## Helge Tverberg on the Bergen Mathematical Institute

We present below a modified version of Tverberg's talk, supplemented with material from the paper:
H Tverberg, A combinatorial mathematician in Norway: some personal reflections. Selected papers in honor of Helge Tverberg, |

**Helge Tverberg: Talk marking the Mathematical Institute's relocation to the Natural Sciences Building 2015.**

On 13 March 2015, the employees, together with some guests, gathered to mark the fact that the entire institute had now been relocated for the first time in many years. I then gave a short talk in which I tried to give an impression of the Institute's early history and it is presented here in written form.

We begin with something that falls outside the scope, but is so interesting that it should still be included: Ljunggren's and Skolem's stay in Bergen in 1930-38. Christian Michelsen, Prime Minister in 1905, died in 1925 and donated his great fortune (he was a ship owner) to found a research institute that started in 1930, the Christian Michelsen Institute. Thoralf Skolem (1887-1963) was a researcher there, while Wilhelm Ljunggren (1905-1973), who had had Skolem as supervisor, was a teacher at the Ulrike Pihl secondary school for girls in Bergen. In this period, Skolem wrote important articles on number theory, logic and algebra, while Ljunggren undertook research in the free time his school work left him and, in 1937, he was awarded a doctoral degree for his thesis on Diophantine equations. It is reasonable to assume that they had contact with each other, but I do not know anything about that. In 1938, both returned to Oslo, Skolem as professor, and Ljunggren as associate professor at the Hegdehaugen school.

Let us record at this stage a little of Skolem's combinatorial contributions. Skolem had an explicit interest in combinatorial results, for example, giving the second proof of Ramsey's theorem, even if his motivation, like Ramsey's, stems from questions in logic. (It is quite interesting comparing the various proofs of Ramsey's theorem: Ramsey's original proof is an excellent instance of how one can refine the structure of a result so as to be able to prove it in many small steps; but Skolem's proof is simpler; and subsequent proofs of Erdos and Szekeres and Erdos and Rado illustrate how a simple change in strategy can effect a reduction in numerical bounds by several orders of magnitude.) But Skolem had an early, substantial interest in combinatorial problems per se, publishing a lengthy account *Untersuchungen Ouber einige Klassen kombinatorischer Probleme* in 1917; for example, Skolem includes a catalogue of connected graphs on up to 8 vertices, each of degree at most 3, clearly with an eye to what we would recognise as design-theoretic properties. He returned to this theme in 1927, with an exposition *Om en del Kombinatoriske problemer* written in connection with Netto's *Lehrbuch der Combinatorik*. Another paper *Über einige besondere Tripelsysteme mit Anwendung auf die Reproduktion gewisser Quadratsummen bei Multiplikation*, in 1931, examined the construction of Steiner triple systems. As interest in the construction of block designs picked up in the 1950s, Skolem realized that there might be interest in some constructions he had previously thought were not new because of their simplicity, and he published two papers *On certain distributions of integers in pairs with given differences* and *Some remarks on the triple systems of Steiner* on the subject in 1957 and 1958. Fundamental to Skolem's approach is the simple idea of partitioning the set of integers {1, 2, ..., 2*m*} into pairs {*a*_{i}, *b*_{i}}, 1 ≤ *i* ≤ *m* such that *a*_{i} - *b*_{i} = *i*, 1 ≤ *i* ≤ *m*. This idea has intrinsic appeal, but, as it happens, these partitions and their natural variants can be used in the construction of a host of structures of combinatorial interest, besides the Steiner triple systems for which Skolem originally wanted them. Although Skolem was correct in his hunch that others might have considered something similar, such are the quirks of mathematical fame and fashion that these Skolem sequences, as they are now known, have spawned a vast literature, complete with internet websites, which threatens to overshadow the rest of Skolem's oeuvre.

In 1948, the University of Bergen was established. The politicians were in a hurry, for the rich Lauritz Meltzer had set a deadline. Meltzer (1861-1943) was a military officer in Bergen turned highly successful investor and entrepreneur who had donated his substantial estate to a future University in Bergen, provided that such an institution be established by a certain date, and after it expired his fortune would go to his relatives in Fredrikstad instead of joining the Meltzer Foundation. This contingency to Meltzer's donation may well have prompted the Norwegian Government to set up the University in Bergen as early as 1948, only five years after his death, as the founding of a university in Bergen had been mooted since at least 1918. From the University's foundation, the Meltzer Fund has been extremely valuable in supporting researchers and students at the University of Bergen. In the spring of 1949, the Mathematical Institute was established, with Ljunggren as the first professor. For the first eight years, only secondary level education was offered since there was no capacity for more, even though assistant teachers were used.

Only in the summer of 1953 were several scientific positions filled when Oddvar Bjørgum and John Olav Stubban became lecturers in applied and pure mathematics respectively. Bjørgum was a trained meteorologist, but eventually became more and more a researcher with, for example, turbulence as an important interest. Stubban was a pure mathematician with a doctorate in geometry from Bergen. My own contact with the Institute started in the autumn of 1954 when I managed to take the examination in descriptive geometry (which was necessary to be able to teach projective drawing at a real gymnasium) before I began 16 months of military service. In December 1958 I became a cand.real with pure mathematics as a major subject and Ernst Selmer as my supervisor. There was still no postgraduate education at Bergen, so I mostly had to read the syllabus on my own. As of 1 January 1959, I was employed as a senior lecturer and a little later Asbjørn Kildal, Kjell Overholt and Kjell Kolden were employed as university lecturers in, respectively, applied, applied and pure mathematics.

It would be a step too far to say anything about all the staff, but Kolden and Stubban deserve some words. Kolden has an unbeatable record as he spent 2^{1}/_{2} years at the time (before World War II) becoming a cand.real, with 4 minors (he read two as self taught) and mathematics major. He had a good memory, so he learned both Greek and Latin and otherwise had great general knowledge, which led to him, at one of our Christmas parties, winning a guessing competition where he was one team and the rest of the institute the other. He had originally been a school teacher, but was displeased with that, so it was probably good for him to get into the University of Bergen, where he was popular with the students. In 1984, he received the King's gold medal.

Stubban was an expert in geometry and an excellent lecturer. Here, however, I'll tell you about the time his former teacher from Belgium, Lucien Godeaux (1887-1975), was a guest lecturer in Bergen and it looked as if there would only be three listeners. Then a fourth person, Helge Dalseide, who the mathematicians ate lunch with each day (he was not at all a mathematician, but a research assistant for a geophysicist), was asked to come along. The unfortunate thing was that Godeaux had the custom of primarily addressing a particular person among the audience, and this became the assistant. He had to sit through several lectures, nod occasionally and give the impression of understanding. Later he said he felt he might have become a good actor. Godeaux was a very productive mathematician, who had produced over 1,000 books and articles.

Bjørgum (who become a professor) unfortunately died in 1961, and then in 1963 his chair was filled by Sigve Tjøtta who had been a lecturer in Bergen since 1960. In what follows we will go into more detail concerning the efforts of both Selmer and Tjøtta as two very important people for the Institute's activities. Ernst Sejersted Selmer was employed as a professor in Bergen in 1957. Then he had been a senior lecturer in Oslo since 1946. He had already made himself known as a computer expert in England (where he fled during the war), and he had much to say for the development of computer technology in Norway. In 1950 he had had a longer stay in the United States where he, among other things, was the first person in Princeton who wrote a program that went through the computer there at the first attempt. He came home from the United States with several patents in this area in his pocket. In Princeton there were many stars; when Selmer got there, he was shown around by Albert Einstein (who was almost run down by Mrs Selmer later). It also included people like von Neumann (the mathematician behind the atomic bomb) and Atle Selberg. Sieve methods had been refined in the hands of Atle Selberg, first acclaimed for his work on an elementary proof of the prime number theorem, and the foremost mathematician from Norway in the last century, although for most of his career he has been at the Institute for Advanced Study in Princeton. Our Norwegian star was holding an appointment in Oslo when he received an invitation from the Institute for Advanced Study in Princeton.

When Selmer came to Bergen he went to the central administration at the university and asked if there was anything special he could contribute; it says a little about his attitude which became extremely valuable to the quite young University of Bergen. He contributed a great deal: the introduction of a new curriculum at the University of Bergen, building up the computing department at the University of Bergen, national efforts in the same area, and building up the district college system. He was also vice-dean and dean for many years, and also editorial secretary for the popular scientific *Nordic Mathematical Journal* for many years. When at last he gave it up, there were eight mathematicians at Kristiansand District College who shared the job between them. Selmer's thesis concerned Diophantine equations of a particular type, which also includes Fermat's equation. When Andrew Wiles proved that Fermat's equation does not have integer roots, and thus solved a several hundred-year-old problem, a group that was later called the Selmer group was important, and there was of course a great satisfaction for Selmer and his wife when they were involved in handing over a great prize, the Rolf Schock Prize, in Stockholm. (There is an even more honourable award for mathematicians, the Field Medal, but Wiles was one year too old to get it).

Selmer, a much valued colleague at the University of Bergen for 31 years, celebrated his 80th birthday in February 2000 - a long video interview with him was screened during the 50th anniversary celebration of the Department of Mathematics, in 1999, creating such interest that a shorter version was later shown as a television segment. The television programme featuring him focussed on the way his work touches the life of all Norwegians through the system of Norwegian national identity numbers he devised so that it would be sensitive to common transcription errors like transpositions and repetitions. His influence here has been substantial, giving rise to a thriving school of coding theory at the University of Bergen which has achieved international recognition.

One important thing that Selmer did was that he created a technical position for Svein Mossige. He would do research but also, instead of teaching, helped other employees with computer calculations. I even had a great deal of benefit from this on several occasions.

In much of the time all of the above was happening, Selmer delivered excellent lectures and had many graduate students. He also embarked on new areas of number theory, inspired by a German mathematician, Gerd Hofmeister, who had previously taken a degree in Bergen and later maintained contact through mutual visits. In these areas, Selmer himself, with legitimate pride, believed that he had written two doctorates after the age of sixty.

Let me also mention another eminent mathematician, whom Selmer brought to Bergen, Oddmund Finn Kolberg (1931-2016). He was an assistant lecturer, lecturer and then professor. He was an excellent lecturer and concentrated mostly on mathematics, unlike Selmer. His field was also number theory, but of a completely different type, which we will not go into further here. For a while it seemed that he wanted to go into new areas of mathematics, but one autumn day in 1969, our office secretary, Mrs Strand, rushed into my office and told me that Kolberg had resigned. We tried to persuade him to stay, but it didn't work. He had several good students who were later employed at the department: Hans Fredrik Aas, Torleiv Kløve, Øystein Rødseth and Reidar Erevik (who died far too young), while the others are now retirees, like my student, Oddvar Iden, also now retired.

But now it's time to move to Tjøtta. He was much younger than Selmer and could for many years devote himself to the purely professional tasks. He was particularly interested in theoretical acoustics and together with his wife, Jaqueline Naze Tjøtta, who was also a mathematician, he was awarded a prize from the French Academy of Sciences in this area. They had many graduate students, and seven of these were later employed at the department: Gerhard Berge, Kristian Dysthe, Knut Eckhoff, Leiv Engevik, Magne Espedal, Aslak Svardal and Alf Øien. He was a dean for many years and had important positions in the Norwegian Research Council.

Selmer and Tjøtta were both knights of St Olav's Order of the First Class. It is strange to note that they were both out of action for critical reasons when they had been honoured: in 1964, the Tjøtta couple were involved in a serious car accident, returning home from a honeymoon in France, and in 1966 Selmer fell down from the roof of his cabin in Hvidsten. Usually you think of mathematics as a harmless profession, but in your spare time things can go wrong. It was very good for both the families and for the University of Bergen that things turned out well for them.

A third person who should also be mentioned is Sven Nissen-Meyer, the first professor of theoretical statistics at the University of Bergen, appointed in 1967. As a young man he was interested in mathematics and physics, but in the interwar period it was impossible to get a position other than school teaching here in Norway; he became a doctor instead and in the USA he slowly progressed through medical statistics into theoretical statistics. In 1967 he came from Berkeley to Bergen and began to build up a statistical environment with Trygve Nilsen as the first university lecturer. His story says something interesting about the difficult interwar period in Norway, because he was not the only one of his kind: the number theorist Trygve Nagell had to go to Uppsala to get a job, algebraist and graph theorist Øystein Ore, with grade 1.0 in his main subject, became professor at Yale, and even after the war, Atle Selberg was bypassed.

In 1959, as mentioned before, Kjell Overholt was employed as a senior lecturer in applied mathematics. He became one of the pioneers in computationally oriented mathematics, and in 1984 a part of the Department of Mathematics became a separate Institute for Informatics. The boundary between these subjects is somewhat fluent, and some employees may just as well belong to either one of the two departments. It is therefore only reasonable that there is still a lot of contact between the two departments, even though they are now in different buildings.

In conclusion, I would like to say a little about the office staff. The Department of Mathematics could hardly have existed without such a skilled one. In this we have been very fortunate in every year; here I would just mention a few: Mrs Strand (her first name Inga was never used), Gerd (Kristiansen), Marie (Skorpa Nilsen) and Mrs (Gidske) Tresselt. Mrs Tresselt worked mainly for the statisticians, so I have nothing special to say about her. Mrs Strand was the first employee and was long thought of as the Institute's "mother". She was very active, also on the social side, and for a long time organised (with helpers) the Christmas party for the employees (and their spouses) which for many years was held at Kronstad Hovedgård, a charming place. Later Gerd came to, also a very nice and helpful lady. Marie, who has been employed for over 25 years and is now an office manager, has also proved to be a pure "gold discovery".

PS. 1. The talk started with a drawing on the board that showed where the department's main site has been situated over the years. Here we mention only the addresses in order: Allégaten 33, Nygårdsgaten 114, Allégaten 34, Johannes Bruns gate 12, Allégaten 41. A glance at the map shows that we have kept ourselves largely the same area all the time. (Note: Allégaten 33 disappeared when no. 41 was built.)

PS. 2. An afterthought: I forgot in the first part to say that Bjørgum also left other traces behind him. When he returned home from Madison's research team in the late 1950s, he was thrilled with Walter Rudin's book *Principles of Mathematical Analysis *(Rudin was employed at the University of Madison) and got it introduced to us where it is still in use. A clear record!

JOC/EFR January 2019

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