Solomon Lefschetz

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Solomon Lefschetz (born September 3, 1884 in Moscow , † October 5, 1972 in Princeton , New Jersey , USA ) was an American mathematician who mainly worked in the field of topology and differential equations .

life and work

The grave of the couple Solomon and Alice H. Lefschetz in Princeton

His parents were Turkish, but his father was an international businessman. Shortly after Lefschetz was born in Moscow, they moved to Paris so that he grew up in a French environment. He studied engineering at the École Centrale Paris with Charles Émile Picard and Paul Émile Appell , where he received his degree in 1905. Since he could not pursue an academic career there as a non-Frenchman, he emigrated to the USA, where he a. 1907 to 1910 worked for the electrical engineering company Westinghouse Electric Corporation in Pittsburgh. In a transformer explosion in the laboratory in November 1907, he lost both hands and part of his forearm. Later he wore artificial hands, which he covered with black gloves. He then taught apprentices in the company in mathematics, but then switched to study at Clark University in Worcester , Massachusetts . In 1911 he received his doctorate from William Edward Story on a topic from algebraic geometry . He became an American citizen the following year.

In 1911 he became an assistant at the University of Nebraska at Lincoln , in 1913 at the University of Kansas at Lawrence , where he became a professor in 1919. In relative isolation, he worked intensively on Henri Poincaré's methods for algebraic topology and showed their importance for algebraic geometry in a series of works, summarized in the book L'analysis situs et la géométrie algébrique . In 1923 he published his famous Fixed Point Theorem , initially for compact orientable manifolds . It says that a continuous mapping of the manifold onto itself has a fixed point if the Lefschetz number does not vanish . In addition, the sum of the Lefschetz numbers gives a topological invariant of . The idea for this came to him from his work on correspondences of the Italian school of algebraic geometry, which Lefschetz admired. Later he expanded the sentence z. B. on manifolds with a margin (1927), so that Brouwer's Fixed Point Theorem resulted, which predicts the existence of a fixed point in continuous mappings of a unit disk. He also simplified his derivation of the sentence. Heinz Hopf gave an interpretation and a simpler proof using homology groups in 1928. Another point of view emphasizes the relationship to the Morse theory , only that one considers vector fields on manifolds (rivers).

A simplified example for the Lefschetz fixed point theorem results from planar continuous mappings that are linearized around their fixed points : Let it be with a matrix ( eigenvalues ). The Lefschetz number for the fixed point is then the sign of ( denote the identity matrix , the determinant). In the case of a source has only eigen values greater than 1, in the case of a sink only eigen values less than 1, in the case of a saddle point eigenvalues larger and smaller than 1 . On a surface of the gender ( holes) such a river has 2 saddle points, 1 source and 1 sink for each hole, so that the sum of the Lefschetz numbers results in a homotopic mapping through a river to identity , the Euler-Poincaré characteristic of the surface.

In 1924 he received a professorship at Princeton on the recommendation of topologist James Alexander , where Oswald Veblen also taught. During this time he developed the algebraic topology of higher-dimensional algebraic varieties from the work of Émile Picard and Henri Poincaré . "Lefschetz pencils" denote bundles of hyperplanes that intersect the variety and are used, similar to Morse theory, to study singular passages ( Picard-Lefschetz formula ).

He developed many concepts of algebraic topology decisively (the word topology even appeared for the first time in 1930 in the title of a book by him) and he wrote an influential textbook of the same name in 1942.

Strangely enough, in the 1930s and 1940s he had an anti-Semitic attitude (although he had Jewish roots himself): He refused to supervise Jewish doctoral students on the grounds that they would not find jobs in the Depression.

During the Second World War he began to be more interested in applications and investigated - partly inspired by Russian works such as those by Lyapunow , Krylow , Andronow , Pontryagin , some of which he also published in English translation - nonlinear differential equations for control theory and the theory of Oscillations, which he continued with topological methods after the war, so that he and his school were one of the fathers of the qualitative theory of differential equations and their stability theory in the 1950s and 1960s (e.g. his book with LaSalle).

From 1944 he visited Mexico regularly and taught there almost annually.

Lefschetz, with a loud and rough tone and a lot of energy and enthusiasm for his field of work, cared little about the elegance of proofs, but saw the main task of mathematics in discovering new things. Gian-Carlo Rota said of him that he almost never published a correct proof or an incorrect theorem.

In 1925 Lefschetz was elected to the National Academy of Sciences and in 1929 to the American Philosophical Society . In 1957 he became a member of the Académie des Sciences . From 1928 to 1958 he was editor of the Annals of Mathematics , which he made into one of the world's leading journals. In 1956 he received the international Antonio Feltrinelli Prize and in 1964 the National Medal of Science . In 1970 he was awarded the Leroy P. Steele Prize of the American Mathematical Society .

His PhD students include Richard Bellman , Albert William Tucker , Ralph Fox , Clifford Dowker , John McCarthy , Clifford Truesdell , Norman Steenrod , John Tukey , Shaun Wylie .

He had been married since 1913. The marriage remained childless.

literature

  • Lefschetz Intersections and transformations of complexes and manifolds , Transactions American Mathematical Society (AMS), Vol. 28, 1926, pp. 1-49, online (Fixedpunktsatz; PDF; 4.3 MB), continued in Vol. 29, 1927, p 429–462, online (PDF; 2.9 MB).
  • ders. L'Analysis situs et la geometrie algebrique , Paris, Gauthier-Villars 1924
  • id. Geometry sur les surfaces et les varietes algebriques , Paris, Gauthier Villars in 1929
  • ders. Topology , AMS 1930
  • ders. Algebraic Topology , New York, AMS 1942
  • id. Introduction to topology , Princeton 1949
  • ders. Algebraic geometry , 2nd ed. Princeton 1964
  • ders. Reminiscences of a mathematical immigrant in the United States , American Mathematical Monthly, Vol. 77, 1970, p. 344
  • ders., Joseph LaSalle The Stability Theory of Ljapunoff - the direct method and its applications , BI University Pocket Book 1967 (English Academic Press 1961)
  • Ders. Differential equations-geometric theory , Interscience, 1957, 2nd ed. 1963
  • ders. Stability of nonlinear control systems , 1965
  • Sundaraman et al. a. The Lefschetz centennial conference , AMS 1984, 3 vol.
  • Albers, Alexanderson Mathematical people , 1985
  • Hodge , in Biographical Memoirs of the Fellows of the Royal Society, Vol. 19, 1973
  • Griffiths, Spencer, Whitehead, in Biographical Memoirs National Academy Sciences Vol. 61

Web links

Individual evidence

  1. ↑ Member History: Solomon Lefschetz. American Philosophical Society, accessed October 27, 2018 .
  2. ^ List of members since 1666: Letter L. Académie des sciences, accessed on January 11, 2020 (French).
personal data
SURNAME Lefschetz, Solomon
BRIEF DESCRIPTION American mathematician
DATE OF BIRTH September 3, 1884
PLACE OF BIRTH Moscow
DATE OF DEATH 5th October 1972
Place of death Princeton, New Jersey