Solomon Lefschetz (3 September 1884-5 October 1972) was a U.S. mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear ordinary differential equations. He was born in Moscow into a Jewish family (his parents were Turkish citizens) who moved shortly after that to Paris. He was educated there in engineering, but emigrated to the USA in 1905. September 3 is the 246th day of the year (247th in leap years). ... 1884 (MDCCCLXXXIV) is a leap year starting on Tuesday (click on link to calendar) of the Gregorian calendar (or a leap year starting on Thursday of the 12-day-slower Julian calendar). ... October 5 is the 278th day of the year (279th in Leap years). ... 1972 (MCMLXXII) was a leap year starting on Saturday (the link is to a full 1972 calendar). ... Motto: E pluribus unum (1789 to 1956) (Latin: Out of Many, One) In God We Trust (1956 to present) Anthem: The Star-Spangled Banner Capital Washington, D.C. Largest city New York City Official language(s) None at federal level; English de facto Government • President • Vice President Federal Republic George... Leonhard Euler is considered by many people to be one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is mathematics. ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... In mathematics, and particularly in analysis, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable. ... Moscow (Russian: Москва́, Moskva, IPA: ) is the capital of Russia and the countrys principal political, economic, financial, educational and transportation center, located on the river Moskva. ... Jews (Hebrew: יהודים, Yehudim) are followers of Judaism or, more generally, members of the Jewish people (also known as the Jewish nation, or the Children of Israel), an ethno-religious group descended from the ancient Israelites and converts who joined their religion. ... The Eiffel Tower, the international symbol of the city For other uses, see Paris (disambiguation). ... Bold text Engineering is the application of scientific and technical knowledge to solve human problems. ...

He was badly injured in an industrial accident in 1907, losing both hands. He moved towards mathematics, receiving a Ph.D. in algebraic geometry from Clark University in Worcester, Massachusetts in 1911. He then took positions in Nebraska and Kansas, moving to Princeton in 1924, where he was soon given a permanent position. He remained there until 1953. Clark University, in Worcester, Massachusetts in the United States, is a private teaching and research institution founded in 1887 by the industrialist Jonas Clark. ...

In the application of topology to algebraic geometry, he followed the work of Charles Émile Picard, whom he had heard lecture in 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). In 1924 he was awarded the Bôcher Memorial Prize for his work in mathematical analysis. Charles Émile Picard (July 24, 1856 - December 11, 1941) was a leading French mathematician. ... A hyperplane is a concept in geometry. ... A Morse function is also an expression for an anharmonic oscillator In differential topology, the techniques of Morse theory give a very direct way of analyzing the topology of a manifold by studying differentiable functions on that manifold. ... In mathematics, a Lefschetz pencil is a construction in algebraic geometry considered by Solomon Lefschetz, in order to analyse the algebraic topology of an algebraic variety V. A pencil here is a particular kind of linear system of divisors on V, namely a one-parameter family, parametrised by the projective... In mathematics, a degenerate case is a limiting case in which a class of object changes its nature so as to belong to another, usually simpler, class. ... In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology and differential geometry behave as they run round a singularity. ... In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. ... In mathematics, the Weil conjectures, which had become theorems by 1975, were some highly-influential proposals from the late 1940s by Andre Weil on the generating functions (known as local zeta-functions) derived from counting the number of points on algebraic varieties over finite fields. ... In mathematics, Alexander Grothendiecks Séminaire de géométrie algébrique was a unique phenomenon of research and publication outside of the main mathematical journals, reporting on work done starting from 1960 and centred on the IHÉS near Paris (the official title was the seminar of Bois... 1924 (MCMXXIV) was a leap year starting on Tuesday (link will take you to calendar). ... The Bôcher Memorial Prize was founded by the American Mathematical Society in 1923 in memory of Maxime Bôcher with an initial endowment of $1,450 (contributed by members of that society). ...

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. In mathematics, the Lefschetz fixed-point theorem counts the number of fixed points of a mapping from a topological space X to itself (subject to some mild conditions on X), by means of traces of the induced mappings on the homology groups of X. The counting is subject to some... On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex. ... In mathematics, the concept of intersection number arose in algebraic geometry, where two curves intersecting at a point may be considered to meet twice if they are tangent there. ... In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree p and q to form a composite cocycle of degree p + q. ... Graph of a differential equation In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ...

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. The rise of Annals, in turn, stimulated American mathematics. The Annals of Mathematics (ISSN 0003-486X), often just called Annals, is a bimonthly mathematics research journal published by Princeton University and the Institute for Advanced Study. ...

See also: Lefschetz hyperplane theorem, Lefschetz duality. In mathematics, the Lefschetz hyperplane theorem states that a hyperplane section W of a non-singular complex algebraic variety V, in complex projective space, inherits most of its algebraic topology from V. This allows certain geometrical questions to be investigated by induction on dimension. ...

External links

• John J. O'Connor and Edmund F. Robertson. Solomon Lefschetz at the MacTutor History of Mathematics archive.
• Solomon Lefschetz at the Mathematics Genealogy Project
• "Fine Hall in its golden age: Remembrances of Princeton in the early fifties" by Gian-Carlo Rota. Contains a lengthy section on Lefschetz at Princeton.