**Wilhelm Magnus**'s parents were Alfred Magnus and Paula Kalkbrenner. Wilhelm attended the Gymnasium in Tübingen from 1916 and was awarded his Abitur in 1925. He had studied Latin throughout the nine years at the Gymnasium but his favourite subjects were mathematics and physics. At this stage he decided that physics was the subject for him and he began his university studies with that in mind. He entered the University of Tübingen in the autumn of 1925 but, even in the year he entered, he was studying advanced mathematics [7]:-

After two semesters at the University of Tübingen, Magnus went to the Johann-Wolfgang-Goethe-University of Frankfurt. He had already decided that he wanted to specialise in mathematics rather than physics when he was taught by Cornelius Lanczos who had been appointed to Frankfurt in 1924. Magnus wrote [1]:-I became interested in Number Theory and, in1925during my first semester at a university, I started to read the 'Disquisitiones Arithmeticae'Ⓣby Gauss. ... the language was not difficult but Gauss's style was and I never really became initiated into his theory of quadratic forms. So I was happy to see the publication of the first volume of Edmund Landau's 'Vorlesungen aber Zahlentheorie'Ⓣwhich appeared shortly afterwards. The publication date given in the three volumes is1927, but the first volume appeared somewhat earlier. Landau was known to be the leading number theorist in Germany, and I started reading his book with great expectations. All went well up to page93, theorem152... I could follow the proof, but I did not see through it, and I felt that if this was number theory it was too difficult for me.

Magnus was also taught by Carl Siegel, who had been appointed to Frankfurt in 1922 to fill Arthur Schönflies's chair, and by Ernst Hellinger, who had been appointed to a chair at the University of Frankfurt in 1914. He wrote about his experiences in their classes in [1]:-When I attended a course taught by Lanczos for the first time, I had already changed my original plan to become a physicist, realizing that I was more at home and at ease in mathematics. Perhaps, this was fortunate, because Lanczos might have made me stay in physics if I had met him earlier. To work in theoretical physics requires an uncanny combination of talents: a specific type of intuitive understanding of the realities of physics and a well developed ability to handle the necessary mathematical tools with complete ease. What made Lanczos such a fascinating teacher for a mathematician was his ability to inject some of the intuition of the physicist into mathematics. Even the supreme clarity of Lanczos's lectures would not have sufficed to produce this effect. What one really could learn from him was the over-riding importance of motivation for the development of a theory.

Magnus completed the work for his first degree in Mathematics, Theoretical Physics and Experimental Physics and was awarded the degree on 18 November 1929 with the grade of "excellent". He was appointed as an assistant at the Mathematical Seminar at Frankfurt from 1 October 1929, holding this position until 30 September 1930.Instinctively, everyone in the class knew that none of us would ever be as powerful a mathematician as Siegel. Contrary to all the talk from psychologists and educators who warn against oppressing the developing student, this need not be a depressing experience at all. The opposite is true: it is beneficial to know early what high standards really mean. And Siegel was encouraging when he felt that this was justified. And his word then carried weight. ... Hellinger was the most widely appreciated teacher among the mathematicians. He, too, was very well prepared. His lectures were highly polished but he never forgot to mention the motivation for a theorem.

Despite the other outstanding teachers he had at Frankfurt, it was Max Dehn, who held the chair of Pure and Applied Mathematics at the University of Frankfurt from 1921 until 1935, who had become Magnus's Ph.D. advisor in 1928. Magnus wrote [1]:-

In 1928 Dehn asked Magnus various questions about one-relator groups. He was able to answer these questions and published his results on one-relator groups in 1930 in the paperMax Dehn was my Ph.D. advisor and I have been influenced deeply by him. I took courses taught by him only in my last year at the university, and they had a lasting effect on me in spite of the fact that they were not as polished and smooth as those which I had attended before. Dehn communicated ideas. One had to be ready for this. In fact, one had to be able to enter into a dialogue with him. Even if one had only a tiny contribution to make, and even if one expressed it in a confused way, this was enough. Dehn always understood. He had the ability which Socrates claimed for himself: to act as a midwife at the birth of an idea. This ability went far beyond mathematics. Dehn had an extensive knowledge of philosophy and of history, and he used it to gain the proper perspective for any particular fact or occurrence. He was very undogmatic and did not belong to any philosophical school, but he always tried to see the significance of ideas and facts within the general framework of human experience.

*Über unendliche diskontinuierliche Gruppen mit einer definierenden Relation*(

*Der Freiheitssatz*) Ⓣ. In this he proves that certain subgroups of one-relator groups are free groups. The results of this paper formed his Ph.D. thesis and he was awarded the degree on 13 January 1931. From 1 November 1930 to 31 July 1932 he was an assistant at the Mathematical Institute at the University of Göttingen. In 1932 he published

*Das Identitätsproblem für Gruppen mit einer definierenden Relation*Ⓣ containing a major result in combinatorial group theory, namely that the word problem for one-relator groups is soluble. In this paper Magnus introduced a method of breaking a one-relator group into simpler one-relator groups. This method became the main tool in attacking one-relator groups in later research.

Magnus was appointed to the staff in Frankfurt serving from 1933 until 1938. During this time he spent nine months of session 1934/35 at Princeton University in the United States supported with a Scholarship from the Rockefeller Foundation. At Princeton he attended a course given by his former teacher Carl Siegel, who was spending a year at the Institute for Advanced Study at Princeton. Magnus writes [7]:-

During this period Magnus introduced Lie ring methods to study the lower central series of free groups. He studied the automorphism groups of free groups in 1934. In 1935 Magnus gave examples of finitely presented groups which were isomorphic to proper factor groups of themselves. Heinz Hopf had originally asked whether such groups exist and, although Jakob Nielsen had shown that free groups of finite rank have this property ten years before Hopf asked the question, nobody - including Jakob Nielsen himself - noticed the question had already been answered.My attendance at Siegel's lectures resulted in the publication of my only number theoretical paper(on the class number of quadratic forms).

He was appointed as an assistant at the University of Königsberg on 1 April 1939. Later that year, on 5 August, he married Gertrud Remy; they had three children, Jutta, Bettina, and Alfred. However his career was to hit problems when he refused to join the Nazi Party and, as a consequence of this, was not allowed to hold an academic post during World War II. Instead he had to work in industry [7]:-

During the war, Magnus also undertook military research in the Department of War Marines situated at Berlin-Wannsee. In 1944 he became an ordinary professor at the University of Königsberg but Königsberg was bombed by the Royal Air Force in August 1944 and during the first three months of 1945 the Russian army bombarded the city resulting in its capitulation on 9 April 1945. By this time 80% of university buildings had been destroyed. The staff fled and many were given positions at the University of Göttingen. Magnus was offered an ordinary professorship at the University of Göttingen in 1946 but he was not to remain there for long.I had joined Telefunken(a radio firm), and there I met Arnold Sommerfeld, the famous physicist, who had been induced by the management of the firm to act as a consultant on theoretical problems. I became fascinated by his work on diffraction problems and by his "radiation condition" which enforces uniqueness for their solutions. Sommerfeld was at least as much - if not more - a mathematician as a physicist. How else would he have taken pride in solving the diffraction problem for a half-plane by introducing a branched covering of physical three-space with edge of a half-plane as a branch line and the half-plane as a branch cut? Now special functions, especially Bessel functions, are used extensively and effectively in Sommerfeld's papers. From him I learned how to apply them as useful tools for the solution of certain problems. I started collecting them. I am not sure how far these functions may have appealed to my collector's instinct, an instinct which manifests itself in many people with application to diverse objects, regardless of any consideration of usefulness.

In 1946 Harry Bateman died and Edmund Whittaker was asked to recommend someone who could undertake the project of organising and publishing Bateman's manuscripts. Whittaker's advice was that Arthur Erdélyi should lead the project and, in 1947, Erdélyi went to the California Institute of Technology. The project was a major one and other collaborators were needed. Magnus left Göttingen in 1947 and joined the Bateman project in 1948, first as a visiting researcher. He wrote [7]:-

Magnus collaborated with Arthur Erdélyi, Fritz Oberhettinger and Francesco G Tricomi on the production of three volumes ofIn the case of Harry Bateman there is very little doubt in my mind that he was, at least in part, motivated by a pure collector's instinct. I became acquainted with his incredible collection of formulas for special functions and definite integrals when working from1948to1950on the handbook of higher transcendental functions, which is commonly known as the "Bateman Project."

*Higher Transcendental Functions*and two volumes of

*Tables of Integral Transforms*. All five volumes appeared in print between 1953 and 1955. For more information, see THIS LINK.

In 1950 Magnus went to the Courant Institute of Mathematical Sciences. There one of his first students was Abe Shenitzer who writes [10]:-

He spent 23 years at the Courant Institute before moving to a chair at the Polytechnic Institute of New York in 1973. He held this post for five years before retiring.I first met Wilhelm Magnus in1950, when I came to the New York University graduate school and enrolled in Magnus's course in algebra. I took an immediate liking to this polite, somewhat shy, and strikingly intelligent man. I was older than most of the students in the class and I approached him without hesitation. The fact that I was a good student helped, and I found myself talking to Magnus about nonmathematical matters as well as mathematical issues. One day he said to me: "You've done more for me than any person can do for another." He was visibly moved and I was utterly perplexed. "Yes," he said, "you are a Jew who was in German concentration camps and I am a German." "But I deal with individual people, not with labels" was my response. This was the beginning of our friendship.

In [11] Magnus's research is described in these terms:-

It was not only in the breadth and depth of research that Magnus excelled. He was also one of the best supervisors of doctoral students, supervising 61 doctoral students during his career. One of his students was Seymour Lipschutz who dedicated a paper to Magnus with the words:-Magnus's mathematical expertise was exceptionally wide ranging. In addition to research in group theory and special functions, he worked on problems in mathematical physics, including electromagnetic theory and applications of the wave equation.

Shenitzer writes [10]:-The author dedicates this paper to his teacher, advisor and friend Wilhelm Magnus(1907-1990); he was a shining example of a kind, considerate and concerned human being.

His teaching is described in [11] as 'outstanding' and this is confirmed by his receipt of the Great Teacher Award of New York University in 1969.Magnus was a creative mathematician and, as he told me, he liked best to work with gifted doctoral students. On the other hand, he was far too intelligent not to function occasionally as a critic who sheds light on a whole area with a single aphoristic remark. When, as a rank beginner, I asked him what made groups of automorphisms important, he replied, "They are the algebraic counterpart of symmetries in geometry." He began his first lecture in a course on geometry with the remark: "The fundamental difference between Hilbert and Euclid is that Hilbert realized that you can study form without substance."

His nine books cover a wide range of mathematical topics such as elliptic functions, tessellations (*Noneuclidean tessellations and their groups* (1974)), combinatorial group theory (a major work *Combinatorial group theory* (1966) written jointly with A Karrass and D Solitar) and mathematical physics.

For more information about these books and extracts from some reviews see THIS LINK.

For a version of Magnus's Preface to *Noneuclidean tessellations and their groups*, see THIS LINK and for the preface to *Combinatorial group theory*, see THIS LINK.

It is unusual for a 20th century mathematician to work in two mathematical areas as far apart as the ones on which Magnus worked. But he was also deeply interested in other topics outside mathematics [10]:-

You can read Magnus's ideas on "What Makes a Mathematician?" at THIS LINK.He was deeply versed in history, philosophy, and literature, but he had a special passion for poetry. His learning was an integral part of his mind. He was the epitome of a cultivated person. He once told me that if he got tired of algebra, then he could always teach a course on Plato. I was present when the philosopher Hans Jonas, the mathematician Fritz John, and Magnus got together to listen to Jonas's report on a conference on gnosticism which he had just attended. There ensued an animated discussion by three people who seemed equally at home in history and in philosophy. An outsider would have found it hard to believe that two of the three participants were eminent mathematicians.

He was awarded a number of honours including a Rockefeller Fellowship in 1934, a Guggenheim Fellowship in 1969 and Fulbright-Hays Senior Research Scholarship in 1973/74. He was a member of the Göttingen Academy of Sciences and the American Mathematics Society. He was awarded an honorary Doctor of Science from the Polytechnic Institute New York in 1980.

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

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