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Biographical Memoirs: Volume 61 SOLOMON LEFSCHETZ September 3, 1884-October 5, 1972 BY PHILLIP GRIFFITHS, DONALD SPENCER, AND GEORGE WHITEHEAD1 SOLOMON LEFSCHETZ was a towering figure in the mathematical world owing not only to his original contributions but also to his personal influence. He contributed to at least three mathematical fields, and his work reflects throughout deep geometrical intuition and insight. As man and mathematician, his approach to problems, both in life and in mathematics, was often breathtakingly original and creative. PERSONAL AND PROFESSIONAL HISTORY Solomon Lefschetz was born in Moscow on September 3, 1884. He was a son of Alexander Lefschetz, an importer, and his wife, Vera, Turkish citizens. Soon after his birth, his parents left Russia and took him to Paris, where he grew up with five brothers and one sister and received all of his schooling. French was his native language, but he learned Russian and other languages with remarkable facility. From 1902 to 1905, he studied at the École Centrale des Arts et Manufactures, graduating in 1905 with the degree of mechanical engineer, the third youngest in a class of 220. His reasons for entering that institution were complicated, for as he said, he had been ''mathematics mad'' since he had his first contact with geometry at thirteen.

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Biographical Memoirs: Volume 61 Since he was not a French citizen, he could neither see nor hope for a career as a pure mathematician. The next best thing was engineering because, as he believed, it used a lot of mathematics. Upon graduating in 1905, Lefschetz decided to go to the United States, for a time at least, with the general purpose of acquiring practical experience. First, he found a job at the Baldwin Locomotive Works near Philadelpia. But he was particularly attracted to electrical engineering, which, at that time, was a nonexistent specialty at the École Centrale. In view of this, in January 1907 he became an engineering apprentice in a regular course at the Westinghouse Electric and Manufacturing Company in Pittsburgh. The course consisted of being shifted from section to section every few weeks. He wound up in the transformer testing section in the late fall of 1907, and in mid-November of that year, he was the victim of a testing accident, as a consequence of which he lost both hands.2 After some months of convalescence, he returned to the Westinghouse Company, where, in 1909, he was attached to the engineering department in the section concerned with the design of alternating-current generators. Meanwhile, Lefschetz had become increasingly dissatisfied with his work there, which seemed to him to be extremely routine. So he resumed, first as a hobby, his mathematical studies that had been neglected since 1903. After a while he decided to leave engineering altogether and pursue mathematics. He left the Westinghouse Company in the fall of 1910 and accepted a small fellowship at Clark University, Worcester, Massachusetts, enrolling as a graduate student. The mathematical faculty consisted of three members: William Edward Story, senior professor (higher plane curves, invariant theory); Henry Taber (complex analysis, hypercomplex number systems); and Joseph de Perott (number

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Biographical Memoirs: Volume 61 theory). At the École Centrale there were two professors of mathematics, Émile Picard and Paul Appel, and each had written a three-volume treatise: Analysis (Picard) and Analytical Mechanics (Appel). Lefschetz plunged into these and, with a strong French training in basic mathematics, was all set to attack a research topic suggested by Professor Story, namely, to find information about the largest number of cusps that a plane curve of given degree may possess. Lefschetz made an original contribution to this problem and obtained his Ph.D. summa cum laude in 1911. In the Record of Candidacy for the Ph.D., it is stated by Henry Taber that it was an "excellent examination, the best ever passed by any candidate in the department," and signed by him under the date June 5, 1911. Clark University had a fine library with excellent working conditions, and Lefschetz made good use of it. By the summer of 1911 he had vastly improved his acquaintance with modern mathematics and had laid a foundation for future research in algebraic geometry. He had also become more and more closely associated with another mathematics student at Clark, Alice Berg Hayes, who became his wife on July 3, 1913, in North Brookfield, Massachusetts. She was to become a pillar of strength for Lefschetz throughout the rest of his life, helping him to rise above his handicap and encouraging him in his work. Lefschetz' first position after Clark was an assistantship at the University of Nebraska in 1911; the assistantship was soon transformed into a regular instructorship. In 1913 he moved to the University of Kansas, passing through the ranks to become a full professor in 1923. He remained at the University of Kansas until 1924. Then, in 1924 came the call to Princeton University, where he was visiting professor (1924-25); associate professor (1925-27); full professor (1927-33); and from 1933 to 1953, Henry Burchard

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Biographical Memoirs: Volume 61 Fine Research Professor, chairman of the Department of Mathematics 1945-53 and emeritus from 1953. The years in the Midwest were happy and fruitful ones for Lefschetz. The almost total isolation played in his development "the role of a job in a lighthouse which Einstein would have every young scientist assume so that he may develop his own ideas in his own way."3 His two major ideas came to him at the University of Kansas. The first idea is described by Lefschetz as follows. Soon after his doctorate he began to study intensely the two-volume treatise of Picard-Simart, Fonctions Algébriques de Deux Variables, and he first tried to extend to several variables the treatment of double integrals of the second kind found in the second volume. He was unable to do 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 unsuspected connection between topology and Severi's theory of the base, constructed in 1906, for curves on a surface. The development of these and related concepts led to a Mémoire, which was awarded the Bordin Prize by the French Academy of Sciences in 1919. The translated prize paper is given in the Bibliography (1921,3). The first half of the Mémoire, with some complements, is embodied in a famous monograph (1924,1). The general idea for the second most important contribution also came to Lefschetz in Lawrence, Kansas, and it is the fixed-point theorem which bears his name. Almost all of Lefschetz' topology arose from his efforts to prove fixed-point theorems. In 1912, L. E. J. Brouwer proved a basic fixed-point theorem, namely, that every continuous transformation of an n-simplex into itself has at least one fixed point. In a series of papers, Lefschetz obtained a much more general result for any continuous transfor-

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Biographical Memoirs: Volume 61 mation of a topological space X into itself where the restrictions on X were progressively weakened. In 1923, he proved the theorem for compact orientable manifolds and, by introducing relative homology groups, he extended it in 1927 to manifolds with boundary; his theorem then included Brouwer's. In 1927, he also proved it for any finite complex and, 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. For example, he created a theory of intersection of cycles (1925,1; 1926,1), introduced the notion of cocycle (which he called pseudo-cycle) and proved the Lefschetz duality theorem (see 1949,1 for an exposition of the fixed-point theorem and the duality theorem). His Topology was published in 1930 (1930,1), and his Algebraic Topology was published in 1942 (1942,1). The former was widely acclaimed and established the name topology in place of the previously used term analysis situs; the latter was less influential but secured the use of the name algebraic topology as a replacement for combinational topology.5 Lefschetz was an editor of the Annals of Mathematics from 1928 to 1958, and his influence dominated the editorial policy that made the Annals into a foremost mathematical journal. In 1943 Lefschetz became a consultant for the U.S. Navy at the David Taylor Model Basin near Washington, D.C. There he met and worked with Nicholas Minorsky, who was a specialist on guidance systems and the stability of ships and who brought to Lefschetz' attention the importance of the applications of the geometric theory of ordinary differential equations to control theory and nonlinear mechanics. From 1943 to the end of his life, Lefschetz'

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Biographical Memoirs: Volume 61 main interest was centered around ordinary nonlinear differential equations and their applications to controls and the structural stabilities of systems. Lefschetz was almost sixty years old when he turned to differential equations, yet he did original work and stimulated research in this field as a gifted scientific administrator. In 1946, the newly established Office of Naval Research funded a project on ordinary nonlinear differential equations, directed by Lefschetz, at Princeton University. This project continued at Princeton for five years past Lefschetz' retirement from the university in 1953. Meanwhile, the Research Institute for Advanced Study was formed in Baltimore, Maryland, as a division of the Glen L. Martin Aircraft Company, and in 1957, Lefschetz established the Mathematics Center under the auspices of the institute and was entrusted with the recruitment of five mathematicians and about ten younger associates. He obtained the cooperation of Professor Lamberto Cesari of Purdue University and appointed Professor J. P. LaSalle of Notre Dame and Dr. J. K. Hale of Purdue to the group, the former as his second in command. After some six years it was necessary to transfer the center elsewhere, and the move, carried out by LaSalle, resulted in their becoming part of the Division of Applied Mathematics at Brown University. The group was later named the Lefschetz Center for Dynamical Systems. LaSalle was director and Lefschetz became a visiting professor, traveling there from Princeton once a week. Lefschetz continued his work at Brown until 1970, two years before his death. In 1944, Lefschetz joined the Institute de Mathematicas of the National University of Mexico as a part-time visiting professor, and this connection continued until 1966. At the Institute, he conducted seminars, gave volunteer courses, and continued his research. He found a number of ca-

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Biographical Memoirs: Volume 61 pable young men there and sent several of them to Princeton University for further advanced training up to the doctorate and beyond. From 1953 to 1966 he spent most of his winters in Mexico City. Lefschetz received many honors. He served as president of the American Mathematical Society in 1935-36. He received the Bôcher Memorial Prize of the American Mathematical Society in 1924, and in 1970 he received the first award of the Steele Prize, also of the American Mathematical Society. He received the Antonio Feltrinelli International Prize of the National Academy of Lincei, Rome, in 1956; the Order of the Aztec Eagle of Mexico in 1964; and the National Medal of Science (U.S.) in 1964. He was awarded honorary degrees by the University of Prague, Prague, Czechoslavakia; University of Paris, Paris, France; the University of Mexico; and Brown, Clark, and Princeton universities. He was a member of the American Philosophical Society and a foreign member of the Academie des Sciences of Paris, the Royal Society of London, the Academia Real de Ciencias of Madrid, and the Reale Instituto Lombardo of Milan. A symposium in honor of Lefschetz' seventieth birthday was held in Princeton in 1954,6 and in 1965 an international conference in differential equations and dynamical systems was dedicated to him at the University of Puerto Rico. The international Conference on Albegraic Geometry, Algebraic Topology and Differential Equations (Geometric Theory), in celebration of the centenary of Lefschetz' birth, was held at the Centro de Investigaci ón del IPN, Mexico City, in 1984. LEFSCHETZ AND ALBEGRAIC GEOMETRY In order to discuss Lefschetz' contributions to algebraic geometry, I shall first describe that field and its evolution

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Biographical Memoirs: Volume 61 up until the period during which Lefschetz worked. Then I will give a somewhat more detailed description of some of his major accomplishments. I will conclude with a few observations about the impact of his work in algebraic geometry. In simplest terms, algebraic geometry is the study of algebraic varieties. These are defined to be the locus of polynomial equations Here the xi are coordinates in an affine space and the Pa are polynomials whose coefficients are in any field K For our purposes, it will be convenient to take K to be the complex numbers, as this was the case in classical algebraic geometry and in almost all of Lefschetz' work. It is worth noting, however, that he was one of the first to consider the case where K is an arbitrary algebraically closed field of characteristic zero. In fact, the so-called Lefschetz principle as expanded in his book Algebraic Geometry (1953, 1) roughly states that any result from the complex case remains valid in this more general situation. In addition to using complex numbers, it is also convenient to add to the above locus the points at infinity. This is accomplished by homogenizing the polynomials Pa and considering the resulting locus V in the complex projective space PN defined by the homogenized equations. Two algebraic varieties V and V' are to be identified if there is a rational transformation that takes V to V' and is generically one to one there. These are called birational transformations, and T estab-

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Biographical Memoirs: Volume 61 lishes an isomorphism between the fields K(V') and K(V) of rational functions on V' and V, respectively. In the nineteenth century the intensive study of algebraic curves —that is, algebraic varieties of dimension one—was undertaken by Abel, Jacobi, Riemann, and others. On an algebraic curve C given by a single affine equation, in the plane, special objects of interest were the abelian integrals where R(x,y) is a rational function. For example, the hyperelliptic integrals are abelian integrals on the hyperelliptic curve y2 = (x-a1) (x-an) In addition to the indefinite integral (3), abelian sums and periods where γ is a closed path on C, were of considerable interest. A major reason for studying abelian integrals and their periods was that these provided an extremely interesting class of transcendental functions, such as the elliptic function p(u) defined up to an additive constant by

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Biographical Memoirs: Volume 61 It was Riemann who emphasized that studying C up to birational equivalence is equivalent to studying the abstract Riemann surface associated to the curve (2). Assuming that f is irreducible, in modern terms is a connected, complex manifold of dimension one for which there is a holomorphic mapping whose image is C and where π: → C is generically one to one. Viewed as an oriented real two-manifold, the Riemann surface has a single topological invariant, its genus g, and we have the familiar picture where δ1 δg, γ1 γg form a canonical basis for H1(, Z). The introduction of greatly clarifies the study of abelian integrals. For example, in terms of a local holomorphic coordinate z on , the rational differential ω = R(x,y)dx above is given by the expression where

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Biographical Memoirs: Volume 61 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. 5. F. Nebeker and A. W. Tucker, "Lefschetz, Solomon," in Dictionary of Scientific Biography, Supplement II, 1991. 6. Algebraic Geometry and Topology, a Symposium in Honor of S. Lefschetz, edited by R. H. Fox, D. C. Spencer, and A. W. Tucker, Princeton University Press, 1957, pp. 1-49. 7. Ibid, note 6.

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Biographical Memoirs: Volume 61 BIBLIOGRAPHY OF S. LEFSCHETZ 1912 Two theorems on conics. Ann. Math. 14:47-50. On the V33 with five nodes of the second species in S4. Bull. Am. Math. Soc. 18:384-86. Double curves of surfaces projected from space of four dimensions Bull. Am. Math. Soc. 19:70-74. 1913 On the existence of loci with given singularities. Trans. Am. Math. Soc. 14:23-41. (Doctoral dissertation, Clark University, 1911.) On some topological properties of plane curves and a theorem of M öbius. Am. J. Math 35:189-200. 1914 Geometry on ruled surfaces. Am. J. Math. 36:392-94. On cubic surfaces and their nodes. Kans. Univ. Sci. Bull. 9:69-78. 1915 The equation of Picard-Fuchs for an algebraic surface with arbitrary singularities. Bull. Am. Math. Soc. 21:227-32. Note on the n-dimensional cycles of an algebraic n-dimensional variety. R. C. Mat. Palermo 40:38-43. 1916 The arithmetic genus of an algebraic manifold immersed in another Ann. Math. 17:197-212. Direct proof of De Moivre's formula. Am. Math. Mon. 23:366-68. On the residues of double integrals belonging to an algebraic surface Quart. J. Pure Appl. Math. 47:333-43. 1917 Note on a problem in the theory of algebraic manifolds. Kans. Univ. Sci. Bull. 10:3-9. Sur certains cycles à deux dimensions des surfaces algébriques. R. C. Accad. Lincei 26:228-34. Sur les intégrales multiples des variétiés algébriques. C. R. Acad. Sci. Paris 164:850-53.

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Biographical Memoirs: Volume 61 Sur les intégrals doubles des variétiés algébriques. Annali Mat. 26: 227-60. 1919 Sur l'analyse situs des variétiés algébriques. C. R. Acad. Sci. Paris 168:672-74. Sur les variétiés abéliennes. C. R. Acad. Sci. Paris 168:758-61. On the real folds of Abelian varieties. Proc. Natl. Acad. Sci. U.S.A. 5:103-6. Real hypersurfaces contained in Abelian varieties. Proc. Natl. Acad. Sci. U.S.A. 5:296-98. 1920 Algebraic surfaces, their cycles and integrals. Ann. Math. 21:225-28. (Correction, Ann. Math. 23:333.) 1921 Quelques remarques sur la multiplication complexe. Comptes Rendus du Congrès International des Mathématiciens, Strasbourg, September 1920. Toulouse: É. Privat. Sur le théorème d'existence des fonctions abéliennes. R. C. Accad. Lincei 30:48-50. On certain numerical invariants of algebraic varieties with application to Abelian varieties. Trans. Am. Math. Soc. 22:327-482. 1923 Continuous transformations of manifolds. Proc. Natl. Acad. Sci. U.S.A. 9:90-93. Progrès récents dans la théorie des fonctions abéliennes. Bull. Sci. Math. 47:120-28. Sur les intégrales de seconde espèce des variétiés algébriques. C. R. Acad. Sci. Paris 176:941-43. Report on curves traced on algebraic surfaces. Bull. Am. Math. Soc. 29:242-58. 1924 L'analysis situs et la géométrie algébrique. Collection de monographies publiée sous la direction de M. Émile Borel. Paris: Gauthier-Villars. (New edition, 1950.)

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Biographical Memoirs: Volume 61 Sur les integrals multiples des variétiés algébriques. J. Math. Pure Appl. 3:319-43. 1925 Intersections of complexes on manifolds. Proc. Natl. Acad. Sci. U.S.A. 11:287-89. Continuous transformations of manifolds. Proc. Natl. Acad. Sci. U.S.A. 11:290-92. 1926 Intersections and transformations of complexes and manifolds. Trans. Am. Math. Soc. 28:1-49. Transformations of manifolds with a boundary. Proc. Natl. Acad. Sci. U.S.A. 12:737-39. 1927 Un théorème sur les fonctions abélinnes. In Memorian N. I. Lobatschevskii, pp. 186-90. Kazan, USSR: Glavnauka. Manifolds with a boundary and their transformations. Trans. Am. Math. Soc. 29:429-62, 848. Correspondences between algebraic curves. Ann. Math. 28:342-54. The residual set of a complex on a manifold and related questions Proc. Natl. Acad. Sci. U.S.A. 13:614-22, 805-7. On the functional independence of ratios of theta functions. Proc. Natl. Acad. Sci. U.S.A. 13:657-59. 1928 Transcendental theory; singular correspondences between algebraic curves; hyperelliptic surfaces and Abelian varieties. In Selected Topics in Algebraic Geometry, vol. 1, chapters 15-17, pp. 310-95. Report of the Committee on Rational Transformations of the National Research Council, Washington. NRC Bulletin no. 63. Washington, D.C.: National Academy of Sciences. A theorem on correspondence on algebraic curves. Am. J. Math. 50:159-66. Closed point sets on a manifold. Ann. Math. 29:232-54. 1929 Géométrie sur les surfaces et les variétiés algébriques. Mémorial des Sciences Mathématiques, Fasc. 40. Paris: Gauthier-Villars.

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Biographical Memoirs: Volume 61 Duality relations in topology. Proc. Natl. Acad. Sci. U.S.A. 15:367-69. 1930 Topology. Colloquium Publications, vol. 12. New York: American Mathematical Society. Les transformations continues des ensembles fermés et leurs points fixes. C. R. Acad. Sci. Paris 190:99-100. (With W. W. Flexner.) On the duality theorems for the Betti numbers of topological manifolds Proc. Natl. Acad. Sci. U.S.A. 16:530-33. On transformations of closed sets. Ann. Math. 31:271-80. 1931 On compact spaces. Ann. Math. 32:521-38. 1932 On certain properties of separable spaces. Proc. Natl. Acad. Sci. U.S.A. 18:202-3. On separable spaces. Ann. Math. 33:525-37. Invariance absolute et invariance relative en géométrie algébrique. Rec. Math. (Mat. Sbornik) 39:97-102. 1933 On singular chains and cycles. Bull. Am. Math. Soc. 39:124-29. (With J. H. C. Whitehead.) On analytical complexes. Trans. Am. Math. Soc. 35:510-17. On generalized manifolds. Am. J. Math. 55:469-504. 1934 Elementary One- and Two-Dimensional Topology. Princeton, N.J.: Princeton University. (Mimeograph.) On locally connected and related sets. Ann. Math. 35:118-29. 1935 Topology. Princeton, N.J.: Princeton University. (Mimeograph.) Algebraicheskaia geometriia: metody, problemy, tendentsii. In Trudy Vtorogo Vsesoiuznogo Matematischeskogo S''ezda, Leningrad, 24-30 June 1934, vol. 1, pp. 337-49. Leningrad-Moscow. Chain-deformations in topology. Duke Math. J. 1:1-18. Application of chain-deformations to critical points and extremals Proc. Natl. Acad. Sci. U.S.A. 21:220-22.

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Biographical Memoirs: Volume 61 A theorem on extremals. I, II. Proc. Natl. Acad. Sci. U.S.A. 21:272-74. On critical sets. Duke Math. J. 1:392-412. 1936 On locally-connected and related sets (second paper). Duke Math. J. 2:435-42. Locally connected sets and their applications. Rec. Math. (Mat. Sbornik) 1:715-17. Sur les transformations des complexes en sphères. Fund. Math. 27:94-115. Matematicheskaia deiatel'nost'v Prinstone. Usp. Mat. Nauk 1:271-73. 1937 Lectures on Algebraic Geometry. Part 1. 1936-37. Princeton, N.J.: Princeton University Press. (Planograph.) Algebraicheskaia geometriia. Usp. Mat. Nauk 3:63-77. The role of algebra in topology. Bull. Am. Math. Soc. 43:345-59. On the fixed point formula. Ann. Math. 38:819-22. 1938 Lectures on Algebraic Geometry. Part 2. 1937-38. Princeton, N.J.: Princeton University Press. On chains of topological spaces. Ann. Math. 39:383-96. On locally connected sets and retracts. Proc. Natl. Acad. Sci. U.S.A. 24:392-93. Sur les transformations des complexes en sphères (note complèmentaire). Fund. Math. 31:4-14. Singular and continuous complexes, chains and cycles. Rec. Math. (Mat. Sbornik) 3:271-85. 1939 On the mapping of abstract spaces on polytopes. Proc. Natl. Acad. Sci. U.S.A. 25:49-50. 1941 Abstract complexes. In Lectures in Topology; The University of Michigan Conference of 1940, pp. 1-28. Ann Arbor: University of Michigan Press. 1942 Algebraic Topology. Colloquium Publications, vol. 27. New York: American Mathematical Society.

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Biographical Memoirs: Volume 61 Topics in Topology. Annals of Mathematics Studies, no. 10. Princeton, N.J.: Princeton University Press. (A second printing, 1951.) Émile Picard (1856-1941): Obituary. American Philosophical Society Yearbook 1942, pp. 363-65. 1943 N. Kryloff and N. Bogoliuboff. Introduction of Nonlinear Mechanics. Annals of Mathematics Studies, no. 11. Translation by S. Lefschetz. Princeton, N.J.: Princeton University Press. Existence of periodic solutions for certain differential equations Proc. Natl. Acad. Sci. U.S.A. 29:29-32. 1946 Lectures on Differential Equations. Annals of Mathematics Studies, no. 14. Princeton, N.J.: Princeton University Press. 1949 Introduction to Topology. Princeton Mathematical Series, no. 11. Princeton, N.J.: Princeton University Press. A. A. Andronow and C. E. Chaikin. Theory of Oscillations. English language edition ed. S. Lefschetz. Princeton, N.J.: Princeton University Press. Scientific research in the U.S.S.R.: Mathematics. Am. Acad. Polit. Soc. Sci. Ann. 263:139-40. 1950 Contributions to the Theory of Nonlinear Oscillations, ed. S. Lefschetz. Annals of Mathematics Studies, no. 20. Princeton, N.J.: Princeton University Press. The structure of mathematics. Am. Sci. 38:105-11. 1951 Numerical calculations in nonlinear mechanics. In Problems for the Numerical Analysis of the Future, pp. 10-12. National Bureau of Standards, Applied Math. Series, no. 15. Washington, D.C.: U.S. Government Printing Office. 1952 Contributions to the Theory of Nonlinear Oscillations, vol. 2, ed. S. Lefschetz. Annals of Mathematics Studies, no. 29. Princeton, N.J.: Princeton University Press.

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Biographical Memoirs: Volume 61 Notes on differential equations. In Contributions to the Theory of Nonlinear Oscillations, vol. 2, pp. 67-73. 1953 Algebraic Geometry. Princeton Mathematical Series, no. 18. Princeton, N.J.: Princeton University Press. Algunos trabajos recientes sobre ecuaciones diferenciales. In Memoria de Congreso Cientifico Mexicana U.N.A.M., Mexico, vol. 1, pp. 122-23. Las grades corrientes en las matemáticas del siglo XX. In Memoria de Congreso Cientifico Mexicana U.N.A.M., Mexico, vol. 1, pp. 206-11. 1954 Russian contributions to differential equations. In Proceedings of the Symposium on Nonlinear Circuit Analysis, New York, 1953, pp. 68-74. New York: Polytechnic Institute of Brooklyn. Complete families of periodic solutions of differential equations Comment Math. Helv. 28:341-45. On Liénard's differential equation. In Wave Motion and Vibration Theory, pp. 149-53. American Mathematical Society Proceedings of Symposia in Applied Math., vol. 5. New York: McGraw-Hill. 1956 On a theorem of Bendixson. Bol. Soc. Mat. Mexicana 1:13-27. Topology, 2nd ed. New York: Chelsea Publishing Company. (Cf. 1930,1.) 1957 On coincidences of transformations. Bol. Soc. Mat. Mexicana 2:16-25. The ambiguous case in planar differential systems. Bol. Soc. Mat. Mexicana 2:63-74. Withold Hurewicz. In memoriam. Bull. Am. Math. Soc. 63:77-82. Sobre la modernizacion de la geometria. Rev. Mat. 1:1-11. Differential Equations: Geometric Theory. New York: Interscience.(Cf. 1962,1.) 1958 On the critical points of a class of differential equations. In Contributions to the Theory of Nonlinear Oscillations, vol. 4, pp. 19-28. Princeton, N.J.: Princeton University Press.

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Biographical Memoirs: Volume 61 Liapunov and stability in dynamical systems. Bol. Soc. Mat. Mexicana 3:25-39. The Stability Theory of Liapunov. Lecture Series no. 37. College Park, Md.: University Institute for Fluid Dynamics and Applied Mathematics. 1960 Controls: An application to the direct method of Liapunov. Bol. Soc. Mat. Mexicana 5:139-43. Algunas consideraciones sobre las matemáticas modernas. Rev. Unión Mat. Argent. 20:7-16. Resultados nuevos sobre casos criticos en ecuaciones diferenciales Rev. Unión Mat. Argent. 20:122-24. 1961 The critical case in differential equations. Bol. Soc. Mat. Mexicana 6:5-18. Geometricheskaia Teoriia Differentsial'nykh Uravnenii. Moskva: Izd-vo Inostrannoi Lit-ry. (Translation of 1957,5.) (With J. P. LaSalle.) Stability by Liapunov's Direct Method. New York: Academic Press. (With J. P. LaSalle.) Recent Soviet contributions to ordinary differential equations and nonlinear mechanics. J. Math. Anal. Appl. 2:467-99. 1962 Differential Equations: Geometric Theory, 2nd rev. ed. New York: Interscience. (Ed. with J. P. LaSalle.) Recent Soviet Contributions to Mathematics. New York: Macmillan. 1963 On indirect automatic controls. Trudy Mezhdunarodnogo Simpoziuma po Nelineinym Kolebaniyam, pp. 23-24. Kiev: Izdat. Akad. Ukrain. SSR. Some mathematical considerations on nonlinear automatic controls. In Contributions to Differential Equations, vol. 1, pp. 1-28. New York: Interscience. Elementos de Topologia. Ciudad de México: Universidad Nacional Autónoma de México.

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Biographical Memoirs: Volume 61 (Ed. with J. P. LaSalle.) Proceedings of International Symposium on Nonlinear Differential Equations and Nonlinear Mechanics, Colorado Springs, 1961. New York: Academic Press. 1964 Stability of Nonlinear Automatic Control Systems. New York: Academic Press. 1965 Liapunov stability and controls. SIAMJ. Control Ser. A 3:1-6. Planar graphs and related topics. Proc. Natl. Acad. Sci. U.S.A. 54:1763-65. Recent advances in the stability of nonlinear controls. SIAM Rev. 7:1-12. Some applications of topology to networks. In Proceedings of the Third Annual Allerton Conference on Circuit and System Theory, pp. 1-6. Urbana: University of Illinois. 1966 Stability in Dynamics. William Pierson Field Engineering Lectures, March 3, 4, 10, 11, 1966. Princeton, N.J.: Princeton University School of Engineering and Applied Science. 1967 Stability of Nonlinear Automatic Control Systems. Moscow: Izdat. ''Mir." (A translation of 1964,1, in Russian.) 1968 On a theorem of Bendixson. J. Diff. Equations, 4:66-101. A page of mathematical autobiography. Bull. Am. Math. Soc. 74:854-79. 1969 The Lurie problem on nonlinear controls. In Lectures in Differential Equations, ed. A. K. Aziz, vol. 1, pp. 1-19. New York: Van Nostrand-Reinhold. The early development of algebraic geometry. Am. Math. Mon. 76:451-60. Luther Pfahler Eisenhart, 1876-1965: A Biographical Memoir. Bio-

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Biographical Memoirs: Volume 61 graphical Memoirs, vol. 40, pp. 69-90. Washington, D.C.: National Academy of Sciences. 1970 Reminiscences of a mathematical immigrant in the United States. Am. Math. Mon. 77(4). 1971 The early development of algebraic topology. Bol. Soc. Brasileira Mat. 1:1-48. AUXILIARY REFERENCES R. C. Archibald, A Semicentennial History of the American Mathematical Society, I, New York, 1938, pp. 236-40. Algebraic Geometry and Topology, a Symposium in Honor of S. Lefschetz , edited by R. H. Fox, D. C. Spencer, and A. W. Tucker, Princeton University Press, 1957, pp. 1-49. Selected Papers by S. Lefschetz, including the book I '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), pp. 198-217 and in The Lefschetz Centennial Conference, I, Mexico 1984, D. Sunderaraman, ed., published as Contemporary Mathematics, 58.1, American Mathematical Society, 1986, pp. 27-46. J. K. Hale and J. P. La Salle, The Contribution of Solomon Lefschetz to the Study of Differential Equations, typed manuscript prepared for Hodge in writing the above article. J. P. La Salle, "Memorial to Solomon Lefschetz," in IEEE Transactions on Automatic Control, vol. AC-18 ( 1973), pp. 89-90. Lawrence Markus, "Solomon Lefschetz, an Appreciation in Memoriam," Bull. Am. Math. Soc., vol. 79 ( 1973), pp. 663-80. William Hodge, "Solomon Lefschetz, 1884-1972," in Yearbook of the American Philosophical Society ( 1974), pp. 186-93. "Lefschetz, Solomon," in National Cyclopedia of American Biography, 56 ( 1975), James T. White and Co., Clifton, New Jersey, pp. 503-4. F. Nebeker and A. W. Tucker, "Lefschetz, Solomon" in Dictionary of Scientific Biography, Supplement II, 1991.