| Élie Joseph Cartan | |
 Professor Élie Joseph Cartan
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| Born | April 9, 1869 Dolomieu, Savoie, France |
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| Died | May 6, 1951 (aged 82) Paris, France |
| Occupation | Mathematician |
| Known for | Lie groups, differential geometry |
| Spouse | Marie-Louise Bianconi |
| Children | Henri, Jean, Louis, Hélène |
| Parents | Joseph Cartan Anne Cottaz |
Élie Joseph Cartan (9 April 1869 – 6 May 1951) was an influential French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications. Events 193 - Septimius Severus is proclaimed Roman Emperor by the army in Illyricum (in the Balkans) Year 1869 ( MDCCCLXIX) is a Common year starting on Friday (link will display the full calendar of the Gregorian calendar (or a Common year Dieudonné Sylvain Guy Tancrède de Dolomieu usually known as Déodat de Dolomieu (Dolomieu near La Tour-du-Pin, June 23, 1750 - Chateauneuf Savoie ( Arpitan: Savouè d’Avâl) is a French department located in the Rhône-Alpes ( Rôno-Arpes This article is about the country For a topic outline on this subject see List of basic France topics. Events 1527 - Spanish and German troops sack Rome; some consider this the end of the Renaissance. Year 1951 ( MCMLI) was a Common year starting on Monday. Events of 1951 January Paris (ˈpærɨs in English; in French) is the Capital of France and the country's largest city A mathematician is a person whose primary area of study and research is the field of Mathematics. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry Henri Paul Cartan ( July 8, 1904 &ndash August 13, 2008) was a son of Élie Cartan, and was as his father was a distinguished Events 193 - Septimius Severus is proclaimed Roman Emperor by the army in Illyricum (in the Balkans) Year 1869 ( MDCCCLXIX) is a Common year starting on Friday (link will display the full calendar of the Gregorian calendar (or a Common year Events 1527 - Spanish and German troops sack Rome; some consider this the end of the Renaissance. Year 1951 ( MCMLI) was a Common year starting on Monday. Events of 1951 January This article is about the country For a topic outline on this subject see List of basic France topics. A mathematician is a person whose primary area of study and research is the field of Mathematics. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group He also made significant contributions to mathematical physics, differential geometry, and group theory. Mathematical physics is the scientific discipline concerned with the interface of Mathematics and Physics. Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups.
He was the father of another influential mathematician, Henri Cartan. Henri Paul Cartan ( July 8, 1904 &ndash August 13, 2008) was a son of Élie Cartan, and was as his father was a distinguished
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Life
Élie Cartan was born in the village of Dolomieu in Isère, the son of a blacksmith. Dieudonné Sylvain Guy Tancrède de Dolomieu usually known as Déodat de Dolomieu (Dolomieu near La Tour-du-Pin, June 23, 1750 - Chateauneuf Isère ( Arpitan: Isera, Occitan: Isèra) is a department, in the Rhône-Alpes ( Rôno-Arpes in Arpitan He became a student at the École Normale Superieure in Paris in 1888 and obtained his doctorate in 1894. École Normale de Musique de ParisThe École normale supérieure (also known as Normale Sup’, Normale, ENS, ENS-Paris, ENS-Ulm or He subsequently held lecturing positions in Montpellier and Lyon, becoming a professor in Nancy in 1903. Montpellier ( Occitan Montpelhièr) is a City in the south of France. ||-||} Lyon, also known as Lyons in English is a city in east-central France. Nancy (nɑ̃si archaic Nanzig Nanzeg is a city and commune in the Lorraine région of northeastern France He took a lecturing position at the Sorbonne in Paris in 1909, becoming professor there in 1912 until his retirement in 1940. He died in Paris after a long illness. He was the father of the mathematician Henri Cartan. Henri Paul Cartan ( July 8, 1904 &ndash August 13, 2008) was a son of Élie Cartan, and was as his father was a distinguished
Work
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. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group He began by working over the foundational material on the complex simple Lie algebras, tidying up the previous work by Friedrich Engel and Wilhelm Killing. In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie Friedrich Engel ( December 26, 1861 &ndash September 29, 1941) was a German Mathematician. Wilhelm Karl Joseph Killing ( May 10 1847 &ndash February 11 1923) was a German Mathematician who made important contributions 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 Algebraic geometry, an algebraic group (or group variety) is a group that is an Algebraic variety, such that the multiplication and inverse
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. In the mathematical fields of Differential geometry and Tensor calculus, differential forms are an approach to Multivariable calculus which is At that time what were called Pfaffian systems (i. 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 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. A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i Cartan added the exterior derivative, as an entirely geometric and coordinate-independent operation. In Differential geometry, the exterior derivative extends the concept of the differential of a function which is a form of degree zero to Differential forms It naturally leads to the need to discuss p-forms, of general degree p. Cartan writes of the influence on him of Charles Riquier’s general PDE theory.
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, a moving frame is a flexible generalization of the notion of an Ordered basis of a Vector space often used to study the extrinsic differential
In the Travaux, he breaks down his work into 15 areas. Using modern terminology, they are these:
- Lie groups
- Representations of Lie groups
- Hypercomplex numbers, division algebras
- Systems of PDEs, Cartan–Kähler theorem
- Theory of equivalence
- Integrable systems, theory of prolongation and systems in involution
- Infinite-dimensional groups and pseudogroups
- Differential geometry and moving frames
- Generalised spaces with structure groups and connections, Cartan connection, holonomy, Weyl tensor
- Geometry and topology of Lie groups
- Riemannian geometry
- Symmetric spaces
- Topology of compact groups and their homogeneous spaces
- Integral invariants and classical mechanics
- Relativity, spinors
Influence and legacy
Most of these topics have been worked over thoroughly by later mathematicians. In Mathematics and Theoretical physics, the idea of a representation of a Lie group plays an important role in the study of continuous Symmetry The term hypercomplex number has been used in Mathematics for the elements of algebras that extend or go beyond Complex number arithmetic 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, the Cartan–Kähler theorem is a major result on the Integrability conditions for differential systems, in the case of Analytic functions In Mathematics, Cartan's equivalence method is a technique in Differential geometry for determining whether two geometrical structures are the same up to a Diffeomorphism In Mathematics and Physics, there are various distinct notions that are referred to under the name of integrable systems. 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 Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry In Mathematics, a moving frame is a flexible generalization of the notion of an Ordered basis of a Vector space often used to study the extrinsic differential In the mathematical field of Differential geometry, a Cartan connection is a flexible generalization of the notion of an Affine connection. In Differential geometry, the holonomy of a connection on a Smooth manifold is a general geometrical consequence of the Curvature of the connection In Differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is the Traceless component of the Riemann curvature tensor. Elliptic geometry is also sometimes called Riemannian geometry. In Differential geometry, Representation theory and Harmonic analysis, a symmetric space is a Smooth manifold whose group of symmetries contains In Mathematics, a compact ( topological, often understood group is a Topological group whose Topology is Compact. In Mathematics, particularly in the theories of Lie groups Algebraic groups and Topological groups a homogeneous space for a group Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 In Mathematics and Physics, in particular in the theory of the Orthogonal groups spinors are elements of a complex vector space introduced to expand the 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.
To look at some of those less mainstream areas:
- the PDE theory has to take into account singular solutions (i. e. envelopes), such as are seen in Clairaut's equation;
- the prolongation method is supposed to terminate in a system in involution (this is an analytic theory, rather than smooth, and leads to the theory of formal integrability and Spencer cohomology);
- the equivalence problem, as he put it, is to construct differential isomorphisms of structures (and discover thereby the invariants) by forcing their graphs to be integral manifolds of a differential system;
- the moving frames method, as well as being connected to principal bundles and their connections, should also use frames adapted to geometry;
- these days, the jet bundle method of Ehresmann is applied to use contact as a systematic equivalence relation. In Mathematics, an envelope of a family of Manifolds (especially a family of Curves is a manifold that is Tangent to each member of In Mathematics, a Clairaut's equation is a Differential equation of the form y(x=x\frac{dy}{dx}+f\left(\frac{dy}{dx}\right In Mathematics, Cartan's equivalence method is a technique in Differential geometry for determining whether two geometrical structures are the same up to a Diffeomorphism In Mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of a Cartesian product X × G 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. In Mathematics, contact geometry is the study of a geometric structure on Smooth manifolds given by a hyperplane distribution in the Tangent bundle
There is a sense, therefore, in which the distinctive side of Cartan's work is still being digested by mathematicians. A mathematician is a person whose primary area of study and research is the field of Mathematics. 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. Calculus of variations is a field of Mathematics that deals with functionals, as opposed to ordinary Calculus which deals with functions. In Mathematics, Bäcklund transforms or Bäcklund transformations relate Partial differential equations and their solutions In Mathematics, more specifically in Abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory
See also
- Cartan connection, Cartan connection applications
- Cartan matrix
- Cartan's theorem
- Cartan subalgebra
- Cartan's equivalence method
- Einstein–Cartan theory
- Integrability conditions for differential systems
- CAT(k) space
References
External links
- Shiing-Shen Chern and Claude Chevalley, Élie Cartan and his mathematical work, Bull. In the mathematical field of Differential geometry, a Cartan connection is a flexible generalization of the notion of an Affine connection. This page covers applications of the Cartan formalism. For the general concept see Cartan connection. In Mathematics, the term Cartan matrix has two meanings Both of these are named after the French Mathematician Élie Cartan. In Mathematics, there are two basic results in Lie group theory that go by the name Cartan's theorem. In Mathematics, a Cartan subalgebra is a nilpotent Subalgebra \mathfrak{h} of a Lie algebra \mathfrak{g} that is In Mathematics, Cartan's equivalence method is a technique in Differential geometry for determining whether two geometrical structures are the same up to a Diffeomorphism Einstein–Cartan theory in Theoretical physics 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 In Mathematics, a CAT( k) space is a specific type of Metric space. Amer. Math. Soc. 58 (1952), 217-250.
- J. H. C. Whitehead, Elie Joseph Cartan 1869-1951, Obituary Notices of Fellows of the Royal Society, Vol. 8, No. 21 (Nov. , 1952), pp. 71-95.
- O'Connor, John J. & Robertson, Edmund F. , “Élie Cartan”, MacTutor History of Mathematics archive
- Élie Cartan at the Mathematics Genealogy Project
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