Biographical Memoirs Volume 61 (1992) / Chapter Skim
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Solomon Lefschetz
Solomon Lefschetz
Pages 270-313
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From page 271... ...
As man and mathematician, his approach to problems, both in life and in mathematics, was often breathtakingly original anct creative. PERSONAL AND PROFESSIONAL HISTORY Solomon Lefschetz was born in Moscow on September 3, ISS4.
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He left the Westinghouse Company in the fall of 1910 and accepted a small fellowship at Clark University, Worcester, Massachusetts, enrolling as a gracluate student. The mathematical faculty consistent of three members: William Edward Story, senior professor (higher plane curves, invariant theory)
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Clark University had a fine library with excellent working conditions, and Lefschetz macle good use of it. By the summer of 1911 he had vastly improved his acquaintance with moclern mathematics and had laid a foundation for future research in algebraic geometry.
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in the second volume. He was unable to clo this directly, and it led him to a recasting of the whole theory, especially the topology.4 By attaching a 2-cycle to the algebraic curves on a surface, he was able to establish a new and unsuspectecl connection between topology and Severi's theory of the base, constructed in 1906, for curves on a surface.
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In 1927, he also proved it for any finite complex anti, in 1936, for any locally connected topological space. In the 1920s and 1930s, as a professor at Princeton University, Lefschetz was wholly occupied with topology, and he established many of the basic results in algebraic topology.
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After some six years it was necessary to transfer the center elsewhere, ant! the move, carried out by LaSalle, resultect in their becoming part of the Division of Applied Mathematics at Brown University.
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The international Conference on Albegraic Geometry, Algebraic Topology anc! Differential Equations (Geometric Theory)
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In simplest terms, algebraic geometry is the stu(ly of algebraic varieties. These are clefined to be the locus of polynomial equations PI(X]
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In the nineteenth century the intensive study of algebraic curves that is, algebraic varieties of dimension onewas undertaken by Abel, Jacobi, Riemann, and others. On an algebraic curve C given by a single define equation, f (x,y)
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(X - as) It was Riemann who emphasized that studying C up to birational equivalence is equivalent to studying the abstract Riemann surface C associated to the curve (21.
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Toward the end of the nineteenth century, the stucly of algebraic curves was extended by Max Noether and others,
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To them, an algebraic surface S was the generic projection into P3 of a smooth algebraic surface S lying in a pN. Thus, S is given by the single affine equation f~x,y,z)
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It is clear that the introduction of homology theory by Poincare was essential for an understanding of rational integrals on a surface, and Poincare's work on "analysis situs" was done while Picard was midstream in his own investigations. In what remains one of the "tour de forces" in the history of mathematics, over a period of about twenty years, Picard was able to arrive at a preliminary understanding of both single and double rational integrals on an algebraic surface.
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module torsion cycles was nondegenerate. · ~ ~ —U7 ~ ~ This is essentially equivalent to what is now known as the algebraic deRham theorem for algebraic surfaces.
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After considerable time it dawned upon me that Picard only dealt with finite 2cycles, the only useful cycles for calculating periods of certain double integrals. Missing link?
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the algebraic surface (7) by considering it as "fibrecI" by the all algebraic curves Cy given by f~x,y,z)
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In a suitable local analytic coordinate system Z~,....,Zn+~ in pn+i centered around Pa' the Lefschetz pencil has the analytic equation 7? ~ ~ ~2 ~1 ' · · · T ~n+1 = t—to, and from this a complete and explicit analysis of the topology of the By as t ~ ta is possible.
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In the case n= I, the cells e'2 are obtained by cutting the Riemann surface We along the retrosections b~ ...., da, y~ ...., fig. As t ~ Id we have a global picture / -my.
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He had marvelous intuition, and so far as T know, all of the results he cIaimecl in algebraic geometry have now been provecl. When ~ was a graduate student at Princeton, it was frequently said that "Lefschetz never stated a false theorem nor gave a correct proof." In the case of the method of Lefschetz' pencils, it was later recognized that he was using t as a complex Morse function, log It- tot being the real Morse function, and this then led to the very beautiful derivation of Lefschetz' theorems, as given by AncireottiFrankel (in Global Analysis, Princeton University Press, 1969)
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To see what it means, we consider the case of algebraic surfaces. Over the punctured t-sphere B = Put, .
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is equivalent to the assertion (18) the intersection form on the space of vanish- (18)
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Here, his results were definitive. To state them we shall specialize to the case of algebraic surfaces, although every
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had been (liscussec! by Picard in his attempt to classify the double integrals (9~.
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of finite characteristic. A suitable cohomology theory is essentially one for which Poincare cluality anti the various Lefschetz theorems stated above, in particular the Hard Lefschetz Theorem, couict be establishect.
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that we have come full circle to the historical roots of algebraic geometry in the study of special transcendental functions arising from abelian integrals, abelian sums, and periods as explained above. In all of these clevelopments, the topological properties of algebraic varieties, as part of the infrastructure of algebraic geometry, play a central role.
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that any map whose degree is different from the degree ~ ~iJn+l of the antipoclal map has a fixed point. If X is a compact polyheclron and q is a non-negative integer, the q th homology group of Xwith rational coefficients is a rational vector space Hq~XJ of finite dimension; an cl a map f: X ~ X induces a homomorphism fq: HqfX)
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relative homology. Like many other results of the time, the I~efschetz Duality Theorem was awkward to state because the correct concepts tract not yet been clevelopecI.
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Thus, the Lefschetz duality theorem appears as a unifying factor, connecting two important but apparently unrelatecl results. Not content with this version of the fixecI-point theorem, Lefschetz continued to seek generalizations.
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as a mapping fof a suitable function space into itself. Anti a solution of the equation is nothing but a fixed point of f To be sure, the function spaces appearing here anti in other places in analysis are far from being compact, ancl so the Lefschetz theorem does not apply directly.
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As for relative homology groups, they are principal ingredients in the axiomatic treatment of the homology theory by Eilenberg and Steenrod, which has been so influential in the development of the subject in the last thirty or so years. LEFSCHETZ AND ORDINARY DIFFERENTIAL EQUATIONS Lefschetz was nearly sixty years old when he turned to differential equations, and he devoted the last twenty-five years of his life to the subject.
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Peixoto provect that the structurally stable systems on a compact surface form an open dense subset of S
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NOTES 1. Phillip Griffiths wrote the section on algebraic geometry, Donald Spencer wrote the sections on personal history and ordinary differential equations, and George Whitehead wrote the section on topology.
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SOLOMON LEFSCHETZ 303 4. Topology can be described as the study of continuous functions, and it is customary to use the work "map" or "mapping" when referring to such functions.
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23:366-68. On the residues of double integrals belonging to an algebraic surface.
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Report on curves traced on algebraic surfaces.
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13:657-59. 1928 Transcendental theory; singular correspondences between algebraic curves; hyperelliptic surfaces and Abelian varieties.
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12. New York: American Mathematical Society.
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1:715-17. Sur les transformations des complexes en spheres.
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Existence of periodic solutions for certain differential equations.
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Complete families of periodic solutions of differential equations. Comment Math.
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Recent Soviet contributions to ordinary differential equations and nonlinear mechanics.
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Proceedings of International Symposium on Nonlinear Differential Equations and Nonlinear Mechanics, Colorado Springs, 1961. New York: Academic Press.
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Lefschetz, including the book L'analysis situs, Chelsea Publishing Company, Bronx, New York, 1971. Sir William Hodge, "Solomon Lefschetz, 1884-1972," in Biographical Memoirs of Fellows of the Royal Society, 19, London, 1973; reprinted in Bulletin of the London Mathematical Society, 6 (1974)
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