NationMaster - Encyclopedia: Élie Cartan

Élie Joseph Cartan (9 April 1869 - 6 May 1951) was a French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications. April 9 is the 99th day of the year in the Gregorian calendar (100th in leap years). ... 1869 is a common year starting on Friday (link will take you to calendar). ... May 6 is the 126th day of the year in the Gregorian Calendar (127th in leap years). ... Global Metrics Human security Major Armed Conflicts: Total Deaths in Battle: 700,000 people Violent Deaths caused by Government (Other than War): Violent Deaths caused by other humans: Juvenile Violent Crime: Political security Nations Holding Multi-party Elections: Percentage Living under a Fully Democratic System of Governance: Free Countries: Percentage... The French Republic or France (French: République française or France) is a country whose metropolitan territory is located in western Europe, and which is further made up of a collection of overseas islands and territories located in other continents. ... A mathematician is a person whose area of study and research is mathematics. ... In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ...

He was born in Dolomieu in Savoie, and became a student at the École Normale Superieure in Paris in 1888. After his doctorate in 1894, he took lecturing positions in Montpellier and Lyon, becoming a professor in Nancy in 1903. He took a lecturing position in Paris in 1909, becoming professor in 1912, and retiring in 1942. He died in Paris. He was the father of the mathematician Henri Cartan. Savoie is a French département. ... Location within France Montpellier ( Occitan Montpelhièr) is a city in the south of France. ... This article is about the French city. ... This article is about the city in France named Nancy. ... Henri Cartan (born July 8, 1904) is a son of Elie Cartan, and is, as his father was, a distinguished and influential mathematician. ...

By his own account, in his Notice sur les travaux scientifiques, the main theme of his works (numbering 186 and published throughout the period 1893-1947) was the theory of Lie groups. He began by working over the foundational material on the complex simple Lie algebras, tidying up the previous work by Engel and Wilhelm Killing. This proved definitive, as far as the classification went, with the identification of the four main families and the five exceptional cases. He also introduced the algebraic group concept, which was not to be developed seriously before 1950. In mathematics, a Lie group is an analytic real or complex manifold that is also a group such that the group operations multiplication and inversion are analytic maps. ... In mathematics, a Lie algebra (named after Sophus Lie, pronounced lee) is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. ... This article is about the role-playing game Engel; Engel is also the German word for Angel; for the Rammstein song see Engel (song). ... In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety. ...

He defined the general notion of anti-symmetric differential form, in the style now used; his approach to Lie groups through the Maurer-Cartan equations required 2-forms for their statement. At that time what were called Pfaffian systems (i.e. first-order differential equations given as 1-forms) were in general use; by the introduction of fresh variables for derivatives, and extra forms, they allowed for the formulation of quite general PDE systems. Cartan added the exterior derivative, as an entirely geometric and coordinate-independent operation. It naturally leads to the need to discuss p-forms, of general degree p. Cartan writes of the influence on him of Riquier’s general PDE theory. A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... In mathematics, the Maurer-Cartan form for a Lie group G is a distinguished differential form on G that carries within itself the basic infinitesimal information about the structure of G. It was much used by Elie Cartan, as a basic ingredient of his method of moving frames. ... In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. ... In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ... In mathematics, and in particular analysis, a partial differential equation (PDE) is an equation involving partial derivatives of an unknown function. ... In mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree. ...

With these basics – Lie groups and differential forms – he went on to produce a very large body of work, and also some general techniques such as moving frames, that were gradually incorporated into the mathematical mainstream. In mathematics, the idea of a frame in the theory of smooth manifolds is understood in terms meaning it can vary from point to point. ...

In the Travaux, he breaks down his work into 15 areas. Using modern terminology, they are these:

Most of these topics have been worked over thoroughly by later mathematicians. That cannot be said of all of them: while Cartan's own methods were remarkably unified, in the majority of cases the subsequent work can be said to have removed his characteristic touch. That is, it became more algebraic. In mathematics and theoretical physics, the idea of a representation of a Lie group plays an important role in the study of continuous symmetry. ... In mathematics, hypercomplex numbers are extensions of the complex numbers constructed by means of abstract algebra, such as quaternions, tessarines, coquaternions, octonions, biquaternions and sedenions. ... In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division is possible. ... In mathematics, a pseudogroup is an extension of the group concept, but one that grew out of the geometric approach of Sophus Lie, rather than out of abstract algebra (such as quasigroup, for example). ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... The term connection (also rendered connexion - this alternative spelling is now generally considered old-fashioned, but it was the house style of The Times of London until at least the late 1970s) has various uses, including: An act of connecting two or more physical entities in a physical sense or... In mathematics, the Cartan connection construction of differential geometry is a flexible generalisation of the connection concept, developed by Élie Cartan. ... In differential geometry, the holonomy of a given structure (for example a Riemannian metric, or more general G-structure) at a point P on a smooth manifold M is the group of all linear maps transforming the tangent space at P that can be induced by a parallel transport around... In differential geometry, the Weyl curvature tensor is the traceless component of the Riemann curvature tensor. ... In mathematics, Riemannian geometry has at least two meanings, one of which is described in this article and another also called elliptic geometry. ... In mathematics, the term symmetric space has several different meanings. ... In mathematics, a compact (topological, often understood) group is a topological group that is also a compact space. ... In mathematics, in particular in the theory of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a manifold or topological space X on which G acts by symmetry in a transitive way; it is not assumed that the action of G is faithful. ... Classical mechanics is a model of the physics of forces acting upon bodies. ... In physics, the term relativity is used in several related contexts: Galileo first developed the principle of relativity, which was the postulate that claimed that the laws of physics be the same for all observers, and advocated a classical view that time was a universal constant. ... In mathematics and physics, in particular in the theory of the orthogonal groups, spinors are certain kinds of mathematical objects (group representations of Spin(N), roughly speaking) similar to vectors, but which change sign under a rotation of radians. ...

There is a sense, therefore, in which the distinctive side of Cartan's work is still being digested by mathematicians. This is constantly seen in areas such as calculus of variations, Bäcklund transformations and the general theory of differential systems; roughly speaking those parts of differential algebra which feel that the existing, Galois theory-led model of symmetry is too narrow and requires something more analogous to a category of relations. An envelope is a packaging product, usually made of flat, planar material such as paper or cardboard, designed to contain a flat object such as a letter. ... In mathematics, a principal G-bundle is a special kind of fiber bundle for which the fibers are all G-torsors (also known as principal homogeneous spaces) for the action of a topological group G. Principal G-bundles are G-bundles in the sense that the group G also serves... In differential geometry, the jet bundle is a certain construction which makes a new smooth fiber bundle out of a given smooth fiber bundle. ... Charles Ehresmann (1905-1979) was a French mathematician who worked on differential topology and category theory. ... A contact is part of the active component of an electric switch. ... A mathematician is a person whose area of study and research is mathematics. ... Calculus of variations is a field of mathematics which deals with functions of functions, as opposed to ordinary calculus which deals with functions of numbers. ... In mathematics, Galois theory is that branch of abstract algebra which studies the symmetries of the roots of polynomials. ...

Categories: 1869 births | 1951 deaths | French mathematicians In mathematics, the Cartan connection construction of differential geometry is a flexible generalisation of the connection concept, developed by Élie Cartan. ... This page covers notations and definitions, sometimes called the Cartan formalism, for the Cartan connection concept. ... In mathematics, the term Cartan matrix has two meanings. ... In the theory of Lie algebras, we say is a Cartan subalgebra of a Lie algebra if is an Abelian subalgebra such that if you look at the weight eigenspaces of the adjoint representation of restricted to , the representation is diagonalizable and the eigenspace of the zero weight vector is... Einstein-Cartan theory extends general relativity to correctly handle spin angular momentum. ... In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms. ...