Paul Koebe

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Paul Koebe, 1930 in Jena

Paul Koebe (born February 15, 1882 in Luckenwalde , † August 6, 1945 in Leipzig ) was a German mathematician who dealt almost exclusively with function theory.

Life

Koebe was the son of a factory owner in Luckenwalde (fire engines for the fire brigade ) and attended the Joachimsthalsche Gymnasium in Berlin. He studied in Kiel (summer semester 1900) and then at the Technical College and the University in Berlin, where he received his doctorate under Hermann Amandus Schwarz in 1905. Another of his teachers was Friedrich Schottky . He then went to Göttingen , where he completed his habilitation in 1907 and in 1910 became an extraordinary professor. From 1911 to 1914 he was associate professor in Leipzig, then full professor in Jena and from 1926 in Leipzig, where he was dean of the mathematics and natural science faculty from 1933 to 1935. In 1922 he received the Ackermann-Teubner Memorial Prize . In November 1933 he was one of the signatories of the professors' commitment at German universities and colleges to Adolf Hitler and the National Socialist state .

Koebe was a member of the Saxon , Prussian , Heidelberg and Göttingen Academy of Sciences and the Finnish Academy of Sciences . Herbert Grötzsch is one of his doctoral students in Leipzig . Heinz Prüfer completed his habilitation with him and was his assistant.

Koebe never married. He died of stomach cancer. He was buried in the family grave at the Evangelical Cemetery in Luckenwalde.

plant

Koebe quickly became famous in 1907 for his proof of the uniformization theorem for Riemannian surfaces prepared by Felix Klein , Schwarz and Henri Poincaré , a topic to which he repeatedly came back in different variations. This uniformization theorem is the generalization of the Riemann mapping theorem on Riemannian surfaces. He solved the 22nd of Hilbert's problems , at that time one of the greatest unsolved problems in mathematics. For the original proof of the main theorem of the uniformization theory, he used a distortion theorem named after him (the "quarter clause"). Koebe also gave a proof of Riemann's 1914 figure theorem that simplified Carathéodory's 1912 proof . At the same time Poincaré also gave a proof of the main theorem of the theory of uniformity in 1907 with his “Method de Balayage”. The theorem states that a simply connected Riemann surface is biholomorphically equivalent (i.e. can be mapped to ...) either to the Riemann sphere, the complex plane or the unit disk. In the case of any Riemann surfaces that result as the quotient spaces of their overlapping area modulo mappings of discrete groups, the overlapping area is simply connected, and the theorem also applies.

One of Koebe's theorems of distortion is the “Koebes ¼-theorem” (quarter theorem) for depicting the unit disk by simple functions : The open circular disk with a radius around the origin is shown in the picture of the interior of the unit disk D by arbitrary (in D) simple functions. The value is the best possible, as the example of the Koebe function shows.

Koebe also examined the conformal mapping of multiple contiguous flat areas to areas bounded by circles. Here he proved the conformal equivalence (that is, existence of simple mappings) to areas bounded by circles (circle normalization problem) for areas that are finitely multiply connected. The investigations were z. B. continued in the school of William Thurston , who investigated geometrical approaches (via spherical packings) to Riemann's mapping theorem or its extensions in the uniformization theorem. In this context, Oded Schramm proved in 1992 a previously open conjecture by Koebe.

Koebe did not hold back with his view of the importance of his achievements. In Germany there were numerous anecdotes about him and his often somewhat rumble manner. His former assistant Cremer, however, attests to him having a sense of humor and emphasizes the liveliness of his lectures. In addition, Cremer emphasizes that Koebe basically wrote his publications, some of which were very detailed, by himself. His interest centered on function theory, although he also wrote a number of papers on Clifford-Kleinian spatial forms. He wasn't interested in applications at all. He "defended" his specialty very fiercely against competitors.

Anecdotes

Due to his weighty self-assessment, Koebe was also the subject of ridicule and practical jokes. For example, it was spread that even the street boys from Koebe's hometown of Luckenwalde would praise the great function theorist, as Hans Freudenthal recalled, who, like Koebe, came from Luckenwalde (but Koebe had only seen it there once from a distance). On the very first day of his mathematics studies in Berlin, Ludwig Bieberbach asked after he found out about his hometown whether he had also been one of these street boys. It was said that Koebe would only stay in hotels anonymously because he was tired of answering the question of whether he was related to the great function theorist, and among colleagues he was briefly referred to as the greatest function theorist from Luckenwalde .

An incident is known that occurred with LEJ Brouwer . Around 1911, Brouwer was concerned with the strict topological justification of the uniformization theorem by Poincaré and Koebe, on which Koebe's fame was based. Following the DMV symposium on automorphic functions in Karlsruhe in September 1911, at which Brouwer presented his work and Koebe also gave a presentation, Koebe himself asserted priority claims in this matter and declared Brouwer's work to be superfluous, as the results were derived from his own sentences would follow. Brouwer then turned to Hilbert and later even to Poincaré, while in vain asked Koebe to present his own proof (which Koebe was unable to produce because he had got lost in his priority claims against the pioneer of topology Brouwer). Brouwer published his own note on this in 1912 in the news of the Göttingen Academy. Brouwer had also mentioned this in a passage as a small concession to Koebe, but found a reformulation in the published version that equaled his recognition of Koebe's priority. According to an anecdote told by Freudenthal, a stranger with a hat pulled down over his face, a turned-up collar and blue glasses would have come to the printer and inspected the artwork. Koebe himself blamed this, according to Freudenthal, on a bad trick that had been played on him. Brouwer was outraged and subsequently checked very carefully what he released for publication.

Edmund Landau asked his colleagues, including Koebe, at a party in Göttingen to anonymously name on a slip of paper the mathematician who had the highest opinion of himself. All the pieces of paper just named Koebe, only one of them said Paul Koebe and rightly so .

Quotes

“There are many areas of mathematics that can be earned by discovering new results. There are mostly long and steep mountain slopes for grumbling goats. The theory of functions can, however, be compared with a lush marshland, particularly suitable for large cattle. ”(Koebe in his presentation at the annual meeting of the German Mathematicians Association in Jena in 1921, quoted from Cremer)

Fonts

literature

  • Ludwig Bieberbach : The work of Paul Koebes. In: Annual report of the German Mathematicians Association, Volume 70, 1967/1968, p. 148 ( online ).
  • Hubert Cremer: Memories of Paul Koebe. In: Annual Report of the German Mathematicians Association, Volume 70, 1967/1968, p. 158.
  • Otto Volk:  Koebe, Paul. In: New German Biography (NDB). Volume 12, Duncker & Humblot, Berlin 1980, ISBN 3-428-00193-1 , p. 287 f. ( Digitized version ).
  • Rainer Kühnau : Paul Koebe and the theory of functions. In: Herbert Beckert , Horst Schumann (Ed.) 100 Years of Mathematical Seminar at the Karl Marx University in Leipzig. German Science Publishers, Berlin 1981.
  • Henri Paul de Saint-Gervais: Uniformization of Riemann Surfaces. Revisiting a hundred-year-old problem. In: Heritage of European Mathematics. European Mathematical Society, 2016.

Web links

Commons : Paul Koebe  - collection of images, videos and audio files

Individual evidence

  1. ^ Gabriele Dörflinger: Mathematics in the Heidelberg Academy of Sciences . 2014, pp. 32–33.
  2. ↑ one-to-one analytic mappings f of a region G around the origin, with f (0) = 0 and first derivative f '(0) = 1
  3. ↑ Koebe treatises on the theory of conformal figure VI , Math. Journal 1920  ( page no longer available , search in web archives )  Info: The link was automatically marked as defective. Please check the link according to the instructions and then remove this notice. , he suspected this in On the Uniformization of Arbitrary Analytical Curves III , Göttinger Nachrichten 1908, 337@1@ 2Template: Toter Link / gdz.sub.uni-goettingen.de  
  4. ↑ See Kenneth Stephenson Circle Packing - a mathematical tale , Notices AMS, Volume 50, 2003, Issue 11, pdf , with references to Koebe's forerunner role
  5. ↑ As Richard Courant found out, for example, around 1910 when he was working on the Dirichlet principle at Hilbert. See Constance Reid: Courant , Springer-Verlag, 1996, ISBN 0-387-94670-5 .
  6. ^ Mathematical Intelligencer, 1984, No. 2
  7. ↑ Dirk van Dalen LEJ Brouwer , Springer 2013, p. 180ff
  8. ↑ Brouwer On the topological difficulties of the continuity proof of the existence theorem of clearly reversible polymorphic functions on Riemann surfaces , Göttinger Nachr., Pp. 604–606
  9. ↑ Koebe himself had previously made corresponding suggestions in writing to Brouwer as to how he could be honored in the planned publication
  10. ↑ Brouwer, Gött. Message p. 604: ... while for the general case only sentences 3 and 4 are still waiting for exhaustive proof. In the meantime, Mr. Koebe has also succeeded in filling this gap completely (with reference to work by Koebe). Before that, it only said that Koebe had told him that he had closed the gap in articles that were yet to be published (Dirk van Dalen Brouwer , p. 185f). The two sentences did not, however, concern the topological core of the continuity method, the subject of discussion at the symposium in Karlsruhe and afterwards.
  11. ↑ Brouwer Werke Volume 2, 575, Mathematical Intelligencer 1984, No. 2, p. 77
  12. ↑ He even insisted that they had to be locked in a safe at the printer. Van Dalen Brouwer p. 187
  13. ↑ Van Dalen Brouwer , p. 187, based on van der Waerden
personal data
SURNAME Koebe, Paul
BRIEF DESCRIPTION German mathematician
DATE OF BIRTH February 15, 1882
PLACE OF BIRTH Luckenwalde
DATE OF DEATH August 6, 1945
Place of death Leipzig