Erich Kähler

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Erich Kähler in Hamburg , 1990

Erich Kähler (born January 16, 1906 in Leipzig ; † May 31, 2000 in Wedel near Hamburg ) was a German mathematician and philosopher.

Life

Kähler studied mathematics, astronomy and physics in Leipzig from 1924 to 1928 and received his doctorate in 1928 under Lichtenstein (and Otto Hölder ) with the thesis "On the existence of equilibrium figures of rotating liquids, which are derived from certain solutions to the n-body problem" . In 1930 he completed his habilitation with Wilhelm Blaschke in Hamburg with the thesis "On the integrals of algebraic differential equations" . In 1929 he became an assistant at the University of Königsberg and worked from 1929 to 1935 at the Mathematical Seminar of the University of Hamburg, from 1930 as a private lecturer, interrupted by a one-year study stay as a Rockefeller scholarship holder in 1931/1932 in Rome , where he taught the Italian geometers Guido Castelnuovo , Francesco Severi , Federigo Enriques and Beniamino Segre met as well as André Weil (who later wrote a book on Kähler manifolds) and Tullio Levi-Civita . A bond with Italy remained from this stay, and so he occasionally published in Italian. In 1934 he accompanied Blaschke to Moscow (partly he traveled with Elie Cartan ), where he met Pawel Sergejewitsch Alexandrow . In 1936 he became a full professor at the University of Königsberg as the successor to Gabor Szegö , where he had been a substitute professor since 1935. According to his own information, he was in the Navy voluntarily from 1937. He received reservist training in the naval artillery in 1938 and 1939 and was drafted into the naval artillery in 1939, which he remained throughout the war. First he was deployed on the German coast and at sea and from 1942 he was on the Atlantic coast as battery chief in the fortress of St. Nazaire . Most recently he was first lieutenant and received the 1st Class War Merit Cross. After the war he was taken prisoner in France (two years on the Ile de Ré and in Mulsanne ). He described the time as a paradise for his research , as he did not have to work as an officer. At the intercession of Frédéric Joliot-Curie (then director of the CNRS ) he was allowed to receive books and he asked Élie Cartan and SS Chern for current mathematical literature. After Kähler, Elie Cartan and André Weil sent him math books. Weil also tried in vain to find him a university position in Sao Paulo in 1947. His wife and children managed to escape from East Prussia in the last days of the war. After a temporary diet lecturer at the University of Hamburg, Kähler was a professor at the University of Leipzig from 1948 (as the successor to Paul Koebe ), which he left in 1958 due to political differences. In Leipzig he gave a five-semester advanced course on algebra, algebraic geometry, function theory and number theory (content of his book Geometria Aritmetica ), sometimes with 10 hours of lectures per week, and had a group of closely connected students who called him masters and his with him worked out ideas in these areas, some of which came from pre-war times. In 1957 he asked for his release because of the conviction of the student pastor Georg-Siegfried Schmutzler for political reasons. After a period from 1958 to 1964 as a full professor at the TU Berlin , he was a full professor at the University of Hamburg from 1964 until his retirement in 1974 as the successor to Emil Artin . From 1964 to 1974 he was director of the Institute for Pure Mathematics there. However, he stayed in his villa in Lankwitz for a long time (and was also an honorary professor at the TU Berlin) and only moved to Wedel in Hamburg in 1974. In Hamburg he gave a lecture on Mathematics I to VIII, but also read about Nietzsche and gave a lecture on mathematics as language and writing . According to his biographer Rolf Berndt, however, he did not find the response he wanted at the Hamburg University, partly due to the fact that he had competition from other outstanding mathematicians and theoretical physicists in Hamburg. Even after his retirement he remained scientifically active, but mainly worked on philosophical questions of mathematics and philosophy. While still in Leipzig, he began to learn Sanskrit, Russian and Chinese.

On February 24, 1955 he was elected a full member of the Berlin Academy of Sciences . In 1957, Kähler was elected a member of the Leopoldina . On November 17, 1966, membership in the Berlin Academy of Sciences was changed to the corresponding one. On May 20, 1969, he was declared a foreign member, and after the reunification in 1990 he was also made a full member. From 1962 he was a member of the Accademia dei Lincei , from 1987 of the Istituto Lombardo Accademia di Scienze e Lettere and from 1949 a member of the Saxon Academy of Sciences.

His doctoral students include Rolf Berndt and Armin Uhlmann .

In 1938 he married the doctor Luise Günther, with whom he had two sons and a daughter (she died in 1970). In 1972 he married the pharmacist Charlotte Kähler. His son Helmuth became an astrophysicist in Hamburg.

plant

After working for three - or n body problem during his doctoral thesis, Kahler dealt with complex analysis . During his study stay in Rome in 1931/1932 he came to the important representatives of algebraic geometry of the "Italian School", Castelnuovo, Enriques and Severi.

During this time the trend-setting idea arose to tie the geometry more closely to algebraic structures and refine them to an arithmetic geometry. Kähler combined the methods of the Italian school of algebraic geometry with methods of differential geometry that he had learned from Blaschke. The Kähler's method of characterizing certain complex Riemannian spaces by a closed differential form is important. Complex manifolds , the metric of which forms a closed differential form, that is to say for which applies, are now called Kahler manifolds . He introduced the Kähler metric and Kählman manifolds in 1932. Kähler manifolds play a fundamental role in the compactification of the extra dimensions in string theory, which is necessary for realistic applications .

Understanding and receiving the work of Kähler, who had strong philosophical inclinations, were made more difficult by the fact that he sometimes used his own terminology. In his great treatise Geometria aritmetica he tries to bring number theory and geometry together by looking at varieties over local rings and not just over bodies. He was thus a forerunner of the theory of schemes by Alexander Grothendieck , who started his program of re-establishing algebraic geometry on the basis of the scheme theory around the same time at the end of the 1950s. Kähler understood Geometria aritmetica as the beginning of a program, which he did not pursue further with the publication of Grothendieck's works. In 1963 he gave a generally understandable overview of his theory. Various ideas contained in Geometria aritmetica were later taken up again in Arithmetic Geometry.

Kähler also dealt with mathematical physics, for example with the Maxwell equations and the Dirac equation in differential form calculus . He developed the differential form calculus of Élie Cartan (Cartan-Kähler theory, Kähler differential forms) and applied it to the theory of systems of differential equations. Other influential work by Kähler concerned the theory of complex functions in two variables.

Kähler was convinced that number theory should play a bigger role in physics. He pursued unconventional ideas. For example, he wanted to replace the Lorentz group in special relativity with a so-called new Poincaré group (according to Hermann Nicolai, it is identical to the de Sitter group). He considered discrete subgroups of this group and associated automorphic forms, which he used to establish relationships with number theory. These ideas were also part of his attempt in later years to develop a comprehensive philosophy on an algebraic basis - he saw the language of mathematics as the basis for solving and treating a wide variety of problems in philosophy, but also in other areas of science and life, whereby he himself how partly used his own terminology in his mathematical work. In the 1970s he gave lectures on philosophy in Hamburg. Much of his philosophical work (such as his Monadology 1975, 1977) remained unpublished.

Fonts

  • Rolf Berndt, Oswald Riemenschneider (editor) Mathematische Werke / Mathematical Works . de Gruyter, Berlin 2003, ISBN 3-11-017118-X
  • Introduction to the theory of systems of differential equations , Hamburger Mathematische Einzelschrift, Teubner 1934
  • Geometria aritmetica , Annali di Matematica, Series IV, Volume 45, 1958, pp. 1-399
  • On the Relationship of Mathematics to Astronomy and Physics , Annual Report DMV, Volume 51, 1941, pp. 52-63 (revised version in the Gauß memorial volume, editor Reichardt, Leipzig 1957)
  • Essence and appearance as mathematical principles of philosophy , Nova Acta Leopoldina, New Series, Volume 30, No. 173, 1965, pp. 9–21
  • Space-time individual , in Heinrich Begehr Mathematik from Berlin , Berlin 1997, pp. 41-105
  • Also spoke Ariadne , Istituto Lombardo, Rend. Sc., A 126, 1992, pp. 105-154
  • Nietzsche's philosophy as the highest stage of German idealism , Spectrum, Volume 22, 1991, pp. 44-46

literature

  • Rolf Berndt : Erich Kähler. In: Annual report of the German Mathematicians Association , Vol. 102, 2000, pp. 178–206.
  • Ernst Kunz , Review by Kähler Mathematische Werke , Mathematical Intelligencer, 2006, No. 1.
  • Horst Schumann : Erich Kähler in Leipzig 1948–1958. In: Herbert Beckert , Horst Schumann (Ed.) 100 Years of Mathematical Seminar at the Karl Marx University in Leipzig. German Science Publishers, Berlin 1981

Web links

Individual evidence

  1. ↑ Stanford Segal, Mathematicians under the Nazis, Princeton UP, 2003, p. 477.Segal interviewed him and described him as right-wing conservative, philosophically influenced by Nietzsche, who in the conversation showed sympathy for Hitler (p. 478). However, he was not a member of the NSDAP.
  2. ↑ Berndt, Annual Report DMV, 2000, p. 180
  3. ^ Segal, Mathematicians under the Nazis, p. 480
  4. ↑ Berndt, Annual Report DMV, 2000, p. 180
  5. ↑ Kähler on a remarkable Hermitian metric , treatises Math. Seminar Universität Hamburg, Volume 9, 1933, pp. 173–186.
  6. ^ For example André Weil in the review of Geometria aritmetica in Mathematical Reviews: The authors seems to have done everything in his power to discourage prospective readers and is only too likely to have succeeded , quoted in Kunz, Review von Kählers Werken, Mathem. Intelligencer 2006, No. 1
  7. ^ Kunz in the review of the works of Kähler, Mathematical Intelligencer 2006, No. 1; Grothendieck points out the importance of Kähler’s work in this area in his Elements de geometrie algebrique .
  8. ↑ Infinitesimal Arithmetic , Univ. Politec. Torino, Rend.Sem.Mat., Volume 21, 1963, pp. 5-29
  9. ↑ Kunz, Math. Intelligencer 2006, No. 1
  10. ↑ Essay in the works of Kähler
  11. ↑ Kähler The Poincaré Group , in J. Chisholm, A. Common Clifford algebras and their application in mathematical physics , NATO Advanced Study Institute, Series C, Volume 183, 1986, pp. 265-272, and in Festschrift for Ernst Mohr, University Library TU Berlin 1985
personal data
SURNAME Kähler, Erich
BRIEF DESCRIPTION German mathematician
DATE OF BIRTH January 16, 1906
PLACE OF BIRTH Leipzig
DATE OF DEATH May 31, 2000
Place of death Hamburg