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Sigurður Helgason

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Born
30 September 1927
Akureyri, Iceland
Died
3 December 2023
Belmont, Massachusetts, USA
Summary
Sigurður Helgason was an Icelandic mathematician who spent most of his career in the United States. He was a geometric analyst who made major contributions to representation theory of group actions on homogeneous spaces, as well as to the theory of generalised radon transforms, and wrote a number of classic textbooks.
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Biography

Sigurður Helgason was the son of Helgi Skúlason (1892-1983) and Kára Sigurðarsóttir Briem (1900-1982). Helgi was born on 22 June 1892 in Oddi, Iceland to Skúli Skúlason, who was cabinet secretary, and Sigríður Helgadóttir. He studied medicine from 1910, graduating in 1915. After practicing for a couple of years, he trained in Freiburg and Berlin in Germany to become a specialist in eye diseases. He worked as a doctor in Reykjavík from 1921 to 1927, and then settled in Akureyri where he lived until 1970. He was an adjunct lecturer in ophthalmology in the Faculty of Medicine of the University of Iceland between 1923 and 1927. Kára Briem was born on 1 April 1900 in Reykjavík where her father was the postmaster. Helgi and Kára married on 31 October 1925 and had three children: Skúli Helgason (1926-1973), Sigurður Helgason (1927-2023), the subject of this biography, and Sigriður Helgadóttir (1933-2003). Skúli Helgason, born 18 June 1926 in Reykjavík, became a doctor in Umea, Sweden, but died young. Sigriður Helgadóttir, born 15 February 1933 in Akureyri, became a pathologist. The city of Akureyri, at the time when Sigurður was born, is described in [19]:-
Akureyri was then the second largest city in Iceland with about 3,000 people living there, whereas the population of Iceland was about 103,000. As with other cities in northern Iceland, Akureyri was isolated, having only a few roads so that horses or boats were the transportation of choice. Its schools, based on Danish traditions, were good.
Mathematics fascinated Helgason from the time he was a young child and he would measure circular objects from home to calculate π. He entered Akureyri Junior College in 1939. Although this school dates back to 1880, for a long time it was linked with Reykjavík Gymnasium with students having to go to Reykjavík for their final three years. It was only in 1930 the school became an independent Gymnasium, the second in Iceland. It offered a six year course with considerable emphasis on languages - Helgason took classes in Danish, English, German, and French. His science subjects were chemistry, astronomy, physics, and mathematics. He was greatly influenced by the astronomer Trausti Einarsson (1907-1984) who had studied at Göttingen. Hejgason said in the interview [21] that his mathematics teacher Einarsson:-
... 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.
Although Sigurður's father, Helgi Skúlason, was based in Akureyri, he travelled to various villages in north Iceland during the summer months to see patients with eye problems. While at the gymnasium, Sigurdur Helgason went with his father on one of these trips. This took them to Raufarhöfn which had a herring factory and Sigurður signed up to work there during the following few summers while still at school. It was a very hard job, with two six-hour shifts each day, but the pay was good. Before taking on the herring factory summer job, he had done farm work during the summer, haymaking and looking after animals. Although Iceland was neutral during World War II, British troops arrived there in 1940. Sigurður sold "War News" newspapers to those who were stationed in Akureyri.

Helgason graduated from Akureyri Gymnasium in 1945. He had, of course, been six years at the Gymnasium, exactly the six years of World War II. He would have liked to have studied university mathematics at Göttingen in Germany but, in 1945, this was out of the question. His second choice was the University of Copenhagen, but although this had operated during World War II, managing to hold most of its classes and conducting some research, Helgason was advised not to begin studying there in 1945 but to wait a year until the University was fully operational. He therefore spent 1945-46 as a student in the School of Engineering at the University of Iceland in Reykjavík. He avoided the technical subjects, and took courses in mathematics, physics and chemistry, and a short course in philosophy.

In 1946 he went to Denmark to begin his studies at the University of Copenhagen. Although his aim was to specialise in mathematics, for the first three years he took courses in mathematics, physics, chemistry and astronomy designed to bring students to a level required to teach these subjects in Danish secondary schools. He then specialised in mathematics taking the advanced courses. There were not many such courses and he could have completed his Candidatus magisterii degree quite quickly, but he was keen to enter for the University of Copenhagen Prize Competition. The topic proposed for the Prize was "To establish a Nevanlinna-type value distribution theory for Dirichlet series". Anyone under 30 years of age could apply, there was one year before solutions had to be submitted and entries were anonymous. A code on each entry allowed the name of the winning candidate to be determined but all other entries remained anonymous. Helgason was declared the winner of the University of Copenhagen Gold Medal in l951. He was awarded the Candidatus magisterii degree in January 1952.

Although by taking part in the competition, he had begun mathematical research, Helgason assumed at this stage that his future would be as a gymnasium teacher of mathematics back in Iceland. His parents, however, suggested that he apply for a fellowship to study for a doctorate in the United States. His application was successful and he received a Fulbright Grant in 1952 which gave him the choice of studying at Harvard University or Princeton University. Helgason was taught at the University of Copenhagen by Børge Jessen who had spent time in 1933-34 at the Institute for Advanced Study at Princeton and had also visited Harvard at that time. While at the Institute for Advanced Study Jessen had collaborated with Salomon Bochner and Aurel Wintner leading to a joint paper with each. He had no hesitation in advising Helgason to study for his doctorate at Princeton; advice for which Helgason was very great grateful.

Helgason flew from Keflavík Airport, the main Icelandic airport, to Idlewild Airport, New York, on a Pan American Airways flight, arriving on 15 March 1952. He took a few courses, for example a topology course by Ralph Fox whom he had already met when Fox had visited Copenhagen. Another student taking this course was Gian-Carlo Rota with whom Helgason would later become very friendly when they were colleagues at MIT. In order to progress to become a Ph.D. candidate, students had to take the oral Ph.D. General Examination. Helgason was examined by Emil Artin, Solomon Lefschetz and Donald Spencer and began his Ph.D. studies with Salomon Bochner as his thesis advisor. He said [21]:-
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.
He was awarded a Proctor Fellowship to fund his second year at Princeton. After two years at Princeton, Helgason submitted his 52-page thesis Banach Algebras and Almost Periodic Functions and was awarded his Ph.D. in 1954. He published his first paper in 1954, namely The derived algebra of a Banach algebra. His address on this paper is Princeton University, but by the time his second paper Some problems in the theory of almost periodic functions was published in 1955, he was already on the staff at the Massachusetts Institute of Technology (MIT). He gives the following acknowledgement on the 1955 paper:-
The author wishes to express his gratitude to Professor S Bochner for his advice and many helpful conversations during the preparation of this paper.
After completing his doctoral studies in 1954, Helgason applied for instructorships at both Harvard and MIT. He went to visit MIT with a friend and liked it, so when he received the offer of a place from them, he accepted immediately before Harvard had responded. The appointment at MIT was for two years as a C L E Moore Instructor. He had quite a light lecturing load, with a few elementary courses and only one advanced course on functional analysis. He began to make a serious study of Harish-Chandra's papers and Élie Cartan's work. He became friendly with John Nash and played a lot of chess with Norbert Wiener.

In the summer of 1955 Chicago had a summer workshop on functional analysis with George Mackey, Irving Segal, Irving Kaplansky and Paul Halmos as lecturers. Saunders Mac Lane, who was chairman at Chicago, visited MIT before the meeting looking for young talent to invite to the workshop. He had some funding and offered financial support to Helgason. He took part in the summer meeting at Chicago which he found a very positive experience. Particularly valuable was becoming friendly with Alexander Grothendieck; they had meals together and went for walks on which they would talk mathematics.

After the two years at MIT, Helgason was appointed as a lecturer at Princeton University in 1956. Although Princeton was not a one year appointment, when he got an offer from the University of Chicago he accepted and left Princeton to became a Louis Block Assistant Professor at the University of Chicago in 1957.

Helgason married Artie Gianopulos on 9 June 1957. Artie Gianopulos had been born on 23 November 1930, the daughter of the locksmith Gus Gianopulos and his wife Anne. Helgason had met Artie at a dance at Bryn Mawr College in 1953. They became engaged to be married in December 1956 and the announcement of their engagement appeared in the Pittsburgh Post-Gazette on Monday, 31 December 1956 [2]:-
Mr and Mrs Gus Gianopulos of Methyl Avenue, Pittsburgh, have announced the engagement of their daughter, Artie Gianopolus, to Dr Sigurdur Helgason, son of Dr and Mrs Helgi Skulason of Akureyri, Iceland. Miss Gianopolus received her bachelor of arts degree from Chatham College in Pittsburgh and her master's degree from Bryn Mawr College. She is presently working as a research social worker at the Marriage Council of Philadelphia. Dr Helgason studied at the University of Copenhagen and received his doctor of philosophy degree in mathematics from Princeton University where he is currently teaching.
Sigurður and Artie Helgason had two children: Thor Helgason, born 15 August 1961, who became an engineer, and Anna Loa Helgason, born 10 October 1967, who became a medic.

After two years at Chicago, Helgason was appointed to the mathematics faculty at MIT but was allowed to spend the year 1959-60 as an Assistant Professor at Columbia University before taking up the Assistant Professorship at MIT in 1960. He was promoted rapidly becoming an Associate Professor in 1961. In 1962 he published his first book, namely Differential Geometry and Symmetric Spaces. For information about this book including extracts from the Preface and from some reviews, see THIS LINK.

He was awarded a Guggenheim Guggenheim Fellowship for 1964-65 to work on the project "Theory of functions on symmetric spaces" and he become a full Professor in 1965.

Helgason wrote Differential Geometry, Lie Groups, and Symmetric Spaces which was published in 1978. He wrote in the Preface:-
The present book is intended as a textbook and reference work on three topics in the title. Together with a volume in progress on "Groups and Geometric Analysis" it supersedes my "Differential Geometry and Symmetric Spaces," published in 1962. Since that time several branches of the subject, particularly the function theory on symmetric spaces, have developed substantially. I felt that an expanded treatment might now be useful.
Although he was already working on "Groups and Geometric Analysis", he first published The Radon Transform in 1980. This work, based mostly on his own papers published between 1959 and 1965, closely followed a lecture course he had given at MIT in the autumn of 1966. Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions was published in 1984 followed by Geometric Analysis on Symmetric Spaces in 1994. His final book was Integral geometry and Radon transforms published in 2010. For more information about these very influential books and about second editions of some of them, see THIS LINK.

Although he spent the rest of his career at MIT, Helgason had a number of research visits to the Institute for Advanced Study in Princeton. He was a Member of the School of Mathematics at the IAS from September 1964 to June 1966, from September 1974 to June 1975, from September 1983 to January 1984 and from February 1998 to May 1998. He also was a member of the Mittag-Leffler Institute in Sweden in 1970-71 and again in the autumn of 1995. In the Spring of 1990 he was on leave as a visiting professor at the University of Copenhagen.

Helgason retired in 2014 after serving for 55 years on the MIT Mathematics Faculty. He received MIT's first Graduate Teaching Award in 1975 and served as the chair of the Graduate Faculty from 1986 to 1993. The Faculty gave the following summary of his research contributions [23]:-
Helgason for many decades led the study of group actions on manifolds. Generations of mathematicians entered the field through his classic 1962 text 'Differential Geometry and Symmetric Spaces', and its greatly expanded 1978 second edition 'Differential Geometry, Lie Groups, and Symmetric Spaces'. His own research contributions included the Plancherel and Paley-Wiener theorems for Riemannian symmetric spaces. He was a superb lecturer, a well-regarded graduate advisor, and a cherished colleague. ... As a part of his geometric work, Helgason helped to create the modern theory of Radon transforms: seeking to reconstruct a function from its integrals along various subspaces. (This is the mathematical basis of many kinds of medical imaging.) Helgason understood that a zoo of special results could be understood as instances of a very general "double fibration". This insight became part of the foundation of his work on the Fourier transform on symmetric spaces, his existence results for solutions of group-invariant differential equations, and his contributions to infinite-dimensional representation theory.
An overview of his contributions is also given in [19]:-
Sigurður Helgason is known worldwide for his first book 'Differential Geometry and Symmetric Spaces'. With this book he provided an entrance to the opus of Élie Cartan and Harish-Chandra to generations of mathematicians. On this the occasion of his 85th birthday we choose to reflect on the impact of Sigurður Helgason's sixty years of mathematical research. He was among the first to investigate systematically the analysis of differential operators on reductive homogeneous spaces. His research on Radon-like transforms for homogeneous spaces presaged the resurgence of activity on this topic and continues to this day. Likewise he gave a geometrically motivated approach to harmonic analysis of symmetric spaces. Of course there is much more - eigenfunctions of invariant differential operators, propagation properties of differential operators, differential geometry of homogeneous spaces, historical profiles of mathematicians.
Among the many awards that he was won for his outstanding work, we mention the Børge Jessen Diploma Award by the Danish Mathematical Society in September l982, the Steele Prize from the American Mathematical Society in 1988, and the Major Knights Cross of the Icelandic Falcon in 1991. He was elected a member of the Icelandic Academy of Sciences in l960, of the American Academy of Arts and Sciences in l970, and of the Danish Royal Academy of Sciences and Letters in l972. In 1997 he was made an honorary member of the Icelandic Mathematical Society. He was awarded an honorary doctorate by the University of Iceland in l986, by the University of Copenhagen in 1988, and by the University of Uppsala in 1996. He became a fellow of the American Mathematical Society in 2012,

He was also honoured with two international conferences held in his honour. The first was the 'Conference on Fourier and Radon Transformations on Symmetric Spaces' held at Roskilde, Denmark on 10-12 September 1992 to mark his 65th birthday. The second was the 'International Conference on Integral Geometry, Harmonic Analysis and Representation Theory' held in Reykjavik, Iceland on 15-18 August 2007 to mark his 80th birthday. The Information concerning this conference states:-
The history of harmonic analysis goes back to the eighteenth century work of Fourier on the heat equation. But it is also closely related to the modern development of number theory, geometry, operator algebras, representation theory and physics, in particular quantum mechanics. The emphasis of this conference is on abstract harmonic analysis, integral geometry and representation theory of Lie groups and the interplay between those fields. This includes generalisations of well known integral transform like the Radon transform to compact and noncompact symmetric spaces, applications of complex methods in real harmonic analysis on homogeneous spaces and algebraic methods in representation theory and its connection to analysis.
He had several hobbies which are mentioned in [23]:-
A self-taught pianist, Sigurður played almost daily up until the day before he died. An avid and skilled chess player, he was an early adopter of computer chess, starting with the (now-obsolete) "Chess Challenger". He particularly enjoyed playing chess with his grandson during the many family gatherings held at his home. He was also an enthusiastic photographer, capturing images of the unique and beautiful parts of Iceland, as well as the more prosaic close-ups of the resident cardinal on his bird feeder in Belmont.
During his career, he advised 17 doctoral students and acted as a mentor many more. He particularly supported Icelandic students [23]:-
Countless Icelandic students and musicians who came to study in Boston knew Sigurður as the unofficial "consulate". He extended remarkable hospitality to them and his home in Belmont was the "home away from home" for many Icelanders. It was a rare Thanksgiving, Christmas or Easter without at least one Icelandic student joining the celebration. He remained very close to his family and friends in Iceland, and made many extended visits.
Following his death at the age of 96 in December 2023, a remembrance of his life was held on 20 January 2024 at the Scandinavian Living Center, in Newton, Massachusetts.
Other Mathematicians born in Iceland
A Poster of Sigurður Helgason

References (show)

  1. C B Allendoerfer, Review: Differential Geometry and Symmetric Spaces, by Sigurdur Helgason, The American Mathematical Monthly 71 (3) (1964), 336.
  2. Artie Gianopulos to Wed Dr Helgason of Iceland, Pittsburgh Post-Gazette (Monday, 31 December 1956).
  3. R L Bishop, Review: Differential Geometry and Symmetric Spaces, by Sigurdur Helgason, Pi Mu Epsilon Journal 3 (8) (1963), 419.
  4. F Gonzalez and E T Quinto, Review: Integral geometry and Radon transforms, by Sigurdur Helgason, Bulletin of the American Mathematical Society 50 (4) (2013), 663-674.
  5. S Helgason, Differential Geometry and Symmetric Spaces (American Mathematical Society, 1962).
  6. S Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces (American Mathematical Society, 1978).
  7. S Helgason, The Radon Transform (Birkhauser, Boston 1980).
  8. S Helgason, Integral geometry and Radon transforms (Springer, New York, 2010).
  9. S Helgason, Integral geometry and Radon transforms (Springer, New York, 2010).
  10. S Helgason, Harish-Chandra and his mathematical legacy: Some personal recollections, Current Science 74 (10) (1998), 921-924.
  11. Helgi Skúlason (1892-1983), Atom (2024).
    https://atom.skagafjordur.is/index.php/helgi-skulason-1892-1983?sf_culture=en
  12. R Hermann, Review: Differential Geometry, Lie Groups, and Symmetric Spaces, by Sigurdur Helgason, SIAM Review 22 (4) (1980), 524-526.
  13. Honorary members, The Icelandic Mathematical Society (2024).
    https://stae.is/isf/heiðursfélagar
  14. Kara Briem - Minnningarord, Morgunbladid (Saturday 30 October 1982).
    https://timarit.is/page/1564339#page/n35/mode/2up
  15. A Koranyi, Review: Groups and Geometric Analysis, by Sigurdur Helgason, American Scientist 73 (5) (1985), 488.
  16. R S Kulkarni, Review: Differential Geometry, Lie Groups, and Symmetric Spaces, by Sigurdur Helgason, Bulletin of the American Mathematical Society 2 (3) (1980), 468-476.
  17. News from the AMS: Sigurdur Helgason, American Mathematical Society (8 January 2024).
  18. G Ólafsson and H Schlichtkrull (eds.), The Selected Works of Sigurdur Helgason (American Mathematical Society, 2009).
    https://bookstore.ams.org/cworks-22
  19. G Ólafsson and R J Stanton, On the Life and Work of S Helgason, in E T Quinto, F Gonzalez and J G Christensen (eds.), Geometric Analysis and Integral Geometry (American Mathematical Society, 2013), 21-44.
  20. E T Quinto, F Gonzalez and J G Christensen (eds.), Geometric Analysis and Integral Geometry (American Mathematical Society, 2013).
  21. Sigurdur Helgason interview, Department of Mathematics, MIT (4 August 2008).
    https://math.mit.edu/~helgason/helgason_interview.pdf
  22. Sigurdur Helgason Obituary, The Boston Globe (10 December 2023).
  23. Sigurdur Helgason, Professor Emeritus of Mathematics, Dies at 96, MIT Mathematics (6 December 2023).
    https://math.mit.edu/about/history/obituaries/helgason.html
  24. Sigurdur Helgason, Some Personal Remarks on the Radon Transform, in E T Quinto, F Gonzalez and J G Christensen (eds.), Geometric Analysis and Integral Geometry (American Mathematical Society, 2013), 3-20.
  25. Sigurdur Helgason, Prabook (2024).
    https://prabook.com/web/sigurdur.helgason/3560834
  26. Sigurdur Helgason's Curriculum Vitae, Department of Mathematics, MIT (2024).
    https://math.mit.edu/~helgason/cv.html
  27. Retirements and Departures, Department of Mathematics, MIT (5 June 2014).
    https://web.mit.edu/annualreports/pres14/2014.06.05.pdf
  28. Skúli Helgason laeknir - Minning, Morgunbladid (Wednesday 24 January 1973).
    https://timarit.is/page/1439325#page/n22/mode/2up
  29. V S Varadarajan, Review: Differential Geometry, Lie Groups, and Symmetric Spaces, by Sigurdur Helgason, American Scientist 68 (1) (1980), 94-95.
  30. J A Wolf, Review: The Radon Transform, by Sigurdur Helgason, American Scientist 69 (5) (1981), 570.
  31. L Zalcman, Review: The Radon Transform, by Sigurdur Helgason, SIAM Review 25 (2) (1983), 275-278.

Additional Resources (show)

Other pages about Sigurður Helgason:

  1. Sigurður Helgason Autobiography
  2. Sigurður Helgason Books

Other websites about Sigurður Helgason:

  1. Mathematical Genealogy Project
  2. MathSciNet Author profile
  3. zbMATH entry

Honours (show)

Honours awarded to Sigurður Helgason

  1. AMS Steele Prize 1988

Cross-references (show)

Written by J J O'Connor and E F Robertson
Last Update June 2024