Solomon Lefschetz

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

Life and work

His parents had Turkish nationality, but his father was active as a merchant. Shortly after the birth of Lefschetz in Moscow, she moved to Paris, so that it grew up in a French environment. He studied engineering at the Ecole Centrale Paris at Charles Émile Picard and Paul Émile Appell, where he received his degree in 1905. Since there but he could not make academic career as a non- Frenchman, he emigrated to the USA, where he first inter alia, 1907-1910 in the electrical engineering company Westinghouse Electric Corporation worked in Pittsburgh. In a transformer explosion in the lab, he lost there in November 1907 both hands and part of the forearm. Later he wore artificial hands that he covered with black gloves. He taught after initially apprentices in the company in mathematics, but then changed her studies at Clark University in Worcester, Massachusetts. In 1911 he received his doctorate with a thesis in algebraic geometry. The following year he became an American citizen.

In 1911 he became an assistant at the University of Nebraska at Lincoln, 1913 at the University of Kansas in Lawrence, where he became professor in 1919. In relative isolation he built in intensive work methods Henri Poincaré algebraic topology and showed in a series of papers, summarized in the book L' analysis situs et la geometry algébrique, their importance for algebraic geometry. In 1923 he published his famous fixed point theorem, first for compact orientable manifolds. He states that a continuous mapping of the manifold to itself at non-vanishing Lefschetzzahl has a fixed point. The sum of the Lefschetzzahlen gives a topological invariant of. The idea for this project came from his work on the correspondences of Lefschetz very revered Italian school of algebraic geometry. Later he extended example, the sentence on manifolds with boundary (1927 ), so that this resulted in the Brouwer fixed point theorem, which predicts the existence of a fixed point at continuous mappings of the unit disc. In addition, he simplified his derivation of the sentence. Heinz Hopf was in 1928 an interpretation and a simpler proof using homology groups. Another way of highlighting its relationship to the Morse theory, except that we consider here vector fields on manifolds (rivers ).

A simplified example of the Lefschetz fixed point theorem is obtained for plane continuous maps that are linearized around their fixed points: It is with a matrix ( eigenvalues ​​). The Lefschetzzahl 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 genus (holes) has such a flux for each hole 2 saddle points, one source and one drain, so that the sum of the Lefschetzzahlen shows the Euler-Poincare characteristic of the surface.

In 1924, he received the recommendation of the topologists James Alexander, a professor at Princeton, where Oswald Veblen taught. During this period he developed from work by Emile Picard and Henri Poincaré algebraic topology of higher dimensional algebraic varieties. " Lefschetz pencils " denote bundles of hyperplanes intersecting the variety, and are similar to the Morse theory to the study of singular points used ( Picard - Lefschetz formula ).

Many concepts of algebraic topology, he developed a significant (the word topology even first appeared in 1930 in the title of a book by him ), and he wrote an influential textbook of the same name in 1942.

Strangely, he had in the 1930s and 1940s, an anti-Semitic in effect setting ( although he himself had Jewish roots): He refused to attend Jewish graduate students, on the grounds that they would find in the depression no jobs.

, During World War II he began to take more interest for applications and examined - partly by Russian work as that of Lyapunov, Krylov, Andronow, Pontryagin excited that he partly published in English translation - nonlinear differential equations for the control theory and the theory of vibrations, which he continued after the war with topological methods, so that he was with his school in the 1950s and 1960s, one of the fathers of the qualitative theory of differential equations and their stability theory (eg his book with LaSalle ).

From 1944 he regularly visited Mexico and taught there for almost a year.

Lefschetz, provided with a loud and harsh tone and a lot of energy and enthusiasm for his field of work, paid little attention to elegance of evidence, but saw the main task of mathematics is to discover new things. Gian- Carlo Rota says of him, he almost never published a correct proof or an incorrect theorem.

From 1928 to 1958 he was editor of the Annals of Mathematics, which he made ​​it into a leading journal. 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 doctoral include Richard Bellman, Albert William Tucker, Ralph Fox, Clifford Dowker, John McCarthy, Clifford Truesdell, Norman Steenrod, John Tukey, Shaun Wylie.

He was married since 1913. The marriage remained childless.