# Åke Vilhelm Carl Pleijel

### Born: 10 August 1913 in Johannes parish, Stockholm, Sweden

Died: 24 September 1989 in Västerled parish, Stockholm, Sweden

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**Åke Pleijel**was the son of Henning Bernhard Mathias

**Pleijel (1873-1962) and Eva Hilda Katarina Bergenström (1877-1957), daughter of Carl Theodor Bergenström, a utility manager, and Caroline Margareta Brandtberg. Henning Mathias was a physicist and electrical engineer who, at the time his son was born, was a professor at the Telegraph Board, and in the following year was made a professor in theoretical electrical engineering at the School of Electronics at the Royal Institute of Technology in Stockholm [3]:-**

Henning married Eva on 6 August 1907 in Klara, Stockholm. They had two children, Märta Laura Carolina Pleijel (1911-2004) and Åke Vilhelm Carl Pleijel, the subject of this biography.In his scientific activities, Henning Pleijel studied technical problems, and primarily he dealt with theoretical electrotechnical issues with telegraph and telephone lines as a specialty. His area of interest meant that his research sometimes led to technical improvements, many of which resulted in financial gains.

Pleijel graduated from the high school in Östermalm, Stockholm on 11 May 1932 and enrolled at Stockholm University on 23 May 32. He was awarded the FK degree (the Filosofie kandidat degree is essentially equivalent to the B.Sc.) on 30 May 1933. He then held various teaching positions at different educational institutions in Stockholm while continuing his studies at Stockholm University. He was awarded the FM degree (the Filosofie magister degree is essentially equivalent to the M.Sc.) on 6 March 1937, and an FL degree (Filosofie licentiat is a Licentiate degree, more advanced than a Master but below a Ph.D.) on 5 November 1938.

After graduating with his first degree, Pleijel began undertaking research advised by Torsten Carleman. He was soon publishing papers in both French and German: *Sur la théorie des corps centrobares* Ⓣ (1934), *Über asymptotische Reihenentwicklungen in der Operatorenrechnung* Ⓣ (1935), *Eine hydrodynamische Randwertaufgabe* Ⓣ (1936), *Eine hydrodynamische Randwertaufgabe* Ⓣ (1937), *Sur les propriétés asymptotiques des fonctions propres des plaques vibrantes* Ⓣ (1939), *Propriétés asymptotiques des fonctions fondamentales du problème des vibrations dans un corps élastique* Ⓣ (1939) and *Sur les propriétés asymptotiques des fonctions et valeurs propres des plaques vibrantes* Ⓣ (1939). He was awarded a Ph.D. on 30 May 1940 for his 101-page thesis *Propriétes asymptotiques des fonctions et valeur propres de certain problèmes de vibrations* Ⓣ.

His main mathematical interest was in the treatment of problems that originated in physics. Tord Ganelius writes in [2] that in his doctoral thesis:-

Pleijel had married while he was teaching mathematics and studying for his doctorate. He married the author Sonja Anna Stina Berg on 27 November 1937. Sonja had been born on 23 September 1909 in Surabaya, Java, Indonesia to Allan Erik Berg and Caroline Christine van der Velde. Åke and Sonja Pleijel had three children, all daughters, Agneta Christina Pleijel (born 26 February 1940), Sonja Carolina Pleijel (born 3 August 1942), and Eva Maria Pleijel (born 31 March 1945). Let us record at this point that Agneta Christina Pleijel became an author, poet, playwright, journalist and literary critic; Sonja Carolina Pleijel studied at Lund University and became a TV producer, director and screenwriter.... Pleijel extended his investigations on the asymptotic behaviour of eigenvalues and eigenfunctions for certain oscillation problems to a number of other areas, including strained plates. During the first decades of the20th century, Hermann Weyl and Richard Courant had studied how self-frequencies depend on the geometry of the swinging body. Carleman introduced a new method that reversed the problem on a so-called Tauberian theorem, a type of theorem that was studied throughout the20th century but which was renewed in relevance through the work of Norbert Wiener around1930.

On 14 September 1940 Pleijel was appointed as a docent in mathematics and an assistant in the mathematics library at the University of Stockholm, a position he held until he was appointed as a docent at Lund University on 25 April 1941. After seven years in Lund, he was back in Stockholm when he was appointed as professor of mathematics on 22 October 1948. It is not quite correct to say that he was in Lund for seven years for, although he held the position in Lund for that time, he spent the last of these years at the Institute for Advanced Study in Princeton, USA. Pleijel, his wife and three daughters sailed on the Drottningholm from Gothenburg, Sweden to New York, USA arriving on 15 September 1947.

While at Princeton, Pleijel worked with Subbaramiah Minakshisundaram (1913-1968), an Indian mathematician with an expertise in parabolic partial differential equations. They collaborated on writing the paper *Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds* (1949). Salomon Bochner writes in a review:-

In fact, the paper contains the famous zeta function, now called the Minakshisundaram-Pleijel zeta function, which is a zeta function encoding the eigenvalues of the Laplacian of a compact Riemannian manifold.This paper offers an extension of a method of Carleman's from a Euclidean to a curvilinear setup, and the results of Carleman have been both generalized and added to; an important tool of investigation is Hilbert's parametrix.

After his year at the Institute for Advanced Study, the family returned to Sweden, sailing from New York to Gothenburg, Sweden departing on 4 December 1948.

Back in Sweden, Pleijel took up his appointment as professor of mathematics at the Royal Institute of Technology in Stockholm on 1 January 1949, where he succeeded Johannes Malmquist, who had retired in 1948. Pleijel had actually been appointed to this position on 22 October 1948 while he was at Princeton.

We quote from [2] regarding the situation in mathematics in Sweden when Pleijel was appointed to the chair in Lund:-

Appointed professor of mathematics at Lund University on 4 August 1952, he took up the appointment on 1 December that year. He served as dean of the Faculty of Mathematical and Natural Sciences from 1 July 1957 to 30 November 1961 and served again in the academic year 1964-1965. TheWhen Pleijel became professor in1949, there were only eight professorships in Sweden in the unrestricted subject of mathematics .... However, the investment in science and technology after1945led to a rapid increase in the number of students and the number of professors in mathematics. In addition, courses were added in special parts of the subject, such as applied mathematics, mathematical statistics and numerical analysis. For the mathematics subjects at the technical colleges, the changes resulted, in among other things, increased demand for courses in mathematics as a subject in the so-called licentiate degree. Pleijel took great interest in the creation of courses suitable for this purpose, but he left the Royal Institute of Technology in Stockholm as early as1952to succeed his future father-in-law, Professor Marcel Riesz, upon his retirement from Lund University.

*1959 International Handbook of Universities*lists Lund University with Åke Pleijel as Dean of the Faculty of Mathematics and Natural Sciences with a staff of 102. It gives the following details about Lund University:-

He became the second President of the Swedish Mathematical Society, holding this position from 1952 to 1957, and served as an editor forFounded1666as a State university. Closed1676during the war with Denmark, reopened1682. The university is operated under the supervision of the Chancellor of Swedish Universities. Governing bodies : det Storre Akademiska Konsistoriet(Consistorium Academicum Majus)and det Mindre Akademiska Konsistoriet(Consistorium Academicum Minus). The Rector and Vice-Rector hold office for a period of three years and are elected by the assembly of academic teachers.

*Mathematica Scandinavica*from 1953 to 1965. He was chair of the Swedish National Committee for Mathematics in 1954. During his time in Lund he was also an inspector for the Småland region. He served on the Fields Medal Committee 1954 which was chaired by Herman Weyl and made awards of Fields Medals to K Kodaira and J P Serre.

At a session on 28 October 1958 the Swedish National Committee for Mathematics decided to accept the invitation, conveyed by the International Mathematical Union, to organise the next International Congress of Mathematicians in Stockholm in 1962. This decision was endorsed by the Swedish Mathematical Society on 30 November 1958 and a joint invitation was issued to the mathematicians of the world, signed by the chairmen of the National Committee and the Society, A Pleijel and G Borg. O Frostman and A Pleijel were entrusted with the initial preparations for the congress together with H Cramer, at that time Chancellor of the Universities of Sweden, and Lars Gårding, all of whom later constituted the Organising Committee.

From 1965 to 1967 Pleijel went to Ethiopia as a representative of UNESCO. During this time he made a considerable contribution to the development of higher education in mathematics in Ethiopia. He also returned to Addis Ababa for two years after his retirement from Uppsala from 1980 to 1982. Pleijel authored several textbooks for Ethiopians and, perhaps most importantly, he managed to interest some young Ethiopians in mathematical research.

On 30 June 1967, Pleijel was appointed as Professor of Mathematics at the University of Uppsala. In fact 1967 was a major year in his personal life since in that year he was divorced from his first wife Sonja and, on 12 December of that year, he married Margit Ingrid Riesz at Västerled, Stockholm. Margit, a Director of Information, had been born on 27 July 1917 to Marcel Riesz and Sofia (Sophie) Justina Albertina Eriksson. It was also Margit's second marriage, having married Stig Hilding Hammar on 13 August 1944.

At the University of Uppsala, Pleijel was dean of the Faculty of Mathematical and Natural Sciences from the autumn term of 1971 to the spring term of 1973. He retired from his professorship in 1979 and was made professor emeritus. On 3 March 1975 he was elected as an Honorary Member of the Royal Society of Edinburgh. Tord Ganelius writes in [2]:-

Pleijel published around 50 papers on topics such as: Differential equations, operational calculus; Ordinary differential equations; Partial differential equations, and Potential theory. Several of the mathematical concepts that Pleijel introduced are now known by his name and have been the subject of many papers. These include, in addition to the Minakshisundaram-Pleijel zeta function we mentioned above, the "Pleijel's nodal domain theorem", the "Pleijel's constant", "Pleijel's estimate", the "Pleijel inequality", and the "Pleijel identity". He has published obituaries of Anders Wiman, Johannes Malmquist, Nils Erik Fremberg and Ivar Fredholm. Following his death in 1989 he was buried in the Northern Cemetery in Lund.Some of Pleijel's work on ordinary differential equations in recent years deals with Weyl's theory of boundary point and boundary circle. In this area, a collaboration developed with the University of Edinburgh, in which several of Pleijel's co-workers in Uppsala participated. One sign of great appreciation from the Scottish partner was the election of Pleijel as member of the Royal Society of Edinburgh.

**Article by:** *J J O'Connor* and *E F Robertson*

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