s conjecture and then on Fourier theory for $G/H$.

Siggi’s key idea on the (spherical) Paley-Wiener theorem for $G/K$ exploited a certain Weyl group symmetry of the inverse transform combined with a contour shift. Henrik and I suspected that we could use a similar idea to handle the Fourier theory of $G/H$. This turned out to be the case after many years of research culminating at the Mittag-Leffler Institute (MLI), in the special year on “Analysis on Lie groups” in 1995–96. Mogens Flensted-Jensen was the main organizer. Siggi was a visitor for the entire fall; so was I, accompanied by my wife Els and son Mark. In that period I got to know him better.

At the end of the fall, darkness came quite early, so we had afternoon tea with candlelight. This created a special atmosphere, with much talking about mathematics and other aspects of our lives. Siggi participated cheerfully, and with his special sense of humor.

Close to Christmas, the Flensted-Jensens invited Siggi and Artie, and me and my family, for dinner in their neighboring apartment on the premises of the MLI. Mark, who had just turned 5, was a bit shy. Siggi immediately put him at ease by giving him a small harmonica as a present. That dinner became unforgettable.

More than once, Siggi told me I should visit his native country, of which he was very proud. His 80th birthday conference in Reykjavik, in the summer of 2007, offered the opportunity. I have a dear memory of Siggi explaining Icelandic history on a trip to the impressive Þingvellir national park.

Siggi’s mathematical work has been a major source of inspiration for me. I am very grateful for this and for the memory of his cheerful, supportive and kind personality.

Erik van den Ban is an emeritus professor at Utrecht University. His email address is e.p.vandenban@uu.nl.

By Jiri Dadok

Sigurður Helgason was a harmonic analyst at heart. In the mid-1950s, he quickly moved from questions of commutative harmonic analysis to the noncommutative side, influenced foremost by Harish-Chandra who was a personal friend of his for almost three decades. The place for his analysis was Lie groups and some of their homogenous spaces (most importantly symmetric spaces). Siggi absorbed their structure by studying the works of E. Cartan, C. Chevalley, and many others. In 1962, he put a portion of his encyclopedic knowledge of these geometric objects into the foundational book Differential Geometry and Symmetric Spaces Hel62. For many years, this advanced book was the only reference book on the subject. “Writing that book was the best thing I did,” he once told me and suggested that I consider writing a book of my own. Since then he issued an updated version of the book Hel78 and proceeded to synthesize his research papers into several books on the analysis on these spaces. To appreciate the wide breadth and depth of his work see his book Groups and Geometric Analysis Hel84.

I chose him as an advisor because I wanted to work in harmonic analysis, a subject that is very useful outside of academic mathematics (hedging my bets, I suppose). I found him slightly formal, mild mannered, polite, but most importantly a fair, honest, and modest man. His office was a study in minimalism, quite neat and decorated with posters showing the natural beauty of his native Iceland. Clearly esthetic aspects were important to him—this certainly shows in his writing which is clear, precise, and well organized. When I was writing my thesis he gave me a copy of Bruhat’s thesis and said that was the way to do it. Later, as an editor of the Indiana Mathematics Journal I wondered how many badly written papers I let slip through that he would have sent back with a copy of Bruhat’s thesis. Now with plagiarism in the news I was reminded of him telling me upon reading the first draft of my thesis: “See this Lemma 3 here, you copied it from my paper which is perfectly fine but you need to acknowledge such things no matter how trivial they seem to you. Others may see it quite differently.” A few years later, as a postdoc I got a letter from him: “Remember that lemma I was lecturing you about? Well, my proof of it is wrong. Do you see how to fix it?” This is how I was gently led to write a paper by neglecting good manners. Mathematicians, like ordinary people, come with different personalities; some are modest, others think very highly of themselves and their work. Siggi was definitely the modest type; he never claimed more than what the statements of his theorems said and these he stated so as to answer the important questions as he saw them. He disliked idle speculations and unnecessary over generalizing. Once as I was at the blackboard of his office expounding on my thoughts he remarked: “You have plenty of ideas that seem reasonable, why don’t you prove some of them!” At another time walking out of a seminar he quipped “Some people don’t know how to differentiate without writing down a long exact sequence.” He was also modest regarding the price of his books. He told me that in his view his publisher was going to charge an excessive amount for the books and that he was able to talk them down a significant amount.

During my third year at MIT, Siggi was visiting Harish Chandra at the Institute for Advanced Study and therefore I didn’t get to know him as well as otherwise would have been possible. So this note of appreciation could have been longer…but I don’t think my regard for him could be any higher.

Jiri Dadok is an emeritus professor at Indiana University. His email address is dadok@indiana.edu.

By Eric Grinberg

In my early graduate school days everyone around me was doing algebraic geometry, a deeply rooted subject with historical foundations and burgeoning contemporary activity. One student was writing a thesis combining algebraic geometry with integral geometry, a newer and less well-known subject. I had heard that, just a few steps down a nearby river, there was a course on integral geometry and the Radon transform, covering the core subject and also emerging applications such as computed tomography (CT). It was said that the professor was a major contributor to and expert in the subject. I joined the class, and thus met Professor Sigurður Helgason. In situ I also met members of the mathematical community which Siggi created; he and they became lifelong mathematical friends. As I followed Siggi’s presentation of the theory of the Radon transform, I was also curious to survey his research explorations. I found particularly fascinating his inversion formulas for Radon transforms on symmetric spaces of rank one and on Grassmannians. While these spaces share some similarities, being constructed from linear spaces and their projectivizations, they are quite different as Riemannian manifolds. Yet their Radon transforms, naturally defined, are invertible by remarkably similar formulas involving the Laplacian. The subject afforded a multitude of interesting questions.

After graduate school, my career path had twenty seven years in the Boston diaspora in store for me. During those years I visited Cambridge regularly and invariably knocked on Siggi’s office door; he was always welcoming, and inspiring mathematical discussions ensued. During many of these office visits, colleagues and students dropped by and joined the discussions. When I resettled in the Boston area I was able once again to attend mathematical events that Siggi organized, and regular invitations to his home began to appear. Artie and Siggi are famous for their generous and gracious annual open house events. (I kept accompanying cards and displayed some in my office.) I also got invites to smaller, more-specialized, less-publicized events, for those in the “Radon transform loop,” as it might be called. One message from Siggi read “I just got some Greenlandic shrimp; come on over.” Though my dietary boundaries exclude Greenlandic shrimp, I went along for the company. I have to disclose that I was also hoping for Icelandic food to be present; I wasn’t disappointed.

Siggi’s writings are vast. Since my graduate school days I don’t think there has been a single month during which I didn’t consult them. He has created a world more vast than J. R. R. Tolkien’s or J. K. Rowlings’s, an inexhaustible supply of structures, of insights to absorb and questions to explore. As we cherish his memory and his writings, we will miss him.

Eric Grinberg is a professor at the University of Massachusetts, Boston. His email address is eric.grinberg@umb.edu.

By Adam Koranyi

I called him Siggi, as did everybody who was on a more or less equal footing with him. But for me he was also a close friend of the kind one finds very few in a lifetime. We overlapped in the academic years 1957 to 1959 at the University of Chicago. He was an Instructor there lecturing on homogeneous spaces, invariant differential operators, and the like. I arrived as a graduate student when I left Hungary as a result of a sudden decision in early 1957, profiting from the chaos following the uprising there.

He was extremely kind to me from the moment we met, and so was his wife Artie. We had a lot of common interests. Siggi and I played compositions of Mozart for four hands on the piano. We had discussions about history, and many other subjects. We also had a common European background, which was sort of a surprise to me, given that my ideas of Iceland came from romantic literature (and were quite wrong).

Still, there was a good bit of local color in the stories Siggi told me later. For example, his father, an eye doctor in Akureyri, periodically went on horseback to the surrounding villages to take care of patients. Or another: Siggi himself, on a high school summer job, was rowing around inside a huge tank of fish oil to let someone else stand in the boat and paint the tank’s inner wall.

In Chicago, I went to his lectures and learned a lot from him. He was an extremely strong and disciplined mathematician always finishing what he started and very rarely making a mistake. His 1962 book was for a long time the only one of its kind. It was the first comprehensive treatment of semisimple Lie groups and symmetric spaces preceded by a detailed treatment of differential geometry. While writing his later books which contained much original material along with general developments, he could produce such things as his very long 1970 paper on conical functions and the Radon transform.

I was in New York once in 1959/60 and saw him at Columbia in the office he was sharing with Harish-Chandra. If I remember well, in the following year he was already settled at MIT and had a house in Belmont in which he spent the rest of his life. I have lots of personal memories about visiting him and his family. Once, in the early seventies he made a great impression on my (then very small) sons by standing on his head.

In the last twelve or fifteen years we kept phoning and emailing each other, but met much less often. For one thing, I stopped going to conferences trying to make a statement that too much travel was bad for the environment (the only result being that I excluded myself from the mathematical community). As time progressed both of us suffered more and more from various consequences of old age and stopped moving around in the world. It is a source of sadness to me that I did not see him even during his final illness.

Adam Koranyi is an emeritus professor at the CUNY Graduate Center. His email address is adam.koranyi@lehman.cuny.edu.

By Peter Kuchment

A great mathematician and human being, Professor Sigurður Helgason has passed away. This is a major loss for the mathematics community, and for me personally.

I first met him “long distance,” by exchanging letters and papers, about 40 years ago, when I was still in the Soviet Union. Our friendship has lasted since. The topics of our exchanges were, among other things, analysis on symmetric spaces, eigenfunctions of invariant differential operators, Huygens’s principle, and such. The articles and books by Helgason were constantly on my desk. Sigurður graciously supplied preprints and reprints of his papers, which were inaccessible in the former Soviet Union (fSU). My attempts at that time to obtain Floquet-theory type results for differential equations invariant with respect to some non-abelian discrete groups (e.g., lattices in symmetric spaces) had failed. It is clear now that I used a wrong approach, which had no chance to work. However, using Helgason’s techniques, I could obtain some analogs for symmetric spaces of the results by L. Ehrenpreis, B. Malgrange, and V. Palamodov on exponential representations of solutions of linear constant coefficient PDEs (what Leon Ehrenpreis called “the fundamental principle”). At some moment, the idea arose of translating one of Helgason’s major books Groups and Geometric Analysis Hel84, which had just appeared, into Russian. My translation appeared in 1986 and became extremely popular in the fSU. It took me a year of heavy work, but I learned a lot from it.

Right before my emigration to the USA in 1989, I accidentally stumbled upon the then rather young and fast expanding applied area of computed tomography. And lo and behold, it turned out that the main integral geometry techniques were related to Helgason’s research (e.g., through his well-known works on the Radon transform and on geometric analysis in general). The same year I had the immense pleasure of meeting him personally for the first time at a conference in California. Since then, we have communicated frequently, also meeting at conferences and taking trips to each other’s institutions. In particular, we gave lectures at each other’s anniversary conferences.

Sigurður Helgason was not only a major figure in mathematics, but also a friendly and highly cultured person (he was an excellent photographer and piano player), always willing to help others. In my case, it was not only our communications by mail that played a huge role for me, but also him writing letters of support when I, practically unknown in the USA as a mathematician, was looking for jobs in the extremely tough job market of the early 1990s. He was also incredibly supportive of my students. For instance, when my PhD student L. Solomatina obtained an important result in geometric analysis, she could only announce it, as was common in the fSU, in a brief communication. The lengthy proofs were extremely hard to publish and thus practically inaccessible. In some cases, the results of such publications were (with good reason) not considered in the West as established facts. Helgason, however, while publishing his own proof in a long article in the Journal of Functional Analysis, attributed the result to Dr. Solomatina. He was also supportive of other graduate students of mine working on integral geometry and tomography.

The memory of Professor Sigurður Helgason, a great person, friend, and mathematician will stay with me till the end of my days.

Figure 2.

AMS Special Session for Helgason’s 85th birthday, with some of the authors, Boston 2012.

Peter Kuchment is a University Distinguished Professor at Texas A&M University. His email address is kuchment@tamu.edu.

By Frank Natterer

I am deeply honored to be invited to share my memories of Sigurður Helgason.

After Hounsfield constructed the first working clinical computerized tomograph in 1972, engineers quickly realized that in order to enhance their results, they had to develop improved algorithms for image reconstruction. Since the principle of computerized tomography (CT) is based on the mathematical tool of the Radon transform, already introduced and analysed by the mathematician Johann Radon already in 1917, they turned to mathematics for help.

When G.T. Herman and I organized the first Oberwolfach Conference on Mathematical Aspects of Computerized Tomography in 1980, the attendees were predominantly applied mathematicians, computer scientists, and engineers.

Despite having been a resounding success, we felt that the conference had neglected the theoretical aspects in image reconstruction. For the next conference in 1986, we therefore decided to extend the focus and set the conference under the header “Theory and Application of the Radon Transform.” By this, we wanted to offer a bridge between pure and applied mathematics. While this may seem an obvious concept these days, it was very unusual then. The workshop notes from that conference proudly mention that exactly 15 talks were held in each category, pure and applied mathematics (numerics/algorithms/applications).

Without too much hope of his attendance, we had invited Helgason, by then a star in pure mathematics for his work on symmetric spaces. To us, he was most well-known for his book The Radon Transform Hel80, a comprehensive, compact and yet rigorous treatment of the Radon transform. In particular, the book summarizes and develops inversion algorithms, and gives a complete proof of the Helgason-Ludwig consistency conditions. To me, and this is how I cite it in my book The Mathematics of Computerized Tomography, his book is the definitive treatise on all analytical problems related to the Radon transform.

Much to our surprise, Helgason accepted our invitation and gave an inspiring talk about applications of the Radon transform in differential equations. It was beautiful to have him with us that time and I remember sitting down with him for long discussions, with very different approaches and goals, and his excellent piano performances.

Quickly, it became clear to us that he had a grasp and overview of the theoretical frameworks that had escaped us before. The conference inspired us to work on the concepts that he had introduced, allowed us to advance our methods and solve problems previously deemed unsolvable.

I am devastated that Sigurður Helgason has passed. The world of tomography and imaging as we know it today is unthinkable without his contributions. He will always be remembered as an outstanding expert and as a traveller between worlds in our community.

Frank Natterer is an emeritus professor at Universität Münster. His email address is nattere@uni-muenster.de.

By Bent Ørsted

Sigurður Helgason had a very special connection to Scandinavian mathematics and in particular to Danish mathematics. After high school in Iceland he went to Copenhagen University to study mathematics from 1946 to 1952; at the Department of Mathematics he did very well and became friends with among others Bent Fuglede (two years his senior) and Svend Bundgaard (fifteen years his senior and a teacher there; in 1954 he became the first professor of mathematics at the new University of Aarhus).

With his books and papers, Helgason had a well-known and enormous influence on the development of analysis on symmetric spaces, not the least for later Danish mathematicians. He has been an Icelandic rock that one could build on, and learn from. He was always supportive, both in inspiring new work, and in participating in conferences—a particularly nice one was in 1992 at Roskilde University in Denmark (Figure 3), organized by Gestur Ólafsson (at the time a professor there). The participants were a veritable array of who’s who in analysis on Lie groups. One of the (many) well-known mathematicians in this picture is Mogens Flensted-Jensen (sitting next to Helgason), famous for his remarkable construction of the discrete series of an affine symmetric space; in 1980 he defended his Danish doctoral degree (a somewhat more advanced degree than the PhD) with Helgason as one of the official opponents. No doubt the work by Helgason on spherical functions had been very influential for Flensted-Jensen, and Helgason was happy about this, as he said in a letter he wrote many years later to Flensted-Jensen.

I am reminded of the dictum attributed to Joseph Bernstein, that in mathematics we have the three key areas of geometry, algebra, and analysis, perhaps even arising historically side-by side, and not linearly in the time-like sense; but in addition there is Lie theory, connecting to all of them and forming in itself a fourth area of mathematics. Certainly root systems are part of such connections, and Helgason was well aware of the role of Lie theory. He wrote a thoughtful historical essay on the work of Sophus Lie, stressing his wish to understand and solve differential equations using symmetries and connections to differential geometry. Differential equations became a central part of Helgason’s work, such as for invariant differential operators $D$ on a homogeneous space $G/K$ for $K$ a closed subgroup of a Lie group $G$ and the study of eigenspaces of functions $\phi$ with $D\phi = \lambda \phi$. Henrik Stetkær and his student Jacob Jacobsen took up Helgason’s question about the irreducibility of such eigenspaces in general as representations of $G$. Stetkær was one of the first of several Danish students to study for the PhD at MIT, entering that supertanker of mathematics.

The famous Helgason conjecture about eigenfunctions on a Riemannian symmetric space being Poisson transforms of hyperfunctions, and the proof by six Japanese researchers KKM$^{+}$78 was the topic of a book by Henrik Schlichtkrull; in this book he also explains the work by Flensted-Jensen on discrete series. Schlichtkrull was a student of Flensted-Jensen in Copenhagen and visited MIT to work there, in particular with Helgason and David Vogan. He also took up another of Helgason’s questions, namely that of Huygens principle for hyperbolic equations; this is the question of the singular nature of the fundamental solution, namely having support in a hypersurface.

Probably one of Helgason’s dearest topics was that of integral transforms, such as the Radon transform, and more generally double fibrations. Furthermore, this is an area of great value for applications such as X-ray analysis and similar methods of imaging.

In conclusion, I think I speak for many Danish mathematicians when I express gratitude for what Helgason did for our area, including supporting special years at the Mittag-Leffler Institute by his presence and being an inspiration through his scholarship and youthful energy.

Figure 3.

Helgason’s 65th Birthday Conference in Roskilde, Denmark, 1992, with Danish friends and some of the authors.

Bent Ørsted is an emeritus professor at Aarhus University. His email address is orsted@math.au.dk.

By Jeremy Orloff

For me, Siggi was a mentor and a role model of how to be a mathematician and an intellectual. What stands out in particular is the great joy he got from mathematics. For decades, he was also a friend and a kind presence in my life. In 1980, I asked Siggi to be my PhD advisor. I certainly found harmonic analysis compelling, but I was really drawn to his style of doing and writing mathematics. It is reassuringly concrete, with all the details firmly in place and efficiently organized, while at the same time maintaining an arc that gives the bigger picture. Finally, there are always the scholarly endnotes that enliven the narrative with historical details and references. Some of my fondest memories from graduate school and the following few years are of proofreading the books he was writing at that time. Looking through his books over the last month, I’m more impressed than ever by his output.

I was not always the most energetic student, yet Siggi was always patient with me. I can remember only a couple of times when he expressed frustration with my behavior. The first was early in my career when I was nursing a broken heart and not getting any work done. I can’t remember how long this went on—I hope it was not more than a month—finally he had had enough and told me rather directly that maybe doing some mathematics would help me. As was often the case, there was much wisdom in his words.

Siggi was supportive of my decision to leave academia and join a software startup. A side benefit was that my wife Kathi and I stayed in the Boston area and got to enjoy the many parties that Siggi and Artie hosted over the years. We both have many wonderful memories of these gatherings. On his passing, Kathi and I find ourselves celebrating Siggi’s friendship and his long life and many accomplishments, while simultaneously feeling a deep sadness for ourselves and especially for Artie, Annie, and Thor.

Jeremy Orloff is a lecturer at MIT. His email address is jorloff@mit.edu.

By Toshio Oshima

Helgason’s conjecture: Any joint eigenfunction of all invariant differential operators on a Riemannian symmetric space is given by the Poisson integral of a hyperfunction.

At the 1970 International Congress of Mathematicians in Nice, Helgason presented his conjecture in the case of the unit disc and suggested it in other cases. At a symposium on hyperfunctions and differential equations held at the Research Institute for Mathematical Sciences (RIMS) at Kyoto University, K. Okamoto introduced this as Helgason’s conjecture. At that time my area of interest was the analysis of differential equations and I did not have any knowledge of Lie groups or homogeneous spaces. I started reading his paper Hel70 and learned many important concepts related to analysis on Riemannian symmetric spaces. Most required results were clearly and concretely written and thus not so hard for me to understand. There appeared several related works on the conjecture but eventually in 1974 it was solved in general by six Japanese mathematicians (Okamoto, Kashiwara, etc., including myself), using a concept of boundary value problems with regular singularities. The full paper KKM$^{+}$78 was published in 1978.

I became a member of the Institute for Advanced Study for two years starting in the fall of 1978, which was enabled by letters of recommendation kindly written by Helgason and M. Sato. I was invited to MIT in May 1979 for a month and stayed at Ashdown House with my wife. I was so happy to have many mathematical discussions with Helgason.

Tambara seminar house: Later, I invited Helgason to Japan several times, and he came to Japan in August 2006 and 2009. I organized workshops on integral geometry and harmonic analysis in the Tambara seminar house belonging to Tokyo University. It is located on a highland surrounded by woods and mountains. He liked walking around this area and said “Let’s go exploring in the jungle.” At night we and young mathematicians talked together drinking wine. Helgason played the piano. He always brought a camera and thanked Japan for the invention of a digital camera. He was a multi-talented mathematician.

Helgason and his papers guided me to a new mathematical field and expanded my perspective at my starting point as a mathematician. I miss him as my great teacher.

Toshio Oshima is an emeritus professor at the University of Tokyo. His email address is oshima@ms.u-tokyo.ac.jp.

By Todd Quinto

Sigurður Helgason was a great mathematician and a kind, generous, welcoming person. He cared about people and creating community.

I came to MIT for my PhD in 1973 excited but nervous about being in one of the best programs in the country. I walked into my first class, Graduate Real Analysis (18.125), with some trepidation. However, I didn’t need to worry because I was learning from Siggi Helgason. He lectured with perfect organization, incredible clarity, and deep insight. He was warm and friendly in class, and he’s the reason I still love the Riesz representation theorems and so much analysis. Later, I was fortunate to learn from him as a coadvisor.

Sigurður Helgason’s many beautiful and elegant theorems for group invariant Radon and X-ray transforms, along with his broad range of fundamental results in harmonic analysis and representation theory, have inspired me. I would like to mention one early result that was fundamental to the field and formative for me. After reading Hel65 as a graduate student in the 1970s, I became fascinated with support theorems. In that article, Helgason proved that if the smooth function $f$ is rapidly decaying at infinity and its Radon hyperplane transform, $Rf$, is zero for all hyperplanes of distance greater than $A$ from the origin, then $f (\mathbf{x}) = 0$ for all $\mathbf{x}$ of distance greater than $A$ from the origin. He first proved this theorem for radial functions by solving an Abel integral equation, then reduced the general case to this case by proving a support theorem for the sphere transform. He produced two good support theorems for the price of one. This was one of the inspirations for my early work using integral equations methods to prove support theorems. His beautiful support theorems motivated my friends and me to continue proving support theorems for many years. The thoroughness, clarity, and depth of all his papers have been an inspiration to me throughout my career.

Siggi Helgason was friendly, open, and personable. During my first term in graduate school, he invited our graduate analysis class to his home one evening. He regaled us with stories, and we talked, played the piano, and fully enjoyed our time with him and his wife, Artie, and young children, Thor and Anna (Annie). Siggi Helgason won the Graduate Student Professor of the Year Award that year. When accepting the award, he looked at his admiring graduate students and said with a twinkle, “I couldn’t have done it without you!” Through the years, I have had the good fortune of meeting Siggi Helgason at conferences, including a 2007 conference in Iceland in honor of his $80^{\text{th}}$ birthday. I enjoyed discussing and learning new integral geometric ideas, seeing the natural wonders of the Iceland he loved so much, and the warm, friendly hospitality in his apartment in Reykjavik. Artie and Siggi Helgason’s holiday parties in Belmont, MA, were not to be missed! His smoked Icelandic lamb was heaven, and the mood was jovial and convivial. I was honored to be one of “the usual suspects” of local former students, including Fulton Gonzalez, John Lewis, and Jerry Orloff, who would catch up over wine or hors d’oeuvres with Siggi and Artie. Getting to know Siggi, Artie, Thor, and Annie through these gatherings was a joy. I thank them for welcoming me into their home and for being good friends.

Sigurður Helgason was a brilliant mathematician who created deep, elegant mathematics, and a mensch who engendered community and valued family and friends. I will miss him.

Todd Quinto is the Robinson Professor of Mathematics at Tufts University. His email address is todd.quinto@tufts.edu.

By François Rouvière

Dear Sigurður, writing these few words to you is a great honor for me. It is a hard task, with so many memories coming back to my mind, scattered over fifty years and more. In 1969, I was a graduate student in research at Purdue University. My thesis advisor François Treves, who had suggested that I study the solvability of invariant differential operators on Lie groups, said that I should meet Helgason at MIT. Our first meeting was truly warm and stimulating thanks to your clear and insightful ideas. You showed me around Boston and you did not fail to stop in front of the statue of Leif Eriksson, the famous Icelandic sailor who “discovered” America a few centuries before Christopher Colombus.

Your 1962 book Differential Geometry and Symmetric Spaces, “the green book,” as well as each one of your subsequent monographs as the years went by, soon became my bedside books. A whole generation of mathematicians has been admiring and grateful for your ability to decipher Élie Cartan (which you read in French) and Harish–Chandra. You translated their difficult works into a piece of clear and readable mathematics. Later on I became indebted to you for discovering the beauty of Radon transforms, which has become my favorite subject for many years.

I have been lucky enough to benefit from your advice and your help on many occasions, though you have never been my advisor in a formal sense. A short anecdote: back in the 70s I proudly sent you my latest preprint about the Casimir operator. You diplomatically thanked me for this interesting work and you added: “However, there is a small item on page 2 which puzzled me.” The “obvious” property I had briefly stated on that page turned out to be wrong and the whole paper was wrecked. Shame on me! I was then stimulated to try a different method.

In 1993, I wanted to insert a birthday dedication to you in my paper about symmetric spaces, and I was able to write it in Icelandic thanks to the friendly assistance of Gestur Ólafsson. I think you were pleased to read a few words of your mother tongue printed in the Journal of Functional Analysis.

I am in debt to you and Mogens Flensted-Jensen for an invitation to the Mittag-Leffler Institute in the fall of 1995. I spent a month working in this wonderful atmosphere and this remains one of the fondest memories of my entire professional life. We also shared a passion for photography and I enjoyed looking at the Icelandic landscapes and the family pictures you often sent me. In your many letters, beautifully handwritten, I am pleased to read again our discussions about symmetric spaces, Radon transforms, as well as photography, news from Iceland, and much more.

Dear Sigurður, master and friend, it is so sad to think this one will be my last letter. Your memory will remain in my thoughts.

François Rouvière is an emeritus professor at the Université Côte d’Azur. His email address is rouviere.francois@wanadoo.fr.

By Boris Rubin

My first encounter with Siggi Helgason was through his nice book The Radon Transform, which was translated into Russian in 1983. I bought it, like many other math books, just in case, because it was small and cheap. Many years later, when I left the former Soviet Union and was thinking about entering new areas of mathematics, I opened that book and was impressed to find many formulas, which were close to my previous interests in fractional integrals. June 1996 became a decisive point in my mathematical life. There was a Centro Internazionale Matematico Estivo (CIME) Session “Integral Geometry, Radon Transforms and Complex Analysis” in Italy, were Sigurður was an invited lecturer. It was a great opportunity to meet him in person. I was dreadfully excited because Sigurður was the greatest expert and I was a newcomer and had a bunch of questions. However, Sig was very patient and friendly. We discussed many questions and he explained many new things. That meeting enlarged my sphere of interests drastically. I realized that integral geometry and fractional integration have a lot in common and can benefit each other. Over the years, I met Sigurður on various occasions, in conferences and seminars. All these meetings were joyful and inspiring. Siggi was a talented listener. I was deeply influenced by his work and personality.

Boris Rubin is a professor at Louisiana State University. His email address is borisr@lsu.edu.

By Henrik Schlichtkrull

The dedication page of Groups and Geometric Analysis reads “To my Danish mathematical friends, past and present.” Being a person that was easy to get along with, Sigurður had many friends from all over the world, but the Danish mathematical community remained close to his heart since his undergraduate student days. In 1946, at the age of nineteen, he moved to Copenhagen. He stayed here for almost six years, until he obtained a masters degree and went on to Princeton. Sigurður often spoke with affection about this time in Copenhagen. The library opened a new world for him as he saw a mathematical journal for the first time. He also mentioned lectures of mathematicians, especially H. Bohr and B. Jessen, as being exceptionally excellent and inspirational for his own style of lecturing later in life.

Years later Sigurður became a frequent visitor to the country and to the mathematics departments in Copenhagen and Aarhus. For numerous Scandinavian students, including myself, Sigurður has been extremely helpful with establishing ties to the mathematical community in Boston. The friendship expressed in Sigurður’s dedication was mutual. Coming from a small country like Iceland or Denmark, it is impressive and inspiring to meet a person with a personality and career like his. He acted in many ways as a beacon for us.

Like most, I knew the name Sigurður Helgason before I met the person. As an undergraduate I became interested in geometry in the mid 1970s and (rather unsuccessfully) started reading Differential Geometry and Symmetric Spaces on my own. In 1977, I attended a summer school in Laugarvatn, Iceland, in which Sigurður was an inspirational main speaker (as well as tour guide). Soon after that I selected symmetric spaces as the topic of my masters thesis, with Mogens Flensted-Jensen as my advisor. Among the Danish mathematical friends of Sigurður he was certainly one of the closest, and he remained so until he passed away three years ago.

A few years later, with the help of Sigurður, I visited MIT during my first year as a PhD student. It was during this year that I got to know him better. Coming to a place like MIT as a student is inspiring. To me it was also sometimes intimidating, but never with Sigurður, who was always very understanding and hospitable. My wife Birgitte and I always felt welcome in his and Artie’s home, and among the Scandinavians who have been to Boston the gatherings in Belmont were legendary. Typically there would be hygge, Scandinavian foods and plenty of music, often performed by Sigurður on the piano and Artie singing.

In 1983 both Sigurður and I were visiting members of the IAS in Princeton. I was busy writing a book which would not have been written without Sigurður’s encouragement. Once again he and Artie opened their home for Birgitte and me, and we have fond memories of the Christmas day when we were invited to join the family party of the Helgason’s.

Sigurður was always interested in all kinds of mathematics, and he enjoyed it when he could share something with others. Patience and kindness were also part of his ways. One day during a longer stay in Copenhagen he was visiting our home. Together with us were our six-year-old son Michael, playing with a set of plastic bricks of various polygonal shapes with which he was fond of assembling closed convex 3-D figures. Sigurður became interested and taught (in fluent Danish) Michael to compute the Euler characteristic. Together they shared the wonder of getting the same number no matter which polytope Michael built.

The impressive career of Sigurður inspired and helped me professionally. His mathematics are admirable and they had a profound impact on my work. But above all Sigurður has left me with affection and admiration of his qualities as a human being. I have been privileged to know him, and for that I am eternally grateful.

Henrik Schlichtkrull is a professor at the University of Copenhagen. His email address is schlicht@math.ku.dk.

By Ragnar Sigurðsson

I first met Sigurður Helgason some time during my undergraduate years at University of Iceland, 1974–1978. I knew then that he was a leading mathematician of the world and greatly admired not only by my teachers but by everyone who knew him. On one of my first days as a PhD student at Lund University in Sweden, I was asked by one of the professors if I was a son of Sigurður Helgason. Of course this question was not serious because he was only telling me that he knew that both Helgason and Sigurðsson are patronymic surnames, son of Helgi and son of Sigurður. Indeed, Sigurður has been a role model for generations of Icelandic mathematicians, so in that sense he is a mathematical father of all of us who are younger than him.

Our acquaintance started to developed into a friendship shortly after I returned to Iceland from Lund. We were both students of the Nordic analysis school so we had many common interests. Sigurður visited Iceland frequently, but he also used to stop over in Reykjavik when he traveled to Europe for conferences and other events. We benefited a lot from these visits, because Artie and he had an apartment located a few hundred meters from the university campus and Sigurður often walked over to our Science Institute to check on how we were doing. He sometimes stayed for hours and our conversations on mathematics were spiced with all kinds of stories about his life, his work, and colleagues around the world.

Sigurður completed his student exam from Akureyri Gymnasium in 1945 with excellent grades, which awarded him a scholarship to study abroad. University of Copenhagen had not recovered after the war, so it was practical for him to stay in Reykjavik for the academic year 1945–1946 and follow the first year program in mathematics and physics with the engineering students. He started in Copenhagen in 1946 and completed his master of science degree by the end of 1951, after which he was awarded a one year Fullbright Fellowship to visit Princeton. He told us once that his plan was to return to Iceland after that year and find a teaching job at some gymnasium. Now when we see the magnitude of his lifetime achievements we are all thankful that his plan changed.

Iceland was dear to his heart and he supported mathematical activity here generously, both at University of Iceland, where he was doctor honoris causa since 1986, and at the Icelandic Mathematical Society, where he was a member since 1947 and honorary member since 1997. One example is that on the occasion of his 90th anniversary Sigurður founded a prize fund for bright mathematics students at the University of Iceland that bears his name. Prize ceremonies are held every year on his birthday.

Sigurður was a great supporter of Nordic cooperation and had many friends in Scandinavia, especially in Denmark. He participated in almost every Nordic Congress of Mathematicians 1957–2005. He represented Iceland as a plenary speaker on five occasions from 1968 to 2005. In an email he wrote to me by the end of 2022 he told me that he had lost track of the congresses and asked about the order of recent and future congresses. In my answer I told him that Iceland will host the next congress in 2027 and reminded him that there will be a certain anniversary that year as well and he replied: “Yes, of course I don’t want to miss the congress on my 100th anniversary!”

As a sixteen-year-old Sigurður was present at Thingvellir on June 17, 1944, when Iceland was declared a republic, changing its status among nations from being an independent autonomy under the Danish Kingdom. He has ever since been one of the best sons of this republic and we will always remember him as one of the greatest scientists we have had.

Ragnar Sigurðsson is a professor at the University of Iceland. His email address is ragnar@hi.is.

By Robert J. Stanton

The AMS Summer Institute 1972 was a remarkable event. I was just beginning research in harmonic analysis on Lie groups. My supervisor, Tony Knapp, had arranged an invitation for me. The milieu was a beautiful campus in New England for three weeks, attending were many of the current and future leaders of this subject—what an experience for a grad student! Moreover, there were distractions: seeing Bert Kostant running the bases in a softball game, Roger Howe playing tennis, Lou Auslander enjoying a game of bridge while telling stories. And there were the lectures—beyond my background, nevertheless inspiring. Helgason gave a series on “Functions on symmetric spaces” and that was my first experience with Siggi.

Figure 4.

Three generations: with student Dadok and teacher Bochner, 1978.

I did not meet him then, but the opportunity arose some six years later when as a junior professor at Rice I was assigned to the colloquium committee. Bochner was the department chair, and we had a new instructor, Jiri Dadok, a former student of Helgason. So it was no problem getting Siggi to visit. A side benefit was the only photo of Sal and Siggi together. That was the start of a friendship with Siggi that lasted some 45 years.

Having common mathematical interests, we both attended many of the same conferences, with him usually one of the highlighted speakers. The special ones were celebrations: his 65th (Roskilde ’92), the Mittag-Leffler Semester (’95), his 80th (Reykjavik ’07), his 85th (Boston ’12).

Hearing I would be in Reykjavik, he wrote asking if I could extend my stay a few days. I said that I had done it already for another week but for a reason new to him. At that time I had been a backpacker for 20 years, and, together with three friends from the States, I was going to hike the trail between Landmannalaugar and Thorsmork. We even had reservations at all the huts on the trail. He wrote back excitedly that he had not done this hike, but his daughter had done it recently. She had the Iceland trail guide for it (in English) and he would send it (to be returned, as it was hers). The guide was a blessing. On the trail we had two tough days—little visibility and many of the trail markers (a 2-meter stick in a cairn) had blown down. When Siggi read my description of the hike, he wrote back “Looks like you got some disagreeable doses of the infamous Icelandic horizontal rain.” A couple years later he added “If you had been hiking on March 27 in the same place as in 2007, you might have walked into a volcanic eruption. It started on the trail.”

Siggi was always generous with his time. We must have exchanged 15 messages regarding that conference and hike. About mathematics he was equally generous with me. I think he sent me all his papers. I especially enjoyed the ones on historical topics. He could explain mid-nineteenth-century mathematical papers with that wonderful prize-deserving skill that he showed in his first book—clarity and historical perspective. I sent him some papers of mine. He would go over them and send me a critique on how to improve them. Concerning one he wrote “In connection with (6.4) it would have been good to have the formula for m(lambda, v). Should Lemma 7.3 have preceded (6.4)? Also it is not quite explicitly clear how the left-hand side of (6.4) is derived from the preceding pages.” I didn’t ask him to do this, Siggi was just generous with his time and had high standards for mathematical writing.

Helgason’s mathematical legacy will speak for itself. I can attest to the man—generous, scholarly, kind—who happened to be a friend.

Robert J. Stanton is an emeritus professor at Ohio State University. His email address is stanton.2@osu.edu.

By David Vogan

I came to MIT as a graduate student in the fall of 1974, with the intention of learning about representation theory from Bert Kostant. As it happened Kostant was on sabbatical that semester, so I had the great good fortune to begin with Helgason’s course on Lie groups. Helgason was a masterful teacher. His exposition always arose from a deeply geometric and analytic understanding of Lie groups. (Since my own understanding of Lie groups was rooted in some weak knowledge of how to multiply two-by-two matrices, this was a constant series of divine oracles.) His handwriting at the blackboard was beautiful. You can still experience this by reading Differential geometry, Lie groups, and symmetric spaces: any sentence in that book must clearly have been beautifully handwritten at its origin.

His Icelandic accent (which I can testify, after hearing hundreds of other possibilities, is the way English was meant to be spoken) was the icing on the cake. Indeed the accent is probably the reason I attended his class faithfully even though it began at 9:00 in the morning.

In his class I first experienced what would be a foundation of my mathematical life: that questions were received with an attentive ear, a smile, and a thoughtful response that never made the questioner feel slow or unwelcome. My own work in representation theory soon became mostly algebraic; but often there was no avoiding integrals and distributions and differential equations and Riemannian metrics. On all of these topics and more he was my library, as a grad student and eventually as a colleague. The library was always open, and the librarian a trusted friend.

Helgason worked in a part of mathematics that was very active and very competitive, particularly from the early 1950s to the early 1980s. Harish-Chandra dominated the field, and working in his shadow was a challenge. I have had the privilege of knowing a number of great mathematicians from that period. Helgason, almost uniquely, managed the challenge in a constantly cheerful and gentlemanly manner. I can recall only a handful of occasions when he seemed on the edge of being unhappy about some mathematical event; and on those occasions Artie quickly restored his customary demeanor.

I’ll end with a story that gives some hint of Helgason’s life beyond mathematics. In the 1990s our church was the venue for a performance by the Icelandic Women’s Chorus. My wife emailed Helgason to tell him about the performance. (Helgason was an early adopter of email, because it was a perfect way to stay in touch with his worldwide network of friends.) Helgason immediately wrote back to say thank you. He knew about the performance already; he was the PR director for the Boston event, and the conductor was to stay at his home.

And he started the email by saying “thank you.” Helgason’s razor-sharp mathematical mind and his eloquent pen can still be enjoyed in his writing; but the mathematical world has lost a gentleman.

David Vogan is the Norbert Wiener Emeritus Professor of Mathematics at MIT. His email address is dav@math.mit.edu.