Heinz Hopf

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Heinz Hopf (right) in Oberwolfach , together with Hellmuth Kneser

Heinz Hopf , actually Heinrich Hopf (born November 19, 1894 in Gräbschen near Breslau , † June 3, 1971 in Zollikon ), was a German - Swiss mathematician and a pioneer of algebraic topology .

Origin and education

Hopf was born in Gräbschen in Silesia as the son of the Jewish brewery owner Wilhelm Hopf and his wife Elisabeth Kirchner. His mother came from a Protestant family and the father converted to Protestantism. After attending the private school for boys of Karl Mittelhaus and the König-Wilhelm Gymnasium at Breslau he studied from 1913 at the Silesian Friedrich-Wilhelm University in Breslau mathematics . At that time Adolf Kneser , Max Dehn , Ernst Steinitz , Erhard Schmidt and Rudolf Sturm taught in Breslau .

Career

His studies were interrupted by the First World War. He volunteered in 1914 and spent the entire war as a lieutenant on the Western Front . He was badly wounded at Verdun . ( Iron Cross 1918). During the recovery period in 1917 he was able to attend courses with Erhard Schmidt in Breslau , and the preoccupation with the problems posed by Schmidt was, in his own words, a ray of hope during the rest of his time as a soldier. After the war he studied in Heidelberg (with Paul Stäckel and Oskar Perron ) and Berlin , with Issai Schur , Ludwig Bieberbach and Erhard Schmidt. In 1925 he received his doctorate from Erhard Schmidt in Berlin with a thesis on the relationship between curvature in differential geometry and topology. The three-dimensional, simply connected Riemannian manifolds of constant curvature were also classified therein (Clifford-Klein space problem, 3 cases hyperbolic, Euclidean, spherical). Under Schmidt he also intensively studied the early topological work of Brouwer and Henri Poincaré .

From 1925 to 1926 he lived in Göttingen , where he heard from Emmy Noether and met the Russian topologist Pawel Sergejewitsch Alexandrow , with whom he had a lifelong friendship. From Emmy Noether in particular, they both learned to treat homology theory in terms of group theory. In his Göttingen habilitation from 1926 he examined homotopy classes of mappings of n-dimensional manifolds in spheres, thus examined mapping classes of vector fields on such manifolds and gave a proof of Solomon Lefschetz's index theorem for vector fields on closed manifolds (the sum of the indices gives an Euler characteristic , i.e. one fundamental topological invariant, known today as the Poincaré-Hopf Theorem ). In 1928 he further developed these ideas from Lefschetz and gave a proof of the Lefschetz fixed point theorem , in which homology groups appeared for the first time. At the same time he defined a product for the cycles in section manifolds using the number of sections, which later turned out to be an early cohomology concept. 1927–1928 he was with Alexandroff as a Rockefeller Fellow in Princeton , where they worked with Oswald Veblen , James Alexander and Lefschetz. In October 1928 he married Anja von Mickwitz (1891-1967). From 1931 Hopf held a professorship at the ETH Zurich as the successor to Hermann Weyl , who moved to Göttingen, a . a. on the recommendation of Schur, whom he later managed to find temporary shelter at the ETH when he was fleeing from the National Socialists in 1936. In Zurich in 1943, Hopf also took on Swiss citizenship.

In 1931 he defined the Hopf invariant as a topological invariant of mappings between spheres of certain (different) dimensions. In 1935 he and Alexandroff published the well-known textbook Topologie in the basic teaching series of Springer Verlag, which is considered to be one of the first textbooks in this field and had a great influence (only the first of the planned three volumes appeared). In 1939 he dealt with the topology of compact Lie groups, introducing the Hopf algebras , which later became of fundamental importance in the theory of quantum groups.

A work from 1941 Fundamental Group and Second Bettian Group is regarded as one of the first papers on homological algebra , which was developed a short time later, especially by Samuel Eilenberg and Norman Steenrod .

In 1940 he used topological aids to prove that real commutative , but not necessarily associative , division algebras have a maximum dimension of 2 above the real numbers; If there is a unit element, the complex numbers are the only such algebra apart from isomorphism (in this case, associativity follows from commutativity).

His students in Zurich include Beno Eckmann , Hans Samelson and Eduard Stiefel .

Appreciations

The Hopf fibers , H spaces , H groups , Hopf invariants , Hopf algebras , the Hopf link and the Hopf-Rinow theorem about the geodetic completeness of Riemann manifolds are named after him .

Hopf received six honorary doctorates (including Sorbonne, Princeton), received the Moscow Lobachevsky Prize , was a member of the Accademia Nazionale dei Lincei , the National Academy of Sciences of the USA and from 1954 to 1958 President of the International Mathematical Union . In 1950 he gave a plenary lecture at the International Congress of Mathematicians in Cambridge (Massachusetts) ( The n-dimensional spheres and the projective spaces in topology ). In 1958 he was elected a member of the Leopoldina , in 1961 in the American Academy of Arts and Sciences and in 1962 in the American Philosophical Society . Since 1949 he was a corresponding member of the Heidelberg Academy of Sciences and since 1966 a corresponding member of the Göttingen Academy of Sciences .

In 2001 the asteroid (25142) Hopf was named after him.

Heinz Hopf Prize

The ETH Zurich awarded every two years to Heinz Hopf Prize for outstanding scientific achievements in the field of pure mathematics . The award ceremony takes place on the occasion of the Heinz Hopf lectures given by the award winner.

Fonts

Many articles by Hopf z. B. from the Mathematische Annalen, Comm.Math.Helvetici are online here: [1] Some essays from the Proc.Nat.Acad. are online here: [2]

literature

Web links

Individual evidence

  1. ↑ a b Robert A. Nowlan: Heinz Hopf: Biography ( Memento from March 20, 2017 in the Internet Archive ) (Eng .; 74 kB)
  2. ↑ On the Clifford-Klein spatial problem , On the Curvatura integra of closed hypersurfaces , Mathematische Annalen Vol. 95, 1926
  3. ↑ Mapping classes of n-dimensional manifolds , vector fields of n-dimensional manifolds , Mathematische Annalen Vol. 96, 1927
  4. ↑ A new proof of the Lefschetz Formula for invariant points , Proc.Nat.Acad.Sciences Vol. 14, 1928 ( Online ; PDF; 421 kB), improved in A generalization of the Euler-Poincaré formula , News of the Göttingen Academy of Sciences, 1928 ( online )
  5. ↑ About the mapping of the three-dimensional sphere on the spherical surface , Mathematische Annalen Vol. 104, 1931
  6. ^ About the rank of closed Liescher groups , Comm.Math.Helv. Vol. 13, 1941/2
  7. ↑ Comm.Math.Helvetici Vol. 14, 1941/2
  8. ↑ Hopf, Systems of Symmetrical Bilinear Shapes and Euclidean Models of Projective Spaces, Quarterly Journal of the Natural Research Society Zurich, Volume 85, 1940, Supplement No. 32, Festschrift Rudolf Fueter , reprinted in Hopf, Selecta, Springer 1964
  9. ↑ M. Koecher, R. Remmert, Isomorphiesätze von Frobenius and Hopf, in: H.-D. Ebbinghaus u. a., Numbers, Springer 1983, p. 156, 162ff
  10. ↑ Hopf, Rinow on the concept of the complete differential geometric surface , Comm.Math.Helv. Vol. 3, 1931, p. 209
  11. ↑ Member History: Heinz Hopf. American Philosophical Society, accessed October 3, 2018 .
  12. ↑ Holger Krahnke: The members of the Academy of Sciences in Göttingen 1751-2001 (= Treatises of the Academy of Sciences in Göttingen, Philological-Historical Class. Volume 3, Vol. 246 = Treatises of the Academy of Sciences in Göttingen, Mathematical-Physical Class. Episode 3, vol. 50). Vandenhoeck & Ruprecht, Göttingen 2001, ISBN 3-525-82516-1 , p. 118.
  13. ↑ Minor Planet Circ. 43192
personal data
SURNAME Hopf, Heinz
BRIEF DESCRIPTION Swiss mathematician
DATE OF BIRTH November 19, 1894
PLACE OF BIRTH Gräbschen , Switzerland
DATE OF DEATH 3rd June 1971
Place of death Zollikon , Switzerland