Solomon Lefschetz

Jewish American mathematician (1884-1972)

Solomon Lefschetz (3 September 1884 - 5 October 1972) was an American mathematician who did fundamental work on algebraic topology and its applications to algebraic geometry.


  • The numerical relations existing between ordinary or so-called Plückerian singularities of a plane curve were determined as early as 1834 by PLÜCKER, but the inverse question has been left almost untouched. It may be stated thus: To show the existence of a curve having assigned Plückerian characters; and is equivalent to the determination of the maximum of cusps κM that a curve of order m and genus p may have. VERONESE ... has solved the question for rational curves.
  • In the development of the theory of algebraic functions of one variable the introduction by Riemann of the surfaces that bear his name has played a well-known part. Owing to the partial failure of space intuition with the increase in dimensionality, the introduction of similar ideas into the field of algebraic functions of several variables has been of necessity slow. It was first done by Emile Picard, whose work along this line will remain a classic. A little later came the capital writings of Poincaré in which he laid down the foundations of Analysis Situs, thus providing the needed tools to obviate the failure of space intuition.
    • (1921). "On certain numerical invariants of algebraic varieties with application to abelian varieties". Transactions of the American Mathematical Society 22 (3): 327–406. ISSN 0002-9947. DOI:10.1090/S0002-9947-1921-1501178-3. (quote from p. 328)
  • As is well known, when one endeavors to pass from one-dimensional birational geometry to the higher dimensions, the difficulties multiply enormously. Many results do not extend at all, or if they do, they are apt to assume a far more complicated aspect or else to demand most difficult proofs.
  • In its early phase (Abel, Riemann, Weierstrass), algebraic geometry was just a chapter in analytic function theory. ... A new current appeared however (1870) under the powerful influence of Max Noether who really put "geometry" and more "birational geometry" into algebraic geometry. In the classical mémoire of Brill-Noether (Math. Ann., 1874), the foundations of "geometry on an algebraic curve" were laid down centered upon the study of linear series cut out by linear systems of curves upon a fixed curve ƒ{x, y) = 0. This produced birational invariance (for example of the genus p) by essentially algebraic methods.

Quotes about Lefschetz

  • Solomon Lefschetz had a reputation for “kibitzing” during the lectures of colleagues; at perhaps the first public talk on computing that von Neumann ever gave, von Neumann said, halfway in, “Well, so far so good” — and Lefschetz added, “and so trivial.” (The students passed down an impious faculty song about their elders; the verse for Lefschetz ended, “When he’s at last beneath the sod, he’ll then begin to heckle God.”)
  • "Lefschetz made a brief trip to Rome, and I asked Severi what he thought of Lefschetz's work. 'È bravo,' Severi told me, meaning more or less 'he is talented.' The word 'bravo,' as far as I can tell, has no equivalent in other languages. 'He is no Poincaré,' Severi added. Poincaré was an eagle, 'un'aquila'—and at this, he raised his hand high. Lefschetz was a sparrow, 'un passero,' and he lowered his hand halfway. But Lefschetz was talented, 'è bravo però, è bravo.'"
    • André Weil, The Apprenticeship of a Mathematician (1992), p. 49
  • I would read Lefschetz's book on analysis situs and algebraic geometry - the book about which Hodge used to say that all the important statements were true and all the others false.
    • André Weil, The Apprenticeship of a Mathematician (1992), p. 154
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