Kurt Mahler 1903-1988

Written by J.H. Coates and A.J. van der Poorten.

Kurt Mahler was born on 26 July 1903 at Krefeld am Rhein in Germany; he died in his 85th year on 26 February 1988 in Canberra, Australia. From 1933 onwards most of his life was spent outside of Germany, but his mathematical roots remained in the great school of mathematics that existed in Germany between the two world wars. Above all Mahler lived for mathematics; he took great pleasure in lecturing, researching and writing. It was no surprise that he remained active in research until the last days of his life. He was never a narrow specialist and had a remarkably broad and thorough knowledge of large parts of current and past mathematical research. At the same time he was oblivious to mathematical fashion, and very much followed his own path through the world of mathematics, uncovering new and simple ideas in many directions. In this way he made major contributions to transcendental number theory, diophantine approximation, p-adic analysis, and the geometry of numbers. Towards the end of his life, Kurt Mahler wrote a considerable amount about his own experiences; see 'Fifty years as a mathematician', 'How I became a mathematician', 'Warum ich eine besondere Vorliebe für die Mathematik habe', 'Fifty years as a mathematician II'. There is also a recent excellent account of his life and work by Cassels (J.W.S. Cassels, 1991, 'Obituary of Kurt Mahler', Acta Arith. (3), 58, 215-228). In preparing this memoir we have freely used these sources. We have also drawn on our knowledge of and conversations with Mahler, whom we first met when we were undergraduates in Australia in the early 1960s.

Krefeld, where Mahler spent the first twenty years of his life, was a town of some 100,000 inhabitants in a predominantly Catholic part of the Prussian Rhineland. His family was Jewish, and had lived in the Rhineland for several generations. His father and several of his uncles worked in the printing and bookbinding trade, beginning as apprentices and slowly saving enough money to start small firms of their own. Kurt and his twin sister Hilde (1903-1934) were the youngest of eight children born to Hermann Mahler (1858-1941) and his wife Henriette, née Stern (1860-1942). Four of the children died young. An elder sister Lydia (who died in 1984) married a printer who was also a musician, and lived in the Netherlands. An elder brother Josef, who joined and eventually took over his father's firm, disappeared together with his wife in a concentration camp during the Second World War.

The family had no academic traditions. None of Kurt's four grandparents went to more than elementary school (Volksschule). However, the four children acquired a love of reading from their father. At the age of 5, Kurt contracted tuberculosis, which severely affected his right knee. The knee was subsequently operated on several times, but it did not heal until he was 35 and left him with a stiff leg, which very much hindered his walking throughout his life. Because of this illness, Kurt only attended school for a total of four years up till the age of 14, but he had some private tuition at home for two additional years. At Easter 1917, shortly before he turned 14, he left elementary school, and attended technical schools for the next two years, with the intention of becoming a precision tool and instrument maker. Mahler always retained a fascination with technical drawing and calligraphy. Most important, these technical schools gave him his first training in algebra and geometry. He very quickly decided that mathematics was what he really liked doing. Already, from the summer vacation of 1917, he began teaching himself logarithms (the arithmetic properties of which turned out to be one of his abiding interests in transcendental number theory) plane and spherical trigonometry, analytic geometry and calculus. In 1918, he became an apprentice in a machine factory in Krefeld, working for one year in the drawing office and then for almost two years in the factory itself. Later, the drafting skills he acquired would be useful; see the papers of L. J. Mordell in the period 1941-45. Mahler said himself that his aim in taking the apprenticeship was that it might eventually allow him to study mathematics at a technical university (Technische Hochschule), thereby avoiding the difficult entrance examination required to enter a traditional university. He did learn a little more elementary mathematics as part of evening classes, but quickly progressed with his mathematical self-education. How successful he was as an auto-didact is illustrated by the fact that he soon acquired and began reading, without any expert guidance, such sophisticated books as Bachmann's Zahlentheorie, Landau's Primzahlen, Knopp's Funktionentheorie, Klein and Fricke's Modulfunktionen and Automorphe Funktionen, and Hilbert's Grundlagen der Geometrie.

In Mahler's own words: 'The great day came in 1921'. He was in the habit of writing little articles about the mathematics he had read. Without his knowledge his father had sent some of Kurt's work to the director of the local grammar school (Realschule). Dr Junker was a mathematician, having written a doctoral thesis in invariant theory under Christoffel. He was evidently impressed by the apprentice's efforts, and sent some of Mahler's work to Klein in Göttingen, who passed it on to his young Assistant, C. L. Siegel. Thus began a lifelong association between Siegel and Mahler: Siegel urged that Mahler should be helped to pass the university entrance examination. Mahler left the factory and spent two years at home preparing for the entrance examination (he cites preparation for papers in German French, and English) with the assistance of teachers at the Realschule, as well as continuing his own reading in mathematics. He passed the examination (he says 'I just scraped through') in the fall of 1923, amidst the political turmoil of German hyper-inflation and the occupation of the Ruhr. Mahler's 1927 Frankfurt doctoral dissertation is dedicated to Dr Josef Junker.

Siegel had moved to the University of Frankfurt am Main and, following his suggestion, Mahler went to study there in 1923, at the age of 20. Frankfurt was then a very stimulating place for study with Dehn, Hellinger, Epstein, Szass and Siegel making up the Mathematics Faculty (see Siegel's lecture [1964, 'Zur Geschichte des Frankfurter Mathematischen Seminars', in Gesammelte Abhandlungen III, Springer-Verlag, Berlin/Heidelberg/New York, 1966, 462-474] on this period at Frankfurt). Mahler was an unusual freshman. In his first semester, he speaks of attending lectures on calculus by Siegel, topology by Dehn and elliptic functions by Hellinger, a seminar on cyclotomy (in which he gave several lectures), and a seminar on the history of mathematics. Mahler was clearly greatly influenced during this period by Siegel, who was the only person whom he recognized as his teacher in mathematical research. In the summer of 1925, when Siegel left for a period of leave overseas, Mahler moved to Göttingen, where he remained until 1933. Göttingen was then still the world's leading mathematical centre, but was going through a period of change because the great era of Hilbert and Klein was almost at an end. Landau seems to have been kind to Mahler, but took little active interest in his work. From Emmy Noether's lectures, he learnt of p-adic numbers, whose study grew to be one of the main themes of his mathematical research. (A few years later, Mahler proudly reports lecturing on his own work on p-adic numbers at Marburg to Hensel). Perhaps most importantly, in Göttingen Mahler met a galaxy of young mathematicians from Europe and the United States, many of whom became leading figures in later years. These included Alexandroff, Hopf, Koksma, Mordell, Popken, van der Waerden, Weil and Wiener. In 1927, Mahler submitted his doctoral dissertation, on the zeroes of the incomplete gamma function, to Frankfurt (he reports that Ostrowski was not very impressed with the dissertation, and advised him 'to do less easy mathematics'.)

For most of his time at Göttingen, Mahler was wholly supported in his studies by his parents and other members of the Jewish community in Krefeld. However, shortly before he was 30, he was awarded a two-year research fellowship by the Notgemeinschaft der Deutschen Wissenschaft, and records that he was even able to save some of the stipend. In the Göttingen years, all the main themes of his later research, with the exception of the geometry of numbers appeared in his papers (which are the first twenty or so papers in his list of publications). Mahler invented a new transcendence method, he discovered his celebrated classification of transcendental numbers, extended the ideas of Hermite's original work in his studies of the approximation properties of e, pioneered diophantine approximation in p-adic fields, and applied his results on p-adic diophantine approximation to prove his well known generalization of Siegel's theorem on integer points on curves of genus 1. Mahler undoubtedly realized that his method could be extended to curves of genus greater than 1, but it was typical of his outlook that he did not have the patience to work through his generalization of Siegel's method. Mahler mentions that his idea of extending the Thue-Siegel theorem to p-adic algebraic numbers came to him on a small island in the North Sea during the Whitsun holidays of 1930, when bad weather had forced him to stay inside!

Mahler had been appointed to his first post, an assistantship in the University of Königsberg, but had not yet taken it up, when Hitler came to power in 1933. He seems to have realized immediately that he must leave Germany. In the summer of 1933, Mahler spent six weeks in Amsterdam with van der Corput and his two pupils Koksma and Popken, whom Mahler had met in Göttingen; they were to remain his lifelong friends. Mahler moved to Manchester for the academic year 1933-34, where Mordell had secured him a small research fellowship called the Bishop Harvey Goodwin Fellowship. Mahler often spoke in later years of Mordell's kindness to him on this and many subsequent occasions, including in helping him to learn English. It seems that the first English lesson Mahler had in Manchester consisted of being put in front of a blackboard immediately on his arrival and told to give a seminar! The next two academic years were spent in Groningen in the Netherlands, supported by a stipend obtained by van der Corput from a Dutch Jewish group. Here a new theme, the geometry of numbers, began to emerge in Mahler's work.

In 1936, he was run into by a bicycle in Groningen, and this accident reactivated the tuberculosis in his right knee. He was unable to walk for some time and had to return to Krefeld, where he had several operations culminating in the removal of the kneecap. These operations together with two three-month periods in a sanatorium at Montana Valis, Switzerland during the summers of 1937 and 1938, finally cured the tuberculosis, but he was left with a permanent limp. Mahler speaks of having to take morphine to lessen the pain after his last operation, and being relieved to find that he could still do mathematical research when he proved that the decimal expansion 0.123 . . . 9101112 . . . is a transcendental number. (In later years, Mahler often stated the view that twentieth-century mathematicians had greatly neglected the study of the arithmetic properties of decimal expansions.) Needless to say, there were other difficulties during these years, which Mahler rarely talked about and certainly did not record in his own written memories. In one incident (which one of us learnt of from Popken, and which Mahler subsequently confirmed in conversation), Mahler was refused entry at the Dutch border and was about to be sent back to Nazi Germany. Fortunately, Koksma had a colleague at the Free University of Amsterdam, G. H. A. Grosheide, who was related to a senior member of the Dutch government. An urgent intercession was made on Mahler's behalf via this channel and he was finally allowed to enter the Netherlands.

In 1937, Mahler returned to Manchester. He thoroughly enjoyed the lively intellectual atmosphere in number theory that Mordell had fostered in the Department. While his own research flourished, the practical side of life could not always have been easy for him. In the period 1937-41, he had two short appointments as a temporary assistant lecturer and a little support from fellowship stipends, but for over two of these years he lived on his own savings. In 1939, he had planned to take up an appointment at the University of Szechuan in China, where his friend Chao Ko was teaching, but he was forced to abandon the idea because of the outbreak of war. However, Mahler had begun to learn Chinese and that study was to remain an important interest and hobby. In 1940, he was interned for three months as an 'enemy alien', first in a tent city near the Welsh border and then in boarding houses on the Isle of Man. Here he lectured to the other internees on the construction of the real numbers by means of Cauchy sequences of rational numbers, as part of a university set up in the internment camp. Mahler records that he later found the same material very suitable for the beginning of first year honours courses in analysis at Manchester. While interned, he was awarded a ScD degree by the University of Manchester.

In 1941, Mahler was appointed to the Assistant Lectureship at Manchester, which Davenport had vacated when he moved to take a chair at Bangor. In the next few years, Mahler developed a geometry of numbers of general sets in n-dimensional space, including his celebrated compactness theorem. His future was now assured. He was promoted to Lecturer (1944), Senior Lecturer (1947) and Reader (1949), and in 1952 the first personal chair in the history of the University was created for him. He became a British subject in 1946 and was elected a Fellow of the Royal Society in 1948. He made his first visit to the United States in 1949, spending most of the time at the Institute for Advanced Study in Princeton. At Christmas 1949, he contracted diphtheria and had another severe bout of illness for three months, but recovered in time to spend the summer lecturing in Colorado, and taking part in the International Congress of Mathematicians at Harvard University.

At Manchester, he lived from 1938 until 1958 at Donner House, a hostel where some 25-30 single staff lived in bedsitting rooms and dined communally. When the hostel was pulled down to make way for more extensive student dormitories, Mahler bought a small house in suburban Manchester, and lived there until his departure for Canberra. However, in later life he complained that he found the burden of looking after his own house rather onerous, and one senses that the fact that he could live at University House (a collegiate institution for postgraduate students and research workers in the Institute of Advanced Studies of the Australian National University) was one of the factors which made him decide to move to Canberra in 1963. Of course, there were many other reasons for this move. Most of the mathematicians he had known at Manchester had moved on to positions around the world, and he was clearly feeling a little isolated there.

In the early 1960s, B.H. Neumann, a colleague of Mahler at Manchester, was invited to set up a new Department of Mathematics in the purely research side of the Australian National University, the Institute of Advanced Studies. Mahler was one of the first of many visitors whom Bernhard Neumann quickly invited. There is no doubt that Mahler immediately liked the warm and stimulating atmosphere in the new department, as well as the beautiful climate of Canberra and the delightful setting of the ANU campus on the edge of what was then a large country town. Mahler himself says he was very happy to accept the offer of a research professorship, which he took up in September 1963. The position gave him great freedom to travel and to pursue his own research, both of which he did with energy and enthusiasm.

However, Mahler was also very concerned with sowing the seeds of his own mathematical knowledge in his new country. As in his own mathematical research, he instinctively felt that the best way to do this was to go back to first principles, and to begin by teaching beginners in the subject. The ANU had begun to award undergraduate degrees only a few years before Mahler arrived, and Hannah Neumann was appointed to head the new Department of Mathematics in the teaching side of the University (the School of General Studies) at about the same time that Mahler took up his chair. Between them, they arranged for Mahler to give two courses to the small number of undergraduates reading mathematics, one in 1963 on elementary number theory, and the second in 1964 on the elliptic modular function j(z). One of us had the good fortune to attend these courses. Mahler started and finished each lecture with extraordinary punctuality; in between, the audience was given a rare insight into his understanding of and enthusiasm for the material of the lecture. As he spoke, he would produce a beautiful written exposition on the blackboard of the key points, which were neatly placed in order in his characteristic rectangular boxes. Although he seemed at first so different and forbidding, we soon discovered that he was very willing to talk about his knowledge of mathematics in general, and to lend us his own mathematical books when we could not find them in the library. Mahler gave lectures at various summer schools in Canberra and elsewhere around Australia, as well as a number of advanced courses on transcendental number theory in the Institute of Advanced Studies. In the end the fascination of what he was doing beguiled us both into research in number theory, and we made our first steps in mathematical research on problems suggested by him.

In 1968, Mahler reached the statutory retiring age for professors, and was forced to retire from the ANU. He then moved to a chair at the Ohio State University in Columbus, Ohio, where the chairman was an old friend, Arnold Ross (whose summer schools for gifted high school students have attracted many young people into mathematical research in Australia, the USA and Germany). In 1972, Mahler returned to Canberra for his 'final retirement', living once more in University House. But his mathematical activity never abated, as is shown by the publication of some forty papers from 1972 until his death. He left the bulk of his estate to the Australian Mathematical Society, which has already used part of it to establish a lectureship in his memory.

Kurt Mahler never married. Indeed, he affirms in notes left with the Royal Society that on his part that was a deliberate decision made on grounds of his poor health. In the event, he outlived his contemporaries. That was of course a source of sadness for him, but also one of wry pride.

Mahler was an excellent photographer – many of his pictures adorn University House at the ANU where he lived for more than twenty years. He remained fascinated by Chinese and was exceptionally proud of having written the paper 'On the generating functions of integers with a missing digit' in Chinese, (K'o Hsüeh Science, 29 (1947), 265-267). His non-mathematical reading comprised mostly science fiction and history.

Mahler received many distinctions during his lifetime. He was elected a Fellow of the Australian Academy of Science in 1965 and received its Lyle Medal in 1977. The London Mathematical Society awarded him its Senior Berwick Prize in 1950, and its De Morgan medal in 1971. In November 1977, he received a diploma at a special ceremony in Frankfurt to mark the golden jubilee of his doctorate. Het Wiskundig Genootschap (the Dutch Mathematical Society) made him an honorary member in 1957, as did the Australian Mathematical Society in 1986.

In a letter to one of us dated 24 February 1988 – received after hearing of his death – Kurt Mahler sets the following problem:

Let f(x) be a polynomial in x with integral coefficients which is positive for positive x. Study the integers x for which the representation of f(x) to the base greater than or equal to 3 has only digits 0 and 1.

Concerning his life's work he also happened to write, in that letter: 'When my old papers first appeared, they produced little interest in the mathematical world, and it was only in recent times that they have been rediscovered and found useful...'. That grossly underrates the impact of his work in the past, but correctly notices the richness of even his minor remarks.

[The original printed version of this Memoir contained a section entitled Mathematical Work, detailing Mahler's work on diophantine approximation and transcendence theory. Due to the restrictions of Web publishing, this section could not be reproduced here.]

In his 'post-retirement years' Mahler made extensive use of his TI-calculator to study digital patterns. His desk was covered with detailed such calculations at the time of his death.