A combinatorial mathematician in Norway: some personal reflections

In the paper, I 'rst try to give some impression of Norwegian contributions to combinatorics in the 20th century. This is followed by some remarks on my own combinatorial experiences. c © 2001 Elsevier Science B.V. All rights reserved. [It] would bring to this country something absolutely new, namely the historic and humanistic side of mathematics. The success of the Institute of the History of Medicine at Johns Hopkins with its liberalizing in!uence over the faculty as well as the students encourages me to try this novelty. Mathematics is something more than an a4air of today and yesterday. It is part of the cultural history of the race. A. Flexner to O. Veblen, 8 September, 1933 1. Mathematics and historiography For G.H. Hardy, mathematical achievement was the most enduring of all; as he put it with characteristic power to provoke in his A Mathematician’s Apology [13] published in 1940, “Archimedes will be remembered when Aeschylus is forgotten”. Hardy’s friend J.M. Lomas challenged Hardy on the issue of personality attaching to mathematical fame one day as they were passing Nelson’s column in Trafalgar Square: would Hardy prefer a statue so high as to be invisible, or one low enough to be recognized? Or, as we might perhaps put in terms of our civic statues here in Bergen, the lofty eminence of Christian Michelsen as against the homely scale of Edvard Grieg. While Hardy chose the 'rst alternative, it is interesting to 'nd that, already in 1933, Abraham Flexner was prepared to advocate including, in e4ect, the second in writing to Oswald Veblen about plans for an Institute for Advanced Study in Princeton, although, in fact, the history of science was not to be taken up by the Institute until the 1950s and, if Hardy’s own features remain recognizable today, it must surely be because his Apology has become a classic of its kind. E-mail address: [email protected] (H. Tverberg). 0012-365X/01/$ see front matter c © 2001 Elsevier Science B.V. All rights reserved. PII: S0012 -365X(01)00222 -9 12 H. Tverberg /Discrete Mathematics 241 (2001) 11–22 Times change, and, with them, standards of historiographical fastidiousness, so that the fat volumes of oKcial histories based on the public doings of monarchs and parliaments have given way to more multifarious, more tentative articles on the oral history of the ordinary person. So too, in scienti'c publications, the traditional festschrift, such as the present volume, or biographical memoir for a learned academy, has come to be supplemented by a more diverse range of personal contributions, even in the last couple of years a spate of mathematical novels, all largely inspired by a wish to make diKcult topics more accessible in an age of mass education. Working in combinatorial mathematics, I have found such recent books as Erdős on Graphs: His Legacy of Unsolved Problems, by Fan Chung and Ron Graham [7], Proofs from The Book, by Martin Aigner and GO unter Ziegler [1], and Bill Tutte’s Graph Theory As I Have Known It [31], all with their engaging touches of personal history, most refreshing reading—truly, as Anthony Hilton observed in a review [14] of the 'rst of these, “mathematics lives as much through the people who do it as it does through the theorems they prove”. So, although my initial instinct, when it was ventured that I might contribute some recollections of combinatorial mathematics in Norway, including my own working life, was that this would be rather precious and pretentious, on second thoughts it seemed that at least making the attempt would be in keeping with this more liberalizing spirit that Flexner anticipated. At least, I can take this opportunity to thank all those who have contributed more seriously to this volume. 2. Norwegian contributions to combinatorics Combinatorial mathematics, while having an ancient pedigree intertwined with and permeating several other branches of mathematics, has, until the last two or three decades, occupied a comparatively humble position on its own. Thus, it is no surprise that a country with a small population like Norway has no great combinatorial tradition—what is most remarkable, of course, is how well Norway has done in number theory, but that is another story. Still, it is worth mentioning here a few names well known more generally, if not primarily for combinatorial research. Of these, the 'rst that comes to mind is Axel Thue (1863–1922). Indeed, Thue’s work in logic might well be classi'ed as combinatorics of words. He also tried cycling his way through the four-colour problem: of a morning, he would try to prove that four colours suf'ced; but, as the day wore on, and without success, he would revert to seeking a counterexample. Thue made a notable contribution to discrete geometry in giving the 'rst proof that the most dense packing of congruent circles is the standard one. Likewise, he hit on a most ingenious pigeon-hole argument in giving a proof of the classic result that primes of the form 4n+ 1 can be written as the sum of two squares. First of all, using the pigeon-hole principle, he showed that each residue class a (mod p), with p prime, can be written as x=y, with x and y both in absolute value less than √ p. Secondly, he notes that for p ≡ 1 (mod 4) there are solutions of the congruH. Tverberg /Discrete Mathematics 241 (2001) 11–22 13 ence a2 ≡ −1 (mod p), as follows from the well-known fact—Wilson’s theorem—that (p− 1)! ≡ −1 (mod p) for all primes p. Thus, for p ≡ 1 mod 4 and a2 ≡ −1 mod p, one gets x2 + y2 ≡ 0 mod p, with 0¡x2 + y2 ¡ 2p, so that x2 + y2 =p. This proof is perhaps the neatest possible of Fermat’s ‘two square’ theorem, although no doubt it does not quite invalidate the spirit of Hardy’s contention, in his Apology, that “there is no proof within the comprehension of anybody but a fairly expert mathematician” (a view recently challenged, in [8], on behalf of a proof by H.J.S. Smith; Thue’s work [30] goes unmentioned in this article, although it will be acknowledged explicitly in the forthcoming second edition of [1] which, for that matter, also went unnoted in [8]). Two further names in the front rank are Viggo Brun (1885–1978) and Thoralf Skolem (1887–1963), both celebrated in number theory, with Skolem distinguished for contributions in logic as well. However, both also contributed to Netto’s Lehrbuch der Combinatorik [20], although Skolem was to complain in [28] that the second edition, and so their notes that had appeared in it, were little known. Brun’s best known work in number theory employs the Sieve of Eratosthenes, which is clearly a combinatorial device. Sieve methods were further re'ned in the hands of Atle Selberg (1917– ), 'rst 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. However, despite the general upsurge of interest in combinatorial questions and methods, Atle Selberg never seems to have strayed into anything more overtly combinatorial—surely a great loss for combinatorics. Skolem had a much more 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 re'ne 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 Erdős and Szekeres and Erdős and Rado illustrate how a simple change in strategy can e4ect 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 [25] 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 recognize as design-theoretic properties. He returned to this theme in 1927, with an exposition [26] written in connection with Netto’s Lehrbuch. Another paper [27], 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 [28,29] on the subject in 1957 and 1958. Fundamental to Skolem’s approach is the simple idea of partitioning the set of integers {1; 2; : : : ; 2m} into pairs {ai; bi}; 16 i6m such that ai − bi = i; 16 i6m. This idea has intrinsic appeal, but, as it happens, these partitions and their natural variants can be used in the construction of a host 14 H. Tverberg /Discrete Mathematics 241 (2001) 11–22 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. While Brun and Skolem remained largely in Norway, a somewhat younger contemporary, Iystein Ore (1900–1968), spent most of his active life in the United States, as it was diKcult to obtain academic employment in Norway in the 1930s. He was originally an algebraist, but he took to combinatorial mathematics, most especially graph theory, on which he wrote two books that enjoyed wide circulation, and, in fact, he also reviewed Netto’s Lehrbuch. He was active in research on the four-colour problem, pushing up the size of the colourable maps, and producing a book on the topic. I did meet him, but only just, at the 15th Scandinavian Congress of Mathematics: we were all much looking forward to his talk on the four-colour conjecture; so it was a great shock to learn instead of his sudden death. Let me conclude this brief history by coming up to the present with Ernst S. Selmer (1920– ), a much valued colleague at the University of Bergen for 31 years, who 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. He too has primarily been concerned with number theory, notably Diophantine equations; and it was a great pleasure to him that the Selmer group played an important role in the proof of Wiles and Taylor of Fermat’s last theorem. However, he also worked on shift registers and various other topics at the interface between combinatorial mathematics and the more traditional areas of algebra and number theory. Selmer revealed something of the application of his work to cryptoanalysis in a fascinating personal talk [24] at the EUROCRYPT ’93 meeting at Lofthus, Norway, in May 1993—indeed, 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 in!uence here has been substantial, giving rise to a thriving school of coding theory at the University of Bergen which has achieved international recognition (Tor Bu has recently contributed a short memoir [6] that touches on some aspects of this development). It is most gratifying that several members of this group have been kind enough to contribute papers to this volume. Moreover, Selmer’s later work in elementary additive number theory has a strong combinatorial interpretation in terms of changing coins and using stamps to make up postage rates. The extent of this research in Bergen can be judged by the contributions Selmer and another long-time colleague, Iystein RHdseth, have made to the revision of Richard Guy’s compilation, Unsolved Problems in Number Theory [12]. Interest in this topic remains active, as the recent paper [21] of Svein Mossige, also here in Bergen, shows. H. Tverberg /Discrete Mathematics 241 (2001) 11–22 15 3. Personal re ections Naturally, I knew nothing of all this when starting my mathematical studies, at the University of Bergen, in 1954—it is true that, for the 'rst three years of my life, Skolem was in Bergen, holding a position at the Christian Michelsen Institute, from 1930 to 1938, but this is not a blessing of which I was conscious (although I have discovered that I now live quite close to where he did at that time, then largely open country, but now built-up). Indeed, Selmer was only appointed professor of mathematics in Bergen in 1957, and it was not until 1978 that the University o4ered courses in combinatorial mathematics. Yet, clearly combinatorics was somehow in the air, for 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 in 1948; and my father and I had taken great pleasure in the famous weighing problem with 12 balls. From 1956, I gained free run of the Departmental Library, which was then so rudimentary that it seemed almost a miracle that it included KO onig’s book [17] on graph theory published in 1936—that book gave me one of my best Summer holidays ever. Moreover, this acquaintance with KO onig’s book served me well by way of introduction years later as I got to know members of the Danish school of graph theory that grew up around Gabriel Dirac, associations that I have always found most agreeable as well as of great mathematical bene't. I do not think that I ever really met Dirac to speak to: I had been looking forward to presenting a proof [37] of Kuratowski’s theorem at a conference in 1985 in Dirac’s honour, remembering that he had given one of the 'rst proofs in a joint paper [10] in 1954, when the distressing news of his untimely death reached me; and, sadly, that conference was to become a memorial to him. (By the way, KO onig is familiar, in [17], with Skolem’s work in connection with Netto’s Lehrbuch, but does not cite [25], with its catalogue of graphs.) 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 realized this youthful plan, but sadly I can no longer 'nd that list. In 1958, I completed my Master’s degree, under the supervision of Professor Selmer, and started work as a lecturer at the university. One was expected to do research, but there was no pressure, and certainly no system of supervision. Thus, I found myself in a rather free, but potentially dangerous, situation, being at liberty to follow my own whims in reading and research. So it was, that, around 1960, I became aware that van der Waerden’s problem, on the minimum permanent of a doubly stochastic matrix of given order, which I had read about in KO onig’s book, was, in fact, still unsolved—van der Waerden had posed it as a problem, although it has come to be referred to more commonly as his conjecture. I thought to reduce the question to a combinatorial problem about certain spatial matrices. It was encouraging that this worked nicely for 3 × 3 matrices. But, 16 H. Tverberg /Discrete Mathematics 241 (2001) 11–22 alas, my combinatorial approach failed for higher orders, as we found by computer studies somewhat later. My attempt [32] was not completely in vain, however, as it attracted the attention of ThHger Bang, in Denmark, who went on to produce an excellent partial result. In the end, the conjecture was settled by Falikman, in [11]: Falikman’s proof was wonderful, using only simple mathematics, clever tricks, and an absolute independence of mind as to what had previously been tried. But Bang replied by showing how Falikman’s proof could be modi'ed to con'rm that the minimum is attained uniquely by a matrix having all entries equal—this is not so well known, as Bang only published it in Danish [2]. I had some greater success with a problem I devised for myself. In 1961, I participated in an instructional conference on Functional Analysis, held at University College, London. The conference included some supporting lectures on classical convexity theory. I found this material fascinating, and read up on it more back in Bergen. Helly’s Theorem was especially fascinating, and, in my reading, I came upon the following application. Let S be a set of 3N points in the plane. Then, there is a point p, not necessarily in S, such that every half-plane containing p contains at least N points from S. It struck me that this would follow simply if it were always possible to split S into N triplets so that the N triangles so formed would have a common point p. For, a half-plane containing p would contain at least 1 vertex from each triangle. However, at the time, I did not see how to prove that such a splitting always existed. But the next year, one evening while at the ICM in Stockholm, I ran into Bryan Birch and Hallard Croft, from the UK. As our group was breaking up on 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 (cf. [4]). That, indeed, was to prove inspiring, and, in 1963, I managed to complete the three-dimensional case. Alas, my proof fell into 7 subcases and seemed hopeless to generalize. It was then that Laurits Meltzer came to my aid. Meltzer (1861–1943) was a military oKcer 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. 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 've 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 in Bergen. Thus it was, in 1964, supported by this fund, I took my manuscript to discuss the problem further with Bryan Birch, then in Manchester, as well as with Richard Rado, at the University of Reading, since Rado had also obtained partial results in higher dimensions. 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 o4, 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! H. Tverberg /Discrete Mathematics 241 (2001) 11–22 17 I explained it to Birch, and, after an agreeable day of mathematical conversation with him, returned to Norway to start writing up the result. In view of my good fortune with that paper [33], I still shudder to think how easily the chain of events leading to it might have been broken. I suppose this re!ects how signi'cant these seemingly fortuitous events were for me as a young researcher. But Radon’s theorem and its generalizations have remained an abiding interest, a skein of ideas I have been pleased to pick up again from time to time and to unravel a little further in several papers. Moreover, from shared interest in this has grown a much valued friendship with JO urgen Eckho4, in Dortmund. Let me add here that it was quite the other way with Andrew Coppel, whom I had known for several years before he surprised me by suddenly taking up the foundations of convex geometry, producing his highly original study [9]. The contact with Rado turned out to be very fruitful. Just at that time, lecturers in Norwegian universities began to be allowed sabbaticals, and so, supported by a grant from the British Council, I was able to join Rado in Reading for the 'rst half of 1966. For me, Reading o4ered a highly stimulating combinatorial atmosphere: Anthony Hilton was there at a comparable stage to me in his research; and Eric Milner was a more senior 'gure, having returned from the University of Malaya—he was to leave shortly after my visit for the University of Calgary, where he was then based for the rest of his life. The 'rst person I met in the Department was perhaps the most illustrious 'gure of all, although very approachable and lively in manner: this was Sir Alexander Oppenheim, who had been a student with G.H. Hardy in Oxford in the 1920s, had known Rado from Cambridge in the 1930s, and had recruited Milner in the 1950s to the University of Malaya, from which Oppenheim himself had just stepped down as Vice-Chancellor. But 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. Eric Milner was especially solicitous and hospitable, and another memory from that time was being taken by him to Oxford to see a splendid performance of Who’s Afraid of Virginia Woolf, starring Richard Burton and Elizabeth Taylor. One contact often leads to another, and, a few years later, in 1969, when Milner was one of the organizers of a combinatorial conference sponsored by NATO in his new position in Calgary, he wanted all NATO countries to be represented. No doubt I was the only combinatorialist in Norway whom he had met. It was a splendid occasion, and certainly a wonderful opportunity for me: I recall thinking, as I listened to Daniel Kleitman and Crispin Nash-Williams, that experiencing their talks [16,18] alone was well worth the whole e4ort and expense of the trip. Crispin Nash-Williams rightly says, in his 'ne obituary [19] of Milner, that “[Milner’s] many friends were devastated when he died on 20 July, [1997]”, the year after he had retired from the University of Calgary as Emeritus Professor. Fortunately, for those of us who knew Eric Milner, Nash-Williams’ obituary achieves a portrait which is exceptionally true to life, catching especially well Milner’s helpful way with fertile problems and suggestions for further research, as well as his great faithfulness in studying the papers of others. 18 H. Tverberg /Discrete Mathematics 241 (2001) 11–22 I have already made reference to the work of Erdős, but it was only in Calgary that I met him for the 'rst time—and we never managed to get him to Bergen. Later, I was to meet him quite often at conferences. I even checked a solution by David Preiss of one of Erdős’ prize problems when we were in Durham, UK, in 1974. Despite the small amount of the prize, I still rather regret not taking a photograph of the award ceremony, with Erdős 'shing out a $10 bill to present to Preiss. I agree with Hilton in his review [14] that there are so many Erdős stories that it will be a shame that many will likely go unrecorded, so here is a personal favourite that I witnessed in Canberra in 1988. Erdős described his $3000-problem on proving that an increasing sequence of integers with positive density contains arbitrarily long (but 'nite) arithmetic sequences, and, pausing, thought to add, “I will, of course, also pay $3000 for a counterexample”. Bernhard Neumann adroitly capped this, volunteering from the front row, “And $6000 for both”. With two recent biographies [15,23] of Erdős now available, it is diKcult to add anything really new to the rich picture one has of him, but a minor incident in Haifa, on opening a car door for him, stays with me as being so much in character. It was a very small car, so Erdős, although not a large man, virtually had to creep out. Yet, the moment he had one foot on the ground, he piped up, “Do you know the following problem?”. I think that most of us would have waited until being fully disengaged from the car and standing upright before relaunching into a professional discussion. As with numerous others, Erdős once helped me to a publication. Leon Mirsky had shown Erdős my proof of Dilworth’s theorem on partitioning a partially ordered set into chains. Erdős enjoyed it, and encouraged me to publish it, although it was little more than a tiny improvement of the proof given by Micha Perles in [22]. Mirsky kindly suggested another little twist to improve the proof further, and Rado came up with yet another helpful idea. I still feel a bit guilty for not thanking them in the paper [34] that emerged from this conclave. However, a few years later an opportunity did arise to dedicate another note [35] to Richard Rado. Reverting to my time in Calgary, one, further, long-lasting e4ect of the conference was that I really got to know Bernt LindstrO om; and I have greatly appreciated his friendship, mathematical and social, in the intervening years. In 1971, I served as the opponent in the public defence of LindstrO om’s doctoral thesis, at the University of Stockholm—still in the Nordic universities a rather special occasion for all involved. Since then, I have been in close contact with the impressive developments in combinatorial mathematics in Sweden, mostly due to LindstrO om, his students, and now increasingly their students in turn. Perhaps the most distinguished of these, Anders BjO orner, contributed a highly informative account [5] of LindstrO om’s work to an issue of European Journal of Combinatorics in May 1993 dedicated to LindstrO om on his 60th birthday, an issue which BjO orner also edited. I was happy that a joint paper [38] with SiniZ sa Vre[ cica was accepted for inclusion in that issue. I have already described how a single, niggling impulse, such as encountering Helly’s theorem, could bear fruit in research. On re!ection, I see that this has happened quite often with me. For instance, in 1971, Robin Wilson gave us a talk in Bergen on graph colouring. This jogged my thinking, inspiring the (re)discovery of the fact that a planar H. Tverberg /Discrete Mathematics 241 (2001) 11–22 19 graph can be four-coloured if each vertex is allowed to share a colour with (at most) one of its neighbours—the result had been published only the previous year, in a paper [3] by Barnette and Stein. Fortunately, the proof was not a rediscovery, and I could avoid appealing to a diKcult result of Tutte. But my paper [36] was, in a sense, killed, as I had to add, in proof, that the real four-coloured theorem had now been proved—I had some apprehension of the transience of this kind of result anyway, describing it as “this interesting theorem (while it lasts)”. However, one thought does still linger with me from that time: if one could 'nd a universal bound on how many times one has to allow neighbours with shared colours, one would have another proof of the four-colour theorem. The sphere has the property, unlike surfaces of higher genus, that, if you remove discs from each of two copies and then glue the remainders together along the two cuts, you obtain another sphere. Given any planar graph G, this procedure then enables you to 'nd another containing as many disjoint copies of G as you please. This means that, were there the sort of universal bound I have in mind, you would obtain a colouring of a graph containing many copies of G where at least one copy would be coloured in the usual proper way. Another lingering thought concerns a beautiful result, due to Bollobas, Milner, and Shelah, which may be cast in terms of marriages as follows. Let some boys make lists of girls they would like to marry. De'ne the popularity of a girl by the cardinality of the set of boys on whose lists she appears. Then every boy can get a wife, provided no list is empty, and every boy has a list which is at least as long as the popularity of any girl on it. There is a double beauty to the result. First of all, it is simple to grasp, and sounds entirely reasonable. Secondly, there are no restrictions on the cardinals involved. However, the proof is not all that easy. I thought I was onto a good idea for a neat proof on realizing that you can reduce the given lists by deleting names until this is no longer possible without breaking the condition in the theorem provided you observe a special constraint. This constraint is just that if you delete names from a list, you must delete so many that its cardinality is reduced. This good idea turned out not to work too well, alas, and it was only with the help of a referee that I could push it through in one of the many cases into which the proof falls. But I still feel that it ought to work. Naturally, although one can always hope for a proof from THE BOOK, it may be foolish to expect that there should always be a nice proof for a nice theorem. But one of my favourite examples is the SchrO oder–Bernstein theorem, that if there are one-to-one mappings from A to B and from B to A, then there is a matching (bijection) between A and B. This sounds technical, but, for the non-mathematician, the Danish mathematician BHrge Jessen o4ers a demonstration on the dance !oor. Of course, the two sets are the boys and the girls at a dancing school, while the mappings express preferences as to the dance partner—and it can only help avoid embarrassment if these are one-to-one. Now, the dancing instructor is very conscientious and has a stratagem that ensures that everyone is on the dance !oor. First of all, the boys chosen by no girl take their partners—no con!ict, of course, since the preferences are one-to-one, even if the girls did not get their choice. Next is the turn of the boys those girls would have 20 H. Tverberg /Discrete Mathematics 241 (2001) 11–22 chosen, and they take their partners. Those partners too might have preferred other boys, so that group of boys follows in with their partners and so it goes, repeating this process, if necessary ad in'nitum. Should anyone still be left, it is the turn of the remaining girls to take their partners. Now, everyone is ready to dance. 4. Research on the margins Let me conclude these personal re!ections by responding to a question that is sometimes posed when I visit outside Norway. How is it possible to survive as a mathematician in a small department at a far corner of the world being the solitary researcher in your speciality? It is a natural enough question to raise, and one can easily imagine the diKculties faced by isolated workers. But I am glad to say that, for me, in Bergen, an answer has been comparatively easy. There are so many factors conducive to a nice mathematical life: the general spirit of the department and congenial colleagues; interaction with lively students; the quantity of time available for study and research; the library and accessibility of materials; sabbaticals and the opportunities for travel—wonderfully stimulating weeks in Oberwolfach, and many sunny months in the hospitable company of Andrew Coppel and his colleagues at the Australian National University, in Canberra, stand out as highlights in my memories. Living in Norway, Professor Joe Gani, Professor Michael Pitman, Professor Helge Tverberg (University of Bergen, Norway) Professor Mike Newman, Professor Derek Robinson (Printed with the permission of the Australian Academy of Science). H. Tverberg /Discrete Mathematics 241 (2001) 11–22 21 I have really been quite lucky in this mix of ingredients; and, in addition, I have had the good fortune to have come by many mathematical friends. But I recognize that all these factors are as precious as they are precarious in times of funding constraints. So, in thanking all those many friends who have contributed to this volume, I hope that your e4orts will, in some measure, also help secure a future for younger mathematical researchers in Norway as part of the international scienti'c community. For, this volume pays respect to the provisions that I have received in my career in my country, and which I should wish coming mathematicians to enjoy equally in their turn.