**Ludwig Bieberbach**'s father was Eberhard Sebastian Bieberbach (1848-1943), who was a medical doctor, and his mother was Karoline (Lina) Ludwig, the daughter of Georg Ludwig who was also a medical doctor. Georg Ludwig was the director of the mental hospital in Heppenheim, founded in 1866, and in 1897 his son-in-law Eberhard Bieberbach took over from him as director of the mental hospital. The Bieberbach family was well-off and, up to the age of eleven, Ludwig was taught by private tutors. In 1905 he entered the Humanistic Gymnasium in Bensheim, a town very close to Heppenheim. Ludwig became interested in mathematics when studying at this Gymnasium, inspired by an excellent mathematics teacher. After completing his studies at the Gymnasium in 1905 and being awarded a certificate for university entrance, he undertook military service for a year (1905-06) stationed in Heidelberg. While he was on his military service, he attended a course of lectures on the theory of functions given by Leo Königsberger at the University of Heidelberg and was enthused both by the material and by the high quality of Königsberger's lecturing. Bieberbach decided to look at the list of announcements of mathematics courses given in different universities which appeared in the Jahresbericht der Deutschen Mathematiker-Vereinigung and, after studying the possibilities, decided that Hermann Minkowski's course on 'Invariant theory' at the University of Göttingen looked the most attractive. Although Göttingen was the leading centre for mathematics at this time, that was not the reason Bieberbach chose to go there, for he was totally unaware of its reputation when he made his decision.

At Göttingen there was an enthusiastic atmosphere for research which had a great influence on Bieberbach. He attended the algebra course by Minkowski which had brought him there, but he was influenced even more strongly by Felix Klein and his lectures on elliptic functions. Another strong influence on the direction of Bieberbach's mathematical interests came from Paul Koebe who was only four years older than Bieberbach. Koebe, an expert on complex function theory, become a dozent at Göttingen in 1907 and also encouraged Bieberbach towards analysis. It was under Klein's direction that Bieberbach researched into automorphic functions for his doctorate which was awarded in 1910 for his thesis *Zur Theorie der automorphen Funktionen* Ⓣ. Another of the dozents at Göttingen was Ernst Zermelo and in 1910 he was chosen to fill the professorship at Zürich left vacant when Erhard Schmidt moved to Erlangen. He was allowed to appoint a young mathematician to assist him, and he asked Bieberbach if he would go to Zürich with him. Already Bieberbach had gained a reputation for his contribution to one of Hilbert's problems.

The first part of Hilbert's eighteenth problem asks whether there are only finitely many essentially different space groups in *n*-dimensional Euclidean space. Arthur Schönflies had been asked by Klein in the late 1880s to investigate the crystallographic space groups and he had classified those in dimensions 2 and 3 in 1891. In 1908, while Bieberbach was Klein's research student, he had been asked by Klein to give lectures on Schönflies' results. From that time on he became interested in answering the first part of Hilbert's eighteenth problem, namely generalising Schönflies' results to *n* dimensions, *n* ≥ 4. He succeeded in 1910 and in that year announced his result with a sketch of the proof. Soon after he arrived in Zürich, Bieberbach left to go to the University of Königsberg where Schönflies, who was the professor there, had arranged a teaching position for him. There he worked out the details of his solution to the first part of Hilbert's eighteenth problem publishing them in two papers *Über die Bewegungsgruppen der Euklidischen Räume* Ⓣ (1911, 1912). He submitted this work to Zürich as his habilitation thesis in 1911. Answering a question from Hilbert's famous collection certainly gave the young Bieberbach an international reputation.

Bieberbach was appointed professor of mathematics in Basel in Switzerland in 1913 and gave an inaugural address *Über die Grundlagen der modernen Mathematik* Ⓣ on the foundations of mathematics. In the following year, on 25 March, he married Johanna (Hannah) Friederike Stoermer (1882-1955); they had four sons, Georg Ludwig (born 1915), Joachim (born 1916), Ulrich (born 1918) and Ruprecht (born 1922). By the time their first son Georg Ludwig was born, on 29 October, Bieberbach had moved to Frankfurt am Main where he had been appointed to the second chair at the new university. Schönflies who had been appointed to the first chair of mathematics at Frankfurt am Main in 1914, the year that the university opened, strongly supported Bieberbach for the second chair. He had also been supported by Georg Frobenius who described him as [1]:-

... someone who attacks, with his unusual mathematical acuity, always the deepest and most difficult problems, and might be the most sharp-witted and penetrating thinker of his generation.

One has to add that Frobenius had a very close knowledge of Bieberbach's work for he too had studied the first part of Hilbert's eighteenth problem, had simplified Bieberbach's first paper on the topic, and had produced ideas which had led Bieberbach to further results. Prior to his appointment, in 1914 Bieberbach had studied the polynomials that are now named after him, which approximate a function that conformally maps a given simply-connected domain onto a disc. Of course World War I began in July of 1914 but, although Bieberbach had done military training ten years earlier, he was not called for military service being unfit on health grounds. It was while Bieberbach was at Frankfurt that he produced the Bieberbach Conjecture for which he is best known today. The Conjecture, which appears as a footnote in his 1916 paper *Über die Koeffizienten derjenigen Potenzreihen, welche eine schlichte Abbildung des Einheitskreises vermitteln* Ⓣ states a necessary condition on a holomorphic function to map the open unit disk of the complex plane injectively to the complex plane. More precisely, he conjectured that if

f(z) =a_{0}+a_{1}z+a_{2}z^{2}+ ...

is a conformal mapping of a unit disk on any domain and *a*_{0} = 0, *a*_{1} = 1, then |*a*_{n}| ≤ *n*. In the 1916 paper Bieberbach showed that |*a*_{2}| ≤ 2, and Charles Loewner showed |*a*_{3}| ≤ 3 in 1923. When Loewner did this Bieberbach told him he had joined the "realm of the immortals". After many years of slow but steady progress, the Conjecture was finally proved by Louis de Branges in 1985. Certainly around the time he made his conjecture, there is considerable evidence that Bieberbach was not greatly liked as a person since he thought so highly of himself. A nice comment in a letter Albert Einstein wrote to Max Born in 1919 says [8]:-

Herr Bieberbach's love and admiration for himself and his muse is most delightful.

However, he continued to increase his influence in German mathematics, becoming secretary of the Deutschen Mathematiker-Vereinigung (German Mathematical Society) in 1920 and having his article *New investigations concerning Functions of a Complex Variable* published in the *Encyklopädie der Mathematischen Wissenschaften* Ⓣ.

Frobenius died in 1917 following which his chair at the University of Berlin was filled by Constantin Carathéodory. However, Carathéodory left in 1919 and again the chair became vacant. The first four mathematicians to be offered the chair (Brouwer, Weyl, Herglotz and Hecke) all turned it down. A new attempt to fill the chair produced two leading contenders, Wilhelm Blaschke and Bieberbach, although the Faculty's strong support for Bieberbach also showed slight doubts [1]:-

If also his exposition now and then shows lack of desirable care, this is far outweighed by the liveliness of his scientific initiative and the large-scale range of his investigations.

On 2 January 1921 Bieberbach accepted the Berlin professorship, delighted to hold a chair held by Frobenius for many years. He took up this eminent chair of geometry on 1 April 1921. After this appointment Bieberbach's research contribution was somewhat reduced, with perhaps his most significant being joint work with Issai Schur, published in 1928, *Über die Minkowskische Reduktiontheorie der positiven quadratischen Formen* Ⓣ. At Berlin he acquired a reputation as an inspiring but rather disorganised teacher, perhaps reflecting the doubt already expressed by the Faculty.

The conversion of Bieberbach to the Nazi cause seems to have been quite sudden. On 30 January 1933 Hitler came to power and on 1 April there was the so-called "boycott day" when Jewish shops were boycotted and Jewish lecturers were not allowed to enter the university. Hirsch writes in [4]:-

Everyone who was there had to make a little speech about the rejuvenation of Germany etc. And Bieberbach did this quite nicely and then he said "A drop of remorse falls into my joy because my dear friend and colleague Schur is not allowed to be with us today."

However, soon after this Bieberbach was converted to the views of the Nazis and energetically persecuted his Jewish colleagues. On 7 April 1933 the Civil Service Law was passed which provided the means of removing Jewish teachers from the universities, and of course also to remove those of Jewish descent from other roles. All civil servants who were not of Aryan descent (having one grandparent of the Jewish religion made someone non-Aryan) were to be retired. Bieberbach, together with his four sons, showed their support for the Nazi cause by taking part in an SA march from Potsdam to Berlin. By November 1933, when he acted as one of Walter Ledermann's examiners, Bieberbach was wearing Nazi uniform when conducting the examination. Also at this time he was teaching a course entitled *Great German mathematicians, a race-theoretic approach* and these lectures formed the basis for three papers he published on the topic. Bieberbach developed the notion of a 'German' synthetic style mathematics as opposed to the abstract 'Jewish' analytic style. He founded a journal *Deutsche Mathematik* to encourage this German style in mathematics but, happily, the journal failed in 1943. His behaviour from 1933 onwards was to make political criteria completely override his mathematical judgement. For example [8]:-

It also seems characteristic of Bieberbach's handling of things that a number of very talented young mathematicians - like Collatz, Grunsky, Rinow, and Wielandt - were not given regular positions at the Institute.

In 1934 Edmund Landau was dismissed from his post at Göttingen. By this time Bieberbach was strongly in favour of such actions against Jewish mathematicians. He wrote [7]:-

A few months ago differences with the Göttingen student body ended the teaching activities of Herr Landau. ... This should be seen as a prime example of the fact that representatives of overly different races do not mix as students and teachers. ... The instincts of the Göttingen students felt that Landau was a type who handled things in an un-German manner.

Bieberbach was managing editor of the Jahresbericht der Deutschen Mathematiker-Vereinigung in 1934 and he published an "open letter" in the journal which was highly critical of Harald Bohr because he had attacked Bieberbach's racist views. Since the letter had been published by Bieberbach without him having obtained approval from the other editors, the Deutschen Mathematiker-Vereinigung was critical of his actions and Bieberbach was forced to resign his editorial position.

On 7 April 1938 Schur was forced to resign from the Commissions of the Prussian Academy of Sciences after Bieberbach had written (on 29 March):-

I find it surprising that Jews are still members of academic commissions.

Helmut Grunsky, who was a doctoral student of Bieberbach's in 1932, became editor of the *Jahrbuch über die Fortschritte der Mathematik* in the mid-1930s. He resisted pressure put on him by Bieberbach not to use Jewish referees. At the beginning of 1938 Bieberbach wrote to Grunsky:-

Above all, may you finally dismiss the Jews from your staff in the New Year. ... I emphasise again that your staff of referees must be in accordance with the regulations which have been obligatory to all Germans since30January1933. ... You see how your conduct harms the good reputation of the Academy.

Bieberbach wrote many papers expressing his racist views. Grunsky wrote the obituary [2] of Bieberbach, yet in this article he does not mention Bieberbach's ideological papers at all. Many mathematicians feel that Bieberbach could not have honestly held the views he did, rather the feeling is that he was ambitious to become the leader of German mathematics and followed a route which he thought would make him successful in this. Schappacher writes [8]:-

... three basic conflicts merged ... the question of how much openness to international mathematical relations was adequate for German mathematicians after years of postwar isolation and ill feelings; the longstanding rivalry between Berlin and Göttingen, and the debate about intuitionism vs. formalism ... In this perspective, Bieberbach's sudden conversion to Nazism appears as the attempt to replay the old battle, taking advantage of the new distribution of power in Germany.

After the end of World War II in 1945 Bieberbach lost all his positions because of his political involvement being dismissed and arrested. Despite this Alexander Ostrowski invited him to lecture at Basel University in 1949. Ostrowski respected Bieberbach's contribution as a mathematician, and considered his political views were irrelevant to his remarkable contributions to mathematics. Many were critical of Ostrowski for making this invitation to Bieberbach. Perhaps there is an irony in the fact that de Branges became the first winner of the Ostrowski Prize for solving the Bieberbach conjecture. In 1951 Bieberbach and Friedrich Wilhelm Levi were on a list to fill the second chair in Berlin. Both were in their 60s but had very different wartime experiences, Levi having been dismissed from the University of Leipzig in 1935 because he was Jewish, and Bieberbach having been the leading Nazi mathematician. Perhaps only one outcome was possible in the circumstances - Friedrich Levi was appointed.

In [3] Grunsky gives a list of 137 papers and books by Bieberbach, and a list of 17 students whose doctorates he supervised before his dismissal in 1945 including Wilhelm Süss (1920) and Helmut Grunsky (1932). The books have been particularly influential and we should indicate the range of topics they cover. His first book was *Einführung in die konforme Abbildung* Ⓣ (1915) which went through many editions (fourth edition 1949, English translation as 'Conformal mapping' in 1953, new updated German edition 1967). This work was followed by *Differentialrechnung* Ⓣ (1917), *Integralrechnung* Ⓣ (1918), *Lehrbuch der Funktionentheorie. I. Elemente der Funktionentheorie* Ⓣ (1921), *Funktionentheorie* Ⓣ (1922), *Theorie der Differentialgleichungen* Ⓣ (1923), *Lehrbuch der Funktionentheorie. II. Moderne Funktionentheorie* Ⓣ (1926), *Vorlesungen über Algebra* Ⓣ (1928), *Analytische Geometrie* Ⓣ (1930), *Projektive Geometrie* Ⓣ (1931), *Differentialgeometrie* Ⓣ (1932), *Einleitung in die höhere Geometrie* Ⓣ (1933), *Carl Friedrich Gauss. Ein deutsches Gelehrtenleben* Ⓣ (1938), and *Galilei und die Inquisition* Ⓣ (1938). All these works went through several editions in which Bieberbach added further material.

After Bieberbach was dismissed following the end of World War II, he continued to produce excellent books such as *Theorie der geometrischen Konstruktionen* Ⓣ (1952) "The style is clear and lively", *Theorie der gewöhnlichen Differentialgleichungen auf funktionentheoretischer Grundlage dargestellt* Ⓣ (1953) "written in the author's well-known lively (and sometimes snappy) style, and it makes good reading. Considerable care has been devoted to the simplification of known proofs and to the detailed discussion of phenomena which usually are given but casual attention", *Analytische Fortsetzung* Ⓣ (1955) "No mathematician who is interested in the Taylor series, in interpolatory theory, or in the study of the singularities of analytic functions, can afford to be without it", and *Einführung in die Theorie der Differentialgleichungen im reellen Gebiet* Ⓣ (1956).

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

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