Solomon Lefschetz

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Solomon Lefschetz
Solomon Lefschetz.jpg
Born (1884-09-03)3 September 1884
Moscow, Russian Empire
Died 5 October 1972(1972-10-05) (aged 88)
Princeton, New Jersey, U.S.
Nationality American
Fields Algebraic topology
Institutions University of Nebraska
University of Kansas
Princeton University
Alma mater École Centrale Paris
Clark University
Doctoral advisor William Edward Story[1]
Doctoral students Edward Begle
Richard Bellman
Felix Browder
Clifford Dowker
George F. D. Duff
Ralph Fox
Ralph Gomory
John McCarthy
Robert Prim
Paul A. Smith
Norman Steenrod
Clifford Truesdell
Albert W. Tucker
John Tukey
Henry Wallman
Shaun Wylie[1]
Known for Lefschetz fixed point theorem
Picard–Lefschetz theory
Lefschetz connection
Lefschetz hyperplane theorem
Lefschetz duality
Lefschetz manifold
Lefschetz number
Lefschetz zeta function
Lefschetz pencil
Lefschetz theorem on (1,1)-classes
Notable awards Bôcher Memorial Prize (1924)
National Medal of Science (1964)
Leroy P. Steele Prize (1970)
Fellow of the Royal Society[2]

Solomon Lefschetz (Russian: Соломо́н Ле́фшец; 3 September 1884 – 5 October 1972) was an American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear ordinary differential equations.[1][2][3][4]


He was born in Moscow into a Jewish family (his parents were Ottoman citizens) who moved shortly after that to Paris. He was educated there in engineering at the École Centrale Paris, but emigrated to the USA in 1905.

He was badly injured in an industrial accident in 1907, losing both hands.[5] He moved towards mathematics, receiving a Ph.D. in algebraic geometry from Clark University in Worcester, Massachusetts in 1911. He then took positions in University of Nebraska and University of Kansas, moving to Princeton University in 1924, where he was soon given a permanent position. He remained there until 1953.

In the application of topology to algebraic geometry, he followed the work of Charles Émile Picard, whom he had heard lecture in Paris at the École Centrale Paris. He proved theorems on the topology of hyperplane sections of algebraic varieties, which provide a basic inductive tool (these are now seen as allied to Morse theory, though a Lefschetz pencil of hyperplane sections is a more subtle system than a Morse function because hyperplanes intersect each other). The Picard–Lefschetz formula in the theory of vanishing cycles is a basic tool relating the degeneration of families of varieties with 'loss' of topology, to monodromy. His book L'analysis situs et la géométrie algébrique from 1924, though opaque foundationally given the current technical state of homology theory, was in the long term very influential (one could say that it was one of the sources for the eventual proof of the Weil conjectures, through SGA7 also for the study of Picard groups of Zariski surface). In 1924 he was awarded the Bôcher Memorial Prize for his work in mathematical analysis.

The Lefschetz fixed point theorem, now a basic result of topology, he developed in papers from 1923 to 1927, initially for manifolds. Later, with the rise of cohomology theory in the 1930s, he contributed to the intersection number approach (that is, in cohomological terms, the ring structure) via the cup product and duality on manifolds. His work on topology was summed up in his monograph Algebraic Topology (1942). From 1944 he worked on differential equations.

He was editor of the Annals of Mathematics from 1928 to 1958. During this time, Annals became an increasingly well-known and respected journal, and Lefschetz played an important role in this.[6]

Lefschetz came out of retirement in 1958, because of the launch of Sputnik, to augment the mathematical component of Glenn L. Martin Company’s Research Institute for Advanced Studies (RIAS) in Baltimore, Maryland. His team became the world's largest group of mathematicians devoted to research in nonlinear differential equations.[7] The RIAS mathematics group stimulated the growth of nonlinear differential equations through conferences and publications. He left RIAS in 1964 to form the Lefschetz Center for Dynamical Systems at Brown University, Providence, Rhode Island.[8]

Selected works[edit]

  • L´Analysis situs et la géométrie algébrique, Paris, Gauthier-Villars 1924[9]
  • Intersections and transformations of complexes and manifolds, Transactions American Mathematical Society (AMS), vol. 28, 1926, pp. 1–49, online ; fixed point theorem, published in vol. 29, 1927, pp. 429–462, online.
  • Géométrie sur les surfaces et les variétés algébriques, Paris, Gauthier Villars 1929[10]
  • Topology, AMS 1930[11]
  • Algebraic Topology, New York, AMS 1942
  • Introduction to topology, Princeton 1949
  • with Joseph P. LaSalle, Stability by Liapunov's direct method with applications, New York, Academic Press 1961[12]
  • Algebraic geometry, Princeton 1953, 2nd edn., 1964
  • Differential equations: geometric theory, Interscience, 1957,[13] 2nd edn., 1963
  • Stability of nonlinear control systems, 1965
  • Reminiscences of a mathematical immigrant in the United States, American Mathematical Monthly, vol.77, 1970, pp. 344–350.


  1. ^ a b c Solomon Lefschetz at the Mathematics Genealogy Project
  2. ^ a b Hodge, W. (1973). "Solomon Lefschetz 1884-1972". Biographical Memoirs of Fellows of the Royal Society. 19: 432. doi:10.1098/rsbm.1973.0016. 
  3. ^ Markus, L. (1973). "Solomon Lefschetz: An appreciation in memoriam". Bull. Amer. Math. Soc. 79 (4): 663–680. doi:10.1090/s0002-9904-1973-13256-2. 
  4. ^ O'Connor, John J.; Robertson, Edmund F., "Solomon Lefschetz", MacTutor History of Mathematics archive, University of St Andrews .
  5. ^ Mathematical Apocrypha: Stories and Anecdotes of Mathematicians and the Mathematical, p. 148, at Google Books
  6. ^ Phillip Griffiths, Donald Spencer and George Whitehead (1992). "Solomon Lefschetz 1884-1972" (PDF). National Academy of Sciences. 
  7. ^ Allen, K. N. (1988, January). Undaunted genius. Clark News, 11(1), p. 9.
  8. ^ About LCDS (Lefschetz Center for Dynamical Systems @ Brown University)
  9. ^ Alexander, J. W. (1925). "Review: S. Lefschetz, L'Analysis Situs et la Géométrie Algébrique". Bull. Amer. Math. Soc. 31 (9): 558–559. doi:10.1090/s0002-9904-1925-04116-6. 
  10. ^ Zariski, O. (1930). "Review: S. Lefschetz, Géométrie sur les Surfaces et les Variétés Algébriques". Bull. Amer. Math. Soc. 36 (9): 617–618. doi:10.1090/s0002-9904-1930-05017-x. 
  11. ^ Smith, P. A. (1931). "Letschetz on Topology". Bull. Amer. Math. Soc. 37 (9, Part 1): 645–648. doi:10.1090/S0002-9904-1931-05201-0. 
  12. ^ Antosiewicz, H. A. (1963). "Review: Joseph LaSalle and Solomon Lefschetz, Stability by Liapunov's direct method with applications". Bull. Amer. Math. Soc. 69 (2): 209–210. doi:10.1090/s0002-9904-1963-10915-5. 
  13. ^ Haas, Felix (1958). "Review: S. Lefschetz, Differential equations: Geometric theory". Bull. Amer. Math. Soc. 64 (4): 203–206. doi:10.1090/s0002-9904-1958-10212-8. 

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