# Helge Arnulf Tverberg

### Born: 6 March 1935 in Bergen, Norway

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**Helge Tverberg**was interested in combinatorial problems from a young age and he recalls in [6]:-

Tverberg was educated in Bergen and, in 1954, he entered the University of Bergen aiming to study mathematical sciences. He passed the examination in descriptive geometry in the autumn of that year which qualified him to teach projective drawing at a gymnasium. At this stage, however, he could not continue with his studies since he had to spend the next sixteen months undertaking compulsory military service. There was an advantage in this, however, for by the time he restarted his mathematical studies, Ernst Selmer was teaching in Bergen.... I recollect participating in a competition to see how many Norwegian words could be formed from the letters in the brand name MELANGE - you were allowed to use each letter at most once, so E could be used twice, and I recall trying to estimate how many 'words' you would need to consider. There had also been some calculations that engaged me when the football pools started in Norway in1948; and my father and I had taken great pleasure in the famous weighing problem with12balls.

In 1956 Ernst Sejersted Selmer (1920-2006) was appointed as the professor of mathematics at the university of Bergen. At this time he was the only professor of mathematics at Bergen. Selmer had received his doctorate from the University of Oslo in 1952 and became a lecturer there before the appointment to Bergen. Selmer did important research in number theory and cryptoanalysis but is perhaps best known for the Selmer group which played an important role in Andrew Wiles' proof of Fermat's Last Theorem. There were no courses on combinatorics at Bergen at this time but in 1956 Tverberg discovered Dénes König's graph theory textbook

*Theorie der Endlichen und Unendlichen Graphen*Ⓣ (1936), the first textbook on graph theory which deals with topics such as Euler trails, Hamiltonian cycles, mazes, trees, directed graphs and factorisations. He explains in [6] how he came across the book:-

Tverberg was awarded the 'cand.real' degree by the University of Bergen in December 1958 and, on 1 January 1959, he was appointed as a senior lecturer at the University. Already by this time Tverberg had two papers in print, namelyFrom1956, I gained free run of the Departmental Library, which was then so rudimentary that it seemed almost a miracle that it included König's book on graph theory published in1936- that book gave me one of my best Summer holidays ever. ... There were perhaps no more than a hundred books in the Library, and I went so far as to draw up a schedule of how much time I should spend reading each one. I must admit that I never realised this youthful plan, but sadly I can no longer find that list. In1958, I completed my Master's degree, under the supervision of Professor Selmer.

*On two inequalities by S Selberg*(1958) and

*A new derivation of the information function*(1958). Douglas Rogers writes about this last mentioned paper in [4]:-

For a list of papers by Tverberg, see THIS LINK.Claude Shannon and Warren Weaver had presented a characterization of the information function in their book 'The Mathematical Theory of Communication', published in1949, reflecting Shannon's war-time research. In the Russian school, Khinchin had published a characterisation under additional assumptions in1953, and Faddeev had been able to pare back these assumptions to those of Shannon and Weaver, in a paper in1956, with both articles cited in the literature in English by1958. Tverberg is familiar with this background, and moves deftly to prove a stronger characterisation by weakening the clearly technical assumption that the information function is continuous, defending this move with judicious dispatch: "If my weakening of the conditions is insignificant from an information-theoretic point of view, I do not think it is so from a purely mathematical one." The whole exercise only takes a couple of pages; and Tverberg's pioneering judgement has been amply confirmed - a survey 'The fundamental equation of information and its generalizations' by W Sander, in1987of generalisations of the functional equation satisfied by the information function ran to108references; and more recently a whole book 'Characterization of Information Measures' by B Ebanks, P Sahoo and W Sander(1998)has been devoted to the topic.

In 1960 Tverberg published in

*Math. Scand.*the paper

*On the irreducibility of the trinomials*

*x*

^{n}±

*x*

^{m}± 1. He writes in the introduction:-

For details of the Mathematics Institute of the University of Bergen and those who worked there, see Tverberg's description at THIS LINK.In an earlier paper in this journal, Selmer studied the polynomialsx^{n}±x^{m}± 1. He gave a complete discussion, as to the possibility of factorisation in the rational field, of the case m =1. The purpose of the note is to extend his results to the general case0 <m<n. I want to express my gratitude towards Professor Selmer, who called my attention to the problem, and whose active interest in it was of great help to me while I was working on the solution.

Tverberg went to London, England, in 1961 to attend the instructional conference on Functional Analysis held at University College. He found the lectures fascinating and was particularly interested in Kelly's Theorem named after Paul Kelly). He thought about this result when back in Bergen and was fascinated by one particular application of it namely:

Now, thought Tverberg, this would follow easily if I could prove thatLetSbe a set of3Npoints in the plane. Then, there is a pointp, not necessarily inS, such that every half-plane containing p contains at leastNpoints fromS.

*S*could be split into

*N*triplets so that the

*N*triangles so formed would have a common point

*p*. He failed in his attempts to prove this and he was still puzzling over this idea in August 1962 when he attended the International Congress of Mathematicians held in Stockholm where he met Bryan J Birch (1931-) and Hallard T Croft (1936-) [6]:-

This now presented Tverberg with a challenge on which, by 1963, he had made some progress by proving the 3-dimensional case. Sadly, however, there was no way his 3-dimensional proof would extend to higher dimensions. He applied to the Meltzer Fund for a scholarship to spend time in England undertaking research in 1964.As our group was breaking upon a street corner after a pleasant meal, I thought to mention my problem to Croft, who had declared his interest in geometry. They laughed, and told me that Birch had already solved the problem. But they added that the further challenge of the obvious analogue in higher dimensions remained open.

Tverberg gives some details of the Meltzer Fund at THIS LINK.

In England in 1964 he visited Bryan Birch at the University of Manchester and Richard Rado at the University of Reading. He writes [6]:-

No two people see the same event in identical ways [8]:-I recall that the weather was bitterly cold in Manchester. I awoke very early one morning shivering, as the electric heater in the hotel room had gone off, and I did not have an extra shilling to feed the meter. So, instead of falling back to sleep, I reviewed the problem once more, and then the solution dawned on me! I explained it to Birch, and, after an agreeable day of mathematical conversation with him, returned to Norway to start writing up the result.

The result of this flash of inspiration led to Tverberg publishingBirch disagrees on this: he remembers that Tverberg was not all that interested in explaining his solution, and rather more in seeing a bit of England on his last day.

*A generalization of Radon's theorem*(1966) which contains what today is known as Tverberg's Theorem:

This remarkable result has proved fundamental to a whole area of research which is surveyed by Imre Bárány and Pablo Soberón in [2]. They write:-Given(r- 1)(d+ 1) + 1points in^{d}, there is a partition of them into r parts whose convex hulls intersect.

Although this is probably the best known of Tverberg's results, he has produced many more and we point the reader to the beautiful papers [4] and [6] in which many are described.Tverberg's theorem has been a cornerstone of combinatorial convexity for over50years. Its impact and influence is only comparable to that of the famous and classic theorems of Carathéodory and Helly. This gem lies at the crossroads of combinatorics, topology, and linear algebra, and it continues to yield challenging and interesting open problems.

For a list of papers by Tverberg, see THIS LINK.

In [1] Imre Bárány tells us something of Tverberg's character:-

Tverberg's wife is mentioned in the above quote; she is Sonja Birgit Olaug Tverberg.I met him in person first time in1980in Oberwolfach. He is tall, very Scandinavian, has a good sense of humour, and is an excellent storyteller. He is interested in all kinds of mathematics. I am impressed. He is an extraordinarily nice, unassuming person apart from being a fine mathematician. He invites me to Bergen and three years later I enjoy his warm hospitality for two weeks there. I get to know his family(wife Sonja and four children). We go for an excursion around Bergen, to the mountains and to the fjords. We take a train, then a bus, then we go by boat, and train again plus plenty of walking and mathematical discussions. I learned later that he prefers train rather than plane as a mode of transportation. On the train he has plenty of time to ponder a mathematical problem. That's why several of his proofs are associated with specific train routes. Helge Tverberg is friendly and generous, he loves intriguing questions and puzzles. He had been running a mathematical puzzle column in the newspaper Bergen Tidende for many years. He knows and loves art and literature and music, jazz in particular. He spent a couple of months in Reading in1966and mentions in his recollection A combinatorial mathematician in Norway: some personal reflections(2001)that "simply being abroad was great too, and I often went into London on weekends, taking much enjoyment in live performances by Duke Ellington's Orchestra, Ella Fitzgerald, and others, which would not have been so common in Bergen in those days."

Douglas Rogers relates several charming anecdotes in [4] concerning Tverberg. Here is one of them:-

In 2001 a part ofThere is a lively awareness of just what students are most likely to remember in a story Tverberg tells against himself. Once, on an autumnal afternoon, when attention might have been flagging, he broke off his exposition to describe his method of catching the flies that tend to seek sanctuary in the warmth of Norwegian homes at that season - you come up behind them with the nozzle of the vacuum cleaner, and just suck them in, which neatly avoids breaking things in the vain attempt to swat them. And he added that, whatever the class might remember of the course, they would be sure to recall this lesson in fly catching. Years later, someone vaguely familiar greeted Tverberg while out skiing, and it turned out that they had been in his class. This erstwhile student confessed regretfully that he could not claim to remember much about the course, but he still recalled that afternoon Tverberg had told them how to catch flies .... But a student attracted to mathematics might well reflect afterwards that theorems too can be caught by a mind alive to the potential of such stratagems.

*Discrete Mathematics*was devoted to papers to honour Tverberg. Douglas Rogers writes [5]:-

The idea of a volume of 'Discrete Mathematics' in honour of Helge Tverberg was not just to honour Helge Tverberg, but rather more to prepare for something like a Norwegian Research Assessment Exercise. ... The biographical pieces also had a dual purpose: certainly to record the key mathematical events in Helge Tverberg's life; but more to paint a contemporary picture of what it might be like to engage professionally in mathematics, a picture on a more human and accessible scale than that of giants of a distant past, such as Abel or Lie or Sylow, which otherwise would not exist. Helge Tverberg, however, was less than fully cooperative, dragging his feet: for someone who was a witty and erudite conversationalist in English, Helge Tverberg's finished composition[6]was scrappy and ill-written. So, the only words which can truly be said to be his are those of the abstract. Indeed, reading the finished composition, Helge Tverberg tried to corral me as co-author, saying "No one is going to believe I wrote this." How wrong he has proved to be. I do not think it could have been more successful even if it had been written in Norwegian.

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

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