Solomon Lefschetz
Encyclopedia
Solomon Lefschetz was an American mathematician
who did fundamental work on algebraic topology
, its applications to algebraic geometry
, and the theory of non-linear ordinary differential equation
s.
into a Jewish family (his parents were Turkish
citizens) who moved shortly after that to Paris
. He was educated there in engineering
at the École Centrale Paris
, but emigrated to the USA in 1905.
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 University of Nebraska and University of Kansas
, moving to Princeton University
in 1924, where he was soon given a permanent position. He remained there until 1953.
In the application of topology to algebraic geometry, he followed the work of Charles Émile Picard
, whom he had heard lecture in Paris at the École Centrale. 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 cycle
s 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.
The Lefschetz fixed point theorem, now a basic result of topology, he developed in papers from 1923 to 1927, initially for manifold
s. 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 equation
s.
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.
Lefschetz came out of retirement in 1958, because of the launch of Sputnik, to augment the mathematical component of Glenn L. Martin Company
’s Research Institute for Advanced Studies
(RIAS) in Baltimore, Maryland. His team became the world's largest group of mathematicians devoted to research in nonlinear differential equations. The RIAS mathematics group stimulated the growth of nonlinear differential equations through conferences and publications. It left RIAS in 1964 to form the Lefschetz Center for Dynamical Systems at Brown University
, Providence, Rhode Island.
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
who did fundamental work on algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
, its applications to algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, and the theory of non-linear ordinary differential equation
Ordinary differential equation
In mathematics, an ordinary differential equation is a relation that contains functions of only one independent variable, and one or more of their derivatives with respect to that variable....
s.
Life
He was born in MoscowMoscow
Moscow is the capital, the most populous city, and the most populous federal subject of Russia. The city is a major political, economic, cultural, scientific, religious, financial, educational, and transportation centre of Russia and the continent...
into a Jewish family (his parents were Turkish
Turkey
Turkey , known officially as the Republic of Turkey , is a Eurasian country located in Western Asia and in East Thrace in Southeastern Europe...
citizens) who moved shortly after that to Paris
Paris
Paris is the capital and largest city in France, situated on the river Seine, in northern France, at the heart of the Île-de-France region...
. He was educated there in engineering
Engineering
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...
at the École Centrale Paris
École Centrale Paris
École Centrale Paris is a French university-level institution in the field of engineering. It is also known by its original name École centrale des arts et manufactures, or ECP. Founded in 1829, it is one of the oldest and most prestigious engineering schools in France and has the special status...
, but emigrated to the USA in 1905.
He was badly injured in an industrial accident in 1907, losing both hands. He moved towards mathematics, receiving a Ph.D.
Doctor of Philosophy
Doctor of Philosophy, abbreviated as Ph.D., PhD, D.Phil., or DPhil , in English-speaking countries, is a postgraduate academic degree awarded by universities...
in algebraic geometry from Clark University
Clark University
Clark University is a private research university and liberal arts college in Worcester, Massachusetts.Founded in 1887, it is the oldest educational institution founded as an all-graduate university. Clark now also educates undergraduates...
in Worcester, Massachusetts in 1911. He then took positions in University of Nebraska and University of Kansas
University of Kansas
The University of Kansas is a public research university and the largest university in the state of Kansas. KU campuses are located in Lawrence, Wichita, Overland Park, and Kansas City, Kansas with the main campus being located in Lawrence on Mount Oread, the highest point in Lawrence. The...
, moving to Princeton University
Princeton University
Princeton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....
in 1924, where he was soon given a permanent position. He remained there until 1953.
In the application of topology to algebraic geometry, he followed the work of Charles Émile Picard
Charles Émile Picard
Charles Émile Picard FRS was a French mathematician. He was elected the fifteenth member to occupy seat 1 of the Académie Française in 1924.- Biography :...
, whom he had heard lecture in Paris at the École Centrale. He proved theorems on the topology of hyperplane
Hyperplane
A hyperplane is a concept in geometry. It is a generalization of the plane into a different number of dimensions.A hyperplane of an n-dimensional space is a flat subset with dimension n − 1...
sections of algebraic varieties, which provide a basic inductive tool (these are now seen as allied to Morse theory
Morse theory
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. According to the basic insights of Marston Morse, a differentiable function on a manifold will, in a typical case, reflect...
, though a Lefschetz pencil
Lefschetz pencil
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...
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 cycle
Vanishing cycle
In mathematics, vanishing cycles are studied in singularity theory and other parts of algebraic geometry. They are those homology cycles of a smooth fiber in a family which vanish in the singular fiber....
s is a basic tool relating the degeneration
Degeneracy (mathematics)
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....
of families of varieties with 'loss' of topology, to monodromy
Monodromy
In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology and algebraic and differential geometry behave as they 'run round' a singularity. As the name implies, the fundamental meaning of monodromy comes from 'running round singly'...
. His book L'analysis situs et la géométrie algébrique from 1924, though opaque foundationally given the current technical state of homology theory
Homology theory
In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. It can be broadly defined as the study of homology theories on topological spaces.-The general idea:...
, 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
Weil conjectures
In mathematics, the Weil conjectures were some highly-influential proposals by on the generating functions derived from counting the number of points on algebraic varieties over finite fields....
, through SGA7). In 1924 he was awarded the Bôcher Memorial Prize
Bôcher Memorial Prize
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 . It is awarded every five years for a notable research memoir in analysis that has appeared during the past six years in a recognized North American...
for his work in mathematical analysis.
The Lefschetz fixed point theorem, now a basic result of topology, he developed in papers from 1923 to 1927, initially for manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
s. Later, with the rise of cohomology theory in the 1930s, he contributed to the intersection number
Intersection number
In mathematics, and especially in algebraic geometry, the intersection number generalizes the intuitive notion of counting the number of times two curves intersect to higher dimensions, multiple curves, and accounting properly for tangency...
approach (that is, in cohomological terms, the ring structure) via the cup product
Cup product
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. This defines an associative graded commutative product operation in cohomology, turning the cohomology of a space X into a...
and duality on manifolds. His work on topology was summed up in his monograph Algebraic Topology (1942). From 1944 he worked on differential equation
Differential equation
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s.
He was editor of the Annals of Mathematics
Annals of Mathematics
The Annals of Mathematics is a bimonthly mathematical journal published by Princeton University and the Institute for Advanced Study. It ranks amongst the most prestigious mathematics journals in the world by criteria such as impact factor.-History:The journal began as The Analyst in 1874 and was...
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.
Lefschetz came out of retirement in 1958, because of the launch of Sputnik, to augment the mathematical component of Glenn L. Martin Company
Glenn L. Martin Company
The Glenn L. Martin Company was an American aircraft and aerospace manufacturing company that was founded by the aviation pioneer Glenn L. Martin. The Martin Company produced many important aircraft for the defense of the United States and its allies, especially during World War II and the Cold War...
’s Research Institute for Advanced Studies
Research Institute for Advanced Studies
The Baltimore-based Research Institute for Advanced Studies , not to be confused with the better-known Institute for Advanced Study in Princeton, New Jersey, was among the several centers for research in the mathematical and physical sciences established throughout the United States after World War...
(RIAS) in Baltimore, Maryland. His team became the world's largest group of mathematicians devoted to research in nonlinear differential equations. The RIAS mathematics group stimulated the growth of nonlinear differential equations through conferences and publications. It left RIAS in 1964 to form the Lefschetz Center for Dynamical Systems at Brown University
Brown University
Brown University is a private, Ivy League university located in Providence, Rhode Island, United States. Founded in 1764 prior to American independence from the British Empire as the College in the English Colony of Rhode Island and Providence Plantations early in the reign of King George III ,...
, Providence, Rhode Island.
See also
- Lefschetz connection
- Lefschetz hyperplane theoremLefschetz hyperplane theoremIn mathematics, specifically in algebraic geometry and algebraic topology, the Lefschetz hyperplane theorem is a precise statement of certain relations between the shape of an algebraic variety and the shape of its subvarieties...
- Lefschetz dualityLefschetz dualityIn mathematics, Lefschetz duality is a version of Poincaré duality in geometric topology, applying to a manifold with boundary. Such a formulation was introduced by Solomon Lefschetz in the 1920s, at the same time introducing relative homology, for application to the Lefschetz fixed-point theorem...
- Lefschetz manifoldLefschetz manifoldIn mathematics, a Lefschetz manifold is a particular kind of symplectic manifold, sharing a certain cohomological property with Kaehler manifolds, that of satisfying the conclusion of the Hard Lefschetz theorem...
- Lefschetz number
- Lefschetz zeta functionLefschetz zeta functionIn mathematics, the Lefschetz zeta-function is a tool used in topological periodic and fixed point theory, and dynamical systems. Given a mapping ƒ, the zeta-function is defined as the formal series...
- Lefschetz pencilLefschetz pencilIn 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...
- Lefschetz theorem on (1,1)-classesLefschetz theorem on (1,1)-classesIn algebraic geometry, a branch of mathematics, the Lefschetz theorem on -classes, named after Solomon Lefschetz, is a classical statement relating divisors on a compact Kähler manifold to classes in its integral cohomology. It is the only case of the Hodge conjecture which has been proved for all...
External links
- http://www.princeton.edu/~mudd/finding_aids/mathoral/pmcxrota.htm"Fine Hall in its golden age: Remembrances of Princeton in the early fifties" by Gian-Carlo RotaGian-Carlo RotaGian-Carlo Rota was an Italian-born American mathematician and philosopher.-Life:Rota was born in Vigevano, Italy...
.] Contains a lengthy section on Lefschetz at Princeton.