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The Legacy of Kurt Mahler The Legacy of Kurt Mahler Jonathan M. Borwein, Yann Bugeaud, and Michael Coons February 25, 1988; it was received after van der Poorten had heard about Mahler’s death. At that time van der Poorten had already begun to collect Mahler’s works. Over twenty-five years have now passed since Mahler’s death, and nearly five years since van der Poorten’s. Recently, in memory of van der Poorten, we1 finished van der Poorten’s project and established the Kurt Mahler Archive. The Archive is hosted by the Centre for Computer-Assisted Research Mathematics and its Applications (CARMA), which is located2 at the University of Newcastle, Australia. It can be found using the following URL: carma.newcastle.edu.au/mahler/index.html Mahler, the Man Mahler was born in Krefeld, Germany, July 20, 1903. He did not come out of an academic family but, nonetheless, from one which loved the printed Kurt Mahler (1903–1988). word; his father ran a printing firm. While first entering academia in a nontraditional way, Mahler I hope that you can continue with your should most appropriately be considered a student preparing of my collected works. When my of Siegel, and this is how the Mathematics Genealogy old papers first appeared, they produced Project has recorded it. With the change of power in little interest in the mathematical world, 1933, Mahler realized that he would need to leave and it was only in recent times that they Germany. After six weeks in Amsterdam, Mahler have been rediscovered and found useful. spent the academic year 1933–34 in Manchester So a collection of all my papers may repair under a fellowship secured by Mordell. He spent this position! most of the next few years in the Netherlands until —(Kurt Mahler, in his last letter to Alf van returning to the University of Manchester in 1937, der Poorten) where he remained for twenty-five years. After The quote at the beginning of this article was that, with the exception of a few years’ hiatus in taken from a letter written by Kurt Mahler to Alf the United States of America, he was an Australian van der Poorten the day before Mahler died on mathematician.3 Jonathan M. Borwein is a Laureate Professor of mathe- 1 matics at the University of Newcastle. His email address is Joy van der Poorten helped immensely with collecting [email protected]. Mahler’s harder-to-get papers as well as with the correspon- dence between Alf van der Poorten and Mahler. We thank Yann Bugeaud is professor of mathematics at the Univer- her greatly for her contribution to the Archive. sité de Strasbourg. His email address is bugeaud@math. 2We are currently putting together a Mahler selecta with the unistra.fr. help of Michael Baake (University of Bielefeld, Germany) and Michael Coons is a senior lecturer and ARC research fel- Reinout Quispel (La Trobe University, Australia). low at the University of Newcastle. His email address is 3There are numerous biographies of Kurt Mahler, which are [email protected]. extremely well written. Thus we will not dwell too much on DOI: http://dx.doi.org/10.1090/noti1241 this here. See [2], [3], [5], [7], [17], [19], [22]. 526 Notices of the AMS Volume 62, Number 5 Mahler and Number Theory in Australia around Australia, as well as a number of ad- Under the invitation of B. H. Neumann, in 1962, vanced courses on transcendental number Mahler visited Australia and, in particular, the then theory in the Institute of Advanced Studies. quite new Australian National University (ANU). He In the end the fascination of what he was do- was impressed enough to quickly join the ANU as ing beguiled us both into research in number a professor in its Institute of Advanced Studies. As theory, and we made our first steps in math- Mahler recollects: ematical research on problems suggested by There was at this time no teaching of number him. theory in the undergraduate school (School —(John Coates and Alf van der Poorten, as of General Studies) at the ANU. I therefore quoted from [19]) gave a course on this subject to second and These two students are well known to those of third year students at the SGS, probably us in number theory, and many of us beyond. the first one ever in Canberra. One of my John Coates went on to become a Fellow of the undergraduates, Coates, asked me to intro- Royal Society and was Sadleirian Professor of duce him to research. I provided him with Pure Mathematics at Cambridge from 1986 to problems to work on, and by the time he ob- 2012. Soon after Mahler moved to Australia, at the tained his BSc, he had already several papers encouragement of George Szekeres, Alf van der published or in print. Poorten visited Mahler regularly in Canberra. A —(Kurt Mahler, as quoted from [19]) friendship based at first on a common interest in 4 One gains much more appreciation for Mahler science fiction turned quickly into a mathematical reading his students’ view of the same time. relationship. Van der Poorten became Mahler’s 5 Mahler was also very concerned with sowing doctoral student, graduating in 1968. Alf van der the seeds of his own mathematical knowl- Poorten went on to positions at the University of edge in his new country. As in his own mathe- New South Wales and then Macquarie University, both in Sydney. matical research, he instinctively felt that the 6 best way to do this was to go back to first prin- Van der Poorten continued Mahler’s legacy of ciples, and to begin by teaching beginners in number theory in Australia, both mathematically the subject. The ANU had begun to award in a very interesting and important collaboration undergraduate degrees only a few years be- with John Loxton, and as a mentor and supporter 7 fore Mahler arrived, and Hanna Neumann of young Australian mathematicians . One such was appointed to head the new Department instance remembered by a former young Australian of Mathematics in the teaching side of the mathematician goes as follows. University (the School of General Studies) I first encountered Alf’s unique style when at about the same time that Mahler took I was a member of the Australian Interna- up his chair. Between them, they arranged tional Olympiad (IMO) team and we came for Mahler to give two courses to the small across his wonderfully entertaining paper on number of undergraduates reading mathe- Apéry’s proof of the irrationality of ζ(3). I as- matics, one in 1963 on elementary number sumed from his name that he was Dutch, and theory, and the second in 1964 on the el- was then pleasantly surprised when I met liptic modular function j(z). One of us had Alf at the IMO team send-off reception and the good fortune to attend these courses. 4 Mahler started and finished each lecture This is entirely evident from the correspondence between with extraordinary punctuality; in between, Mahler and van der Poorten. The amount of space devoted the audience was given a rare insight into to science fiction is roughly asymptotic to that devoted to mathematics. his understanding of and enthusiasm for 5Alf van der Poorten was enrolled as a student at the the material of the lecture. As he spoke, he University of New South Wales, not at the ANU. would produce a beautiful written exposi- 6There is a comprehensive biography of Mahler’s life and tion on the blackboard of the key points, work written by David Hunt [6] in Number Theory and Re- which were neatly placed in order in his lated Fields. In memory of Alf van der Poorten. Based on the characteristic rectangular boxes. Although proceedings of the international number theory conference, he seemed at first so different and forbidding, Newcastle, Australia, March 12–16, 2012 [1]. 7 we soon discovered that he was very willing In keeping with van der Poorten’s avid support of young to talk about his knowledge of mathematics Australian mathematicians, in his memory his family funds the Alf van der Poorten Travelling Fellowships, which aim in general, and to lend us his own mathemat- to assist young pure mathematicians (who have earned a ical books when we could not find them in PhD in Australia) in travelling in Australia and overseas so the library. Mahler gave lectures at various that they can enrich their mathematical research through summer schools in Canberra and elsewhere contact with other mathematicians. May 2015 Notices of the AMS 527 to find that he lived and worked in Sydney! Kisin (2007), Terence Tao (2009),8 and Peter Sarnak He then and there told me what the p-adic (2011). numbers were and immediately offered me a job at any time in the future! I took him up The Mahler Archive on his offer at the end of my first year at Mel- The way most of us interact with Mahler’s legacy bourne University, and spent six weeks in his is now through his work. With the advent of the office annex learning about elliptic curves, digital age, looking up papers is a much easier the Riemann–Roch theorem, the Weil conjec- task than in previous times. With this in mind, tures, and, of course a lot of great stuff about we have made the Kurt Mahler Archive freely recurrence relations (Skolem–Mahler–Lech) available online (see the first part of this article for and continued fractions. I have always ap- the details and URL). It contains (in PDF format) preciated the time he spent talking to me—it every mathematical article published by Mahler, was clear he cared a great deal about young as well as a host of links to biographies and Australian mathematicians.
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