Paul Koebe

Quick Info

15 February 1882
Luckenwalde, Germany
6 August 1945
Leipzig, Germany

Paul Koebe was a German mathematician who worked on complex functions. His colleagues found him brilliant but difficult.


Paul Koebe's father was Otto Hermann Koebe (24 June 1852 - 4 November 1932), known as Hermann, and his mother was Karoline Emma Krämer, known as Emma. Hermann Koebe set up a metal foundry with a copper smithy for pump manufacture in Poststrasse, Luckenwalde in 1878. After building a new factory in the industrial area, he produced the first suction and pressure pumps, later also steam and electric motor pumps. He received his first patent for the "Triumph" pump, which could be operated by only one man. Hermann Koebe products have been used worldwide. A 1902 advertisement reads:-
Hermann Koebe Factory for fire extinguishers, Luckenwalde. Koebe's pump "Triumph" can be blown out and blasted on in a few seconds by one man without any physical exertion. Business founded in 1878.
Paul Koebe was born in Luckenwalde, as was his father. This town is in Brandenburg, on the river Nuthe, about 50 km south of Berlin.

The fire extinguisher business was highly successful and Hermann Koebe was well-off and able to give his son a good education. Koebe received his elementary school education in Luckenwalde and then attended the Joachimsthalsches Gymnasium in Berlin. This school, for gifted boys, was founded in 1607 but in 1880 it moved into a new building in the Kaiserallee district (named the Bundesallee district since 1950). There were several buildings in Italian High Renaissance style with accommodation for teachers and pupils, sports facilities etc. He entered this school in 1891 and there he studied religion, Latin and modern languages, history and geography, mathematics and science. The course, which was based more on practical applications than that of the more academic gymnasium, still qualified Koebe to enter university. Koebe was awarded his secondary school certificate in 1900.

He first studied mathematics and physics at Kiel University which he entered in 1900. There he attended lectures by the physicists Ludwig Claisen (1851-1930) and Philipp Eduard Anton von Lenard (1862-1947), and the mathematicians Goetz Martius (1853-1927), and Paul Stäckel. After one semester he moved to Berlin University where he was to study for five years. At Berlin his thesis was directed by Hermann Schwarz. On 28 July 1904 he took his written Ph.D. examinations and a year later he submitted his thesis Über diejenigen Funktionen eines Arguments, welche ein algebraisches Additionstheorem besitzen . The additional examiner for the oral on his thesis was Friedrich Schottky who had been appointed to Berlin in 1902 while Koebe was in the middle of his studies. Koebe was awarded his doctorate from the Friedrich-Wilhelms-Universität of Berlin on 24 June 1905. His thesis was dedicated to "My dear parents." He wrote in the Introduction to the thesis:-
In his lectures at the University of Berlin on the theory of elliptic functions, Weierstrass took on the task: To determine all analytical functions of an argument that have an algebraic addition theorem. Weierstrass carried out the complete solution of this task in his lectures. To determine the multiform functions with algebraic addition theorem, he used different methods at different times. It is essential to show that the number of branches of such a function is finite.

This thesis contains, among other thing, the communication of what I believe is a simple and natural process by which it can be shown that every multiform function with an algebraic addition theorem is an algebraic function of a unique function. It is undoubtedly the case that Weierstrass must have come extremely close to this method of proof; for the tools which are used in this method of proof closely follow the methods which he himself worked out for the purpose mentioned, the knowledge of which I owe to a lecture given by H A Schwarz.

In the seventh volume of the 'Acta Mathematica', Edvard Phragmén published a method he found for solving the Weierstrass problem mentioned above. Although the proof I have found is unmistakably related to Phragmén's, there seem to be remarkable differences between the two methods.
Koebe also studied for three semesters at the Charlottenburg Technische Hochschule. He writes in his thesis:-
In Berlin or in Charlottenburg I heard lectures by professors and lecturers: F E Blaise, Benedict Friedländer, Georg Frobenius, Lazarus Fuchs, Stanislaus Jolles, Guido Hauck, Kurt Hensel, Georg Hettner, Johannes Knoblauch, Edmund Landau, R Lehmann, Max Planck, Gustav Röthe, Friedrich Schottky, H A Schwarz, Issai Schur, Carl Stumpf, Emil Warburg, and Ulrich von Wilamowitz-Möllendorff.
Koebe undertook research at Göttingen for his habilitation presenting his thesis in 1907. He remained at Göttingen as a docent until 1910 having been promoted to extraordinary professor. Hermann Weyl writes [7]:-
Between 1907 and 1910 Koebe was a dominant figure in Göttingen. The uniformization theorem in all its various forms, which he and Poincaré first proved at that time, occupies a central position in the theory of analytic functions. Koebe never tired of composing new variations on this theme. In a limited field he had great powers of intuition; he was a constructive geometer of the first water. Klein and Poincaré, more than twenty years before, had tried to prove the theorem for the special case of algebraic functions by the so-called continuity method. Later the Riemann problem had been attacked by the same procedure, with equally unsatisfactory results. Abandoning this unwieldy instrument, Koebe and Poincaré now reached their goal by combining H A Schwarz's idea of the universal covering surface with simple estimates of the Harnack type for harmonic functions.
It was in 1907 that Koebe achieved this most famous result on the uniformization of Riemann surfaces, it being a major contribution to Hilbert's Twenty Second Problem. This problem was about uniformization, asking whether every algebraic or analytic curve can be expressed in terms of single-valued functions. Koebe resolved the problem in the one-dimensional case. Shortly after 1900 Koebe had established the general principle of uniformization which had been originally conceived by Klein and Poincaré. Koebe's proof of the uniformization theorem in the papers Über die Uniformisierung reeller algebraischer Kurven (1907) and Über die Uniformisierung beliebiger analytischer Kurven (1907) has been described as:-
... arguably one of the great theorems of the century.
For a list of Koebe's papers, see THIS LINK.

Ludwig Bieberbach was nearly five years younger than Koebe who was a docent at Göttingen during the time that Bieberbach was undertaking research for his doctorate advised by Felix Klein. Koebe proved to be a major influence on the direction of Bieberbach's research. Both Bieberbach and Koebe were keen to emphasise their own importance and to somewhat exaggerate the importance of their results, as happened in an incident described in [4]:-
One of Bieberbach's theorems had to do with bounds on the amount a certain class of complex-analytic maps rotates geometric figures. Paul Koebe had proved an earlier theorem about bounds on the distortions caused by such maps, and Bieberbach's introduction to his paper in volume 4 of the 'Mathematische Zeischrift' (1919) explicitly said that Koebe's "distortion theorem" contributed nothing to his "rotation theorem." There are two questions here: the existence of bounds of a certain type (the qualitative question), and obtaining explicit, perhaps best possible, bounds (the quantitative question). In 1920, in volume 6 of the same journal, Koebe ... said that, on the contrary, the qualitative rotation theorem was an immediate corollary of his distortion theorem, though the quantitative one was not. In 1921 (same journal, volume 9) Bieberbach publicly replied, sort of admitting Koebe was right, but saying that quantitative results were his aim, and anyway, both Koebe's theorem and his rotation theorem flowed directly from another theorem of his: "My conjecture that my 'surface theorem' is the true root of all results known up until now about the behaviour of univalent mappings has thus found complete confirmation." In 1922 Edmund Landau took up the matter in his advanced seminar and wrote Koebe and Bieberbach a joint letter. Edmund Landau said that, as a consequence of this further study, he had come to the conclusion that Koebe was more correct than Bieberbach in their public exchange, but not correct enough!
It would appear that Bieberbach was more angry with Edmund Landau after this incident than he was with Koebe.

Although clearly an outstanding mathematician, nevertheless, Koebe had a reputation for stealing the ideas of others, particularly younger colleagues. Constance Reid writes [3]:-
... Koebe was considered a conceited and disagreeable man with a reputation for picking up the ideas of younger people and, because he was so quick, being able to finalise and publish them first. He was, nevertheless, an outstanding mathematician ...
For example, Richard Courant obtained his doctorate from Göttingen in 1910 under Hilbert's supervision. His thesis was entitled Über die Anwendung des Dirichletschen Prinzipes auf die Probleme der konformen Abbildung . In [8] Ben Yandell summarises quotations from Courant in [3]:-
Reid quotes Courant to the effect that Koebe stole the idea of his thesis, not yet published, of which the two had spoken. Koebe rushed off a paper on the same subject and thus claimed independent credit for the result. Then Koebe showed up at a seminar where Courant was scheduled to speak on his thesis and by virtue of seniority took the position of speaking first. This was not well received by Courant's friends, according to Reid. They rigged up an elaborate apparatus and hid it in a chamber pot under the lectern of a class Koebe was teaching. An alarm sounded at erratic intervals as Koebe lectured. When he finally pulled the mechanism out of the chamber pot, it was greeted with much laughter. One of the mischief makers, Kurt Hahn, saw to it that an article describing the incident appeared in the local paper.
Another famous mathematician whom Koebe seems to have stolen ideas from was L E J Brouwer. In [5] this episode is related in detail with several of Brouwer's letters being quoted. We give only a brief summary below and we point the reader to [5] for the full story. First we note Koebe himself wrote a report of a session of the Deutschen Mathematiker-Vereinigung, introducing the report as follows:-
For its annual meeting in September 1911, the German Mathematical Society decided to make the latest development in the theory of automorphic functions the subject of a special session. This session filled the morning of September 27th. The introductory lecture by Klein, which gave a general orientation, was followed by the lectures and presentations by Brouwer, Koebe, Bieberbach and Hilb.
After the session, a discussion took place which Koebe did not accurately describe in his report, rather turning it to his advantage. On 24 February 1912 Brouwer wrote to Hilbert, beginning his letter as follows (see [5]):-
I am asking you for help and protection in a very unpleasant matter.
After explaining the background to the problem, he writes:-
Only after longer private discussions, in which also Bieberbach, Felix Bernstein and Rosenthal participated, Koebe learned after the talks, from 27 till 29 September, from me which partial result (by the way, formulated by Klein in the 'Mathematische Annalen' 21, and at that time called the Weierstrass theorem) can be obtained via his 'Verzerrungssatz' and which part, to be settled by my contribution, still remains necessary. And, as the above mentioned gentlemen know precisely, in these discussions I have mentioned all the details of my present note.

However, already at that time several warning voices said to me: 'All that, you have explained now to Koebe, you will only with the greatest effort be able to claim as your property, as soon as he has understood you', and indeed Koebe displayed some symptoms that seemed to bear out those voices. So when I was at home again I wanted to refrain from any publication on that particular topic, which is anyway rather far from my interest and with which I only casually concerned myself at the request of Klein, in order to avoid an unpleasant fight with Koebe.

Only after Blumenthal prodded me, and after I had, moreover, heard that Klein would like to see a publication from my hand, the note of the thirteenth of January came forth.
On 9 March, Koebe wrote again to Hilbert:-
After sending you my latest letter, I got the enclosed card from Koebe. It neither brings the retraction of his false claims, demanded by me, nor the promised proofs of his note that he owes me. I now must give up hope of his return to common sense, and I therefore ask you to have my note for the 'Gottinger Nachrichten' printed. All the same, it is important to me to answer here, for your information, Koebe's objections against my note . ... Here Koebe is moving around in a circulus vitiosus: for on the one hand he demands me to praise his as yet unpublished work extensively, on the other hand he tries to prevent me from learning its contents.
There is a strange story about this note of Brouwer's. After it was published Brouwer realised that there had been changes to one of his footnotes, the changed footnote giving Koebe much greater credit than his original did. People at the time believed that Koebe himself had made the change in Brouwer's note. Dalen gives the following quote from the writings of Hans Freudenthal [5]:-
Oral tradition tells a cloak-and-dagger story about this footnote: On some dark afternoon in March an unidentified person wearing a large hat, a turned up collar, and blue glasses called at Dieterick'sche Univ.-Buchdruckerei W. Fr. Kaestner in Göttingen, the printing office of 'Göttinger Nachrichten', and asked for the printer's proof of the next issue. He got it, and after a while he gave it back and left. The identity of this person has never been determined, nor is it known whether he made any change in Brouwer's reading proof, which of course disappeared after printing. I do not know how much of this story is true. To a trustworthy friend of mine who years later asked him about this incident, Koebe explained it as a trick somebody played on him. Though the revised edition of the footnote gives information which at that time was not publicly available, the hypothesis that it was a practical joke cannot at all be excluded in the Göttingen ambiance. Koebe was a picturesque character whose honesty and frankness forbade him to disguise his greatness as a mathematician; in order to escape embarrassing admiration he travelled incognito, and he often said that in his birthplace Luckenwalde the street boys called after him "There goes the famous function theorist!"
We should note that Brouwer does not come out of this exchange much better than Koebe for Brouwer also chooses in his letters to modify things said in the discussion on 27 September 1911 to be more favourable to him.

The above quotes give a very negative impression of Koebe as a person, and this does appear to be the general view of his colleagues at the time. There is, however, a more positive view expressed by Huber Cremer (1897-1983) who was Koebe's assistant from 1927 to 1931.

You can read an English version of Cremer's article [10] at THIS LINK.

Koebe was appointed to Leipzig University in 1910 as an extraordinary professor of mathematics. He became an ordinary professor in 1914 when he accepted a position at Jena University. He returned to Leipzig, this time as an ordinary professor, in 1926. He was dean of the Faculty of Mathematics and Natural Sciences at Leipzig from 1933 to 1935.

The article [9] describes his contributions in some detail and gives a list of 68 publications by Koebe [we list 70 publications at THIS LINK]. These are not, however, a collection of great works on a par with his proof of the uniformization theorem. Koebe's style was pompous and chaotic and, as we have shown above, Koebe anecdotes were famous in Germany between the two wars. He did make other important contributions, however, and his circle domain conjecture is still being attacked. A special case was proved in 1993 by Z-X He and O Schramm.

Freudenthal writes in [1]:-
He tended to deal broadly with special cases of a general theory by a variety of methods ...
Freudenthal, who like Koebe was born in Luckenwalde, also tells us that Koebe's life-style was, as his mathematics, chaotic. It is unclear from what Freudenthal writes whether he is implying that Koebe required a wife to help organise his life but certainly he had no wife, remaining a bachelor all his life. His high quality as a mathematician was recognised with several major honours: he was elected: Corresponding member of the Royal Society of Sciences in Göttingen 1915-1919; Corresponding member of the Academy of Sciences in Göttingen 1919-1945; Corresponding member of the Prussian Academy of Sciences in Berlin 1925-1945; Full member of the Mathematics and Natural Sciences class of the Saxon Academy of Sciences in Leipzig 1927-1945; and Corresponding member of the Heidelberg Academy of Sciences 1942-1945. He also received an honorary award from the Prussian Academy of Sciences (1914), the mathematical prize of the King of Sweden (1927) and the Ackermann-Teubner Prize (1922) from the University of Leipzig for his three memoirs Über die Uniformisierung der algebraischen Kurven , published in volumes 67, 69, and 72 of the Mathematische Annalen.

He died from gastric cancer and was buried in the family tomb in the Evangelical Cemetery in Luckenwalde.

References (show)

  1. H Freudenthal, Biography in Dictionary of Scientific Biography (New York 1970-1990).
    See THIS LINK.
  2. H Beckert and H Schumann (eds.), 100 Jahre Mathematisches Seminar der Karl-Marx-Universität Leipzig (Berlin, 1981).
  3. C Reid, Courant (Springer Science & Business Media, 2013).
  4. S L Segal, Mathematicians under the Nazis (Princeton University Press, 2003).
  5. D van Dalen, L E J Brouwer - Topologist, Intuitionist, Philosopher: How Mathematics Is Rooted in Life (Springer Science & Business Media, 2012).
  6. O Volk, Koebe, Paul, Neue Deutsche Biographie 12 (1979), 287-288.
  7. H Weyl, Mind and Nature: Selected Writings on Philosophy, Mathematics, and Physics (Princeton University Press, 2009).
  8. B Yandell, The Honors Class: Hilbert's Problems and Their Solvers (CRC Press, 2001).
  9. L Bieberbach, Das Werk Paul Koebes, Jahresberichte der Deutschen Mathematiker-Vereinigung 70 (1968), 148-158.
  10. H Cremer, Erinnerungen an Paul Koebe, Jahresbericht der Deutschen Mathematiker-Vereinigung 70 (1968), 158-161.
  11. W Hayman, Function theory 1900-1950, in Development of mathematics 1900-1950, Luxembourg, 1992 (Basel, 1994), 369-384.
  12. Prof Dr phil. habil. Paul Koebe, Professorenkatalog der Universität Leipzig, University of Leipzig.

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Written by J J O'Connor and E F Robertson
Last Update September 2020