# Lars Göran Borg

### Quick Info

Born
19 November 1913
Kumla, Närke, Sweden
Died
6 January 1997
Bromma, Stockholm, Sweden

Summary
Göran Borg was a Swedish mathematician who worked in differential equation theory.

### Biography

Göran Borg was the son of Johan Erik Zakarias Borg (1880-1953) and Elna Peterson (1887-1974). Erik Borg married Elna on 18 October 1910. Their daughter Gunni Lovisa Borg (1912-1989) was born on 2 April 1912 in Kumla and their son, Göran Borg, the subject of this biography, was born about 18 months later.

On 1 October 1939, Göran Borg married Gunborg Maria Sjølinder (born 26 August 1910 - died 23 March 1992) at Nora bergsförsamling, Örebro, Sweden. Gunborg Maria was the daughter of Natanael Sjølinder (born 6 April 1875) and Hulda Fredrika Gustafsson (born 19 January 1879). Göran and Gunborg Borg had three children, Erik Lars Natanael Borg (born 10 December 1940), Sven Erik Göran Borg (born 30 January 1943) and Anna-Karin Borg. We note that Erik Borg (born 1943) became a famous Swedish audiologist.

Borg studied at the University of Uppsala and by the time his first child was born in 1940 he had the degree fil.lic. (Licentiate of Science). His wife at this time has the degree med.lic. By the time his second son was born in 1943, Borg had the degree fil.mag. (Master of Science). At Uppsala his doctoral studies were supervised by Arne Beurling. Borg was awarded his Ph.D. in 1945 for his thesis Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe: Bestimmung der Differentialgleichung durch die Eigenwerte . He published a 97-page paper, which was essentially his thesis, in Acta Mathematica in 1946. Harry Pollard writes in a review of the paper [6]:-
The usual Sturm-Liouville problem consists in determining the spectrum for a differential equation y'' + (\lambda + \phi(x)) = 0 when the boundary conditions are given. The present paper studies the inverse problem of recovering φ(x) when the spectrum and the boundary conditions which give rise to it are known. For example, the problem of discovering the equations of a string from a knowledge of the fundamentals and overtones is of the inverse kind. The conclusions are as follows. (i) The solution of the inverse problem is not unique in general. (ii) The solution is unique if it is prescribed that \phi(x) = \phi(\pi - x), provided that the boundary condition has one of the forms $y(0) = y(\pi) = 0$ or $y'(0) = y'(\pi) = 0$. (iii) To an arbitrary Sturm-Liouville problem corresponds an associated one with the following property: if the spectrum of the associated problem is known in addition to the prescribed one, then φ(x) is uniquely determined. The chief tools used in the proofs are the theory of Fourier series and von Koch's theory of infinite determinants. The paper concludes with a study of physical applications; in particular, it is shown that under appropriate conditions the string problem mentioned above has a unique solution.
A detailed discussion of the contents of Borg's thesis is given in [4].

Borg attended the Tenth Scandinavian Mathematical Congress held in Copenhagen, Denmark, in July 1946 and delivered the paper Inverse problems in the theory of characteristic values of differential systems which was published in the Proceedings of the Congress. In this, Borg surveyed the results of his thesis and indicated extensions to problems which are not self-adjoint.

The publication of his thesis was not Borg's first paper, for he had published Über die Stabilität gewisser Klassen von linearen Differentialgleichungen in 1944. After the award of his doctorate, Borg was appointed as a docent at the Mathematical Institute of the University of Uppsala in 1945. He published papers such as Bounded solutions of a system of differential equations (1948), On a Liapounoff criterion of stability (1949), On the completeness of some sets of functions (1949), and Über die Ableitung der S-Funktion (1950).

In July 1950, Borg and his wife Gunborg sailed on the Gripsholm from Gothenburg, Sweden, to New York, USA, arriving on 27 July. They gave their address in the United States as c/o Lars Ahlfors, Harvard University, Boston. At this time Ahlfors was the Chairman of the Mathematics Department at Harvard University. Borg was going to the International Congress of Mathematicians held in Cambridge, Massachusetts from 30 August 30 to 6 September 1950. He was one of eight Swedish mathematicians attending this Congress; others included Arne Beurling, Lennart Carleson and Harald Cramér. At the Congress Borg presented the paper An inversion formula to the Mathematical Physics and Applied Mathematics Section on 4 September and it was published in the Proceedings of the Congress. Borg and his wife sailed from New York on the Stavangerfjord on 12 September 1950, the ship being bound for Oslo, Norway.

In 1953 Borg was appointed as Professor of Mathematics at the Royal Institute Technology in Stockholm. Jan Boman writes in [2]:-
After gymnasium I went to a physics program at the Royal Institute of Technology, KTH, in Stockholm. After graduation from KTH in 1955 I began higher studies in mathematics at KTH; a number of inspiring mathematics teachers at KTH such as Åke Pleijel, Tord Ganelius, and Göran Borg may have contributed to my choice of mathematics. In 1959 I took the degree of Teknologie Licentiat with mathematics as major subject. My advisor was Göran Borg, a former Beurling student who had made ground-breaking contributions to the inverse spectral problem for Sturm-Liouville operators.
Borg attended the International Congress of Mathematicians held in Amsterdam from 2 September - 9 September 1954. He gave the talk On Spectral Properties of a System of Infinitely Many Differential Equations. He gives the following abstract:-
The report will contain generalizations of spectral representation theorems and converse spectral theorems, corresponding to those, given by Martjenko, Gelfand, Levitali, Krein, Levinson and the present author for the second order ordinary differential equation.
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 for Mathematics and the Swedish Mathematical Society, Åke Pleijel and Göran Borg repecticely.

MathSciNet lists eleven papers by Borg, the last A condition for the existence of orbitally stable solutions of dynamical systems being published in 1960. From the early 1950s he seems to have moved away from mathematics research and taken on many administrative tasks. He was Dean of the Royal Institute Technology Section of Technical Physics 1958-1963, Head of the Department of Mathematics 1967-1968, and Rector of Royal Institute Technology from 1968 to 1974. He also made many important contributions outside the Institute, particularly in linking university applied mathematics with mathematical applications to industry. Let us note three of these in particular.

First we mention the Swedish Natural Science Research Council [7]:-
One of the more important government research councils in Sweden is the Natural Science Research Council controlling allocated federal research funds used primarily for temporary support, filling in "gaps" in the research front, strengthening weak points in research, providing "emergency" money, etc. Thus this particular research support is particularly significant because it is so carefully directed, rather than because of the money involved. The support to fundamental research in natural sciences and mathematics includes fellowships, scholarships, and specific institutional support. ... new Council membership as of July 1, 1962: ... Professor Göran Borg ... .
The Swedish Institute of Applied Mathematics was founded in 1971 with Borg playing an important role in forming the Institute and serving on the Board from the start. Germund Dahlquist tell us about this Institute in [3]:-
The Swedish Institute of Applied Mathematics, the ITM, was formed in 1971 to strengthen the Swedish efforts to develop applicable mathematics important to industry and government . ITM is an acronym for the Swedish name "Instituted for Tillampad Mathematik". The word "institute" is perhaps a misleading name for the strongly decentralized activity of the ITM. The institute operates mainly as a link for information and project cooperation between researchers, chiefly graduate students, in different specialties at the eleven universities of our country. (We include here the five Institutes of Technology. ) The population of Sweden is approximately 8,000,000. In the sixties, some Swedish industrialists realized that there was much to be gained from increased research in applicable mathematics with a strong link to real world problems in the operating, planning and construction departments of their companies. A great deal had recently happened, with vast potentialities for the future: think of computers, control theory, operations research etc. They felt that problems existed, for which no single company could establish resources sufficient for the development of the necessary new methods. At the same time, several university mathematicians also felt the need for improving the existing occasional contacts between university and industry. In particular Professors Ulf Grenander (mathematical statistics) and Göran Borg (mathematical analysis) were instrumental in the formation of the ITM. Grenander is nowadays active at Brown University, but he still stimulates the ITM with his ideas and comments on its activities. Borg is a member of the board of the Institute.
The third contribution of Borg's that we wish to highlight is his role as a Board Member of the L M Ericsson Telephone Company 1972-1980. While in that role he took part in the 7th International Teletraffic Congress held in Stockholm 13-20 June 1973. His role was mentioned in the Opening Address by the Chairman Dr C Jacobaeus [1]:-
As a representative of the Swedish Telephone Industries, the other main sponsor, we welcome Professor Dr Göran Borg of L M Ericsson. Göran Borg is a board member of L M Ericsson and he is also Dean of the Royal Institute of Technology here in Stockholm. Being a mathematician himself, Professor Borg has a good insight and knowledge of our problems and working methods. He has also taken a deep interest how these matters are treated at the university level.
At this Opening Session, there was the following 'Address by Professor Dr Göran Borg, Board Member of the L M Ericsson Telephone Company':-
Ladies and Gentlemen, As a representative for one of the main sponsors, the Swedish Telephone Industries, I want to underline the importance we in the industry lie on the research in telecommunication traffic. It is, of course, evident already from the fact that we over the years have spent and are spending a lot of money in this area. We think also that all efforts here have been most rewarding. As a basis for the system engineering it has given us the means to develop above all switching systems with a good economy. It has had an importance that has been felt throughout the industry and of course also for the many undertakings that the Swedish telecommunications industry has abroad. For myself, as Dr Jacobaeus said, as Dean of the Royal Institute of Technology and as mathematician it has been of a special interest to follow an activity that so evidently links pure science with technology. The interplay between the engineers and the mathematicians is very intense. In fact many engineers have acquired a good competence in mathematics and many mathematicians have got an expert knowledge in switching systems and their behaviour. I think that this kind of cooperation has moved the frontiers of telecommunication ahead. Certainly, we would like to see the same in other areas of technology. With these words I would like to join in with all other wishes of welcome to the delegates and their associates. I wish this congress very good success with its undertakings and hope that the results that will be reached here shall be to the future benefit for the development of telecommunications.
In fact Borg served on numerous other committees, for example, on 12 January 1968 he was appointed to the Committee for Television and Radio in Education (TRU Committee). At this time he was rector of the Royal Institute Technology.

In 1976 Borg retired from his professorship at the Royal Institute Technology, Stockholm, but continued to undertake several other roles, for example as a committee chairman of the SNS Development Council during 1978-1980. SNS, the Centre for Business and Policy Studies, was founded in 1948 by a group of young business leaders with a common belief in social science as a tool for economic and social progress. Since its foundation SNS has played a central role in shaping Swedish policy and bringing Swedish policy lessons to the international policy making community. Borg was one of the contributors to the publication Strategier för förnyelse: kunskap som nydanare av teknik och industri (Stockholm Studieförb, 1980). The problems of Swedish industry in the 1970s was a constantly recurring theme in the media. In March 1978 a committee was set up by the government to make recommendations and it published Vägar till ökad välfärd (1979):-
The engineering chapters were to a large extent based on analysis work within a working group led by Professor Göran Borg at the Royal Institute of Technology (KTH).
Borg was honoured with election as chairman of the Swedish Mathematical Society, serving 1957-1960, election to the Royal Swedish Academy of Engineering in 1962 and being awarded the Honorary degree of Doctor of Laws by the University of Dundee, Scotland, on 12 July 1971.

### References (show)

1. 7th International Teletraffic Congress, Opening session. https://itc-conference.org//_Resources/Persistent/e1efae8459e8ca8fd5007d9c4fa4befe43154282/jacobaeus731.pdf
2. B Boman, Mathematical Reminiscences, Inverse Problems and Imaging 4 (4) (2010), 571-577.
3. G Dahlquist, The Swedish Institute of Applied Mathematics: A Link Between Industry, Society and University, in M Zweng, T Green, J Kilpatrick, H Pollack and M Suydam (eds.), Proceedings of the Fourth International Congress on Mathematical Education (Springer Science & Business Media, 2012), 79.
4. L Garding, Göran Borg, in Mathematics and Mathematicians. Mathematics in Sweden before 1950 (American Mathematical Society, Providence, R.I., 1994), 266.
5. Göran Lars Borg, prabook.com. https://prabook.com/web/goran_lars.borg/1299279
6. H Pollard, Review: Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe. Bestimmung der Differentialgleichung durch die Eigenwerte, by Göran Borg, Mathematical Reviews MR0015185 (7,382d).
7. Swedish Natural Science Research Council, in International Science Notes (Department of State, Bureau of Oceans and International Environmental and Scientific Affairs, 1963), 10-11.