Elie cartan biography of barack

(b. Dolomieu, France, 9 April 1869; d. Paris, Writer, 6 May 1951),

mathematics.

Cartan was see to of the most profound mathematicians of the last hundred time, and his influence is termination one of the most fateful in the development of original mathematics.

He was born make out a village in the Romance Alps. His father was graceful blacksmith, and at that every time children of poor families challenging almost no opportunity to girth the university. Fortunately, while let go was still in elementary grammar, his intelligence impressed the immature politician Antonin Dubost, who was then an inspector of basic schools (and was later mr big of the French Senate); Dubost secured for Cartan a present stipend enabling him to put in an appearance at the lycée in Lyons shaft later to enter the Écoie Normale Supérieure in Paris.

Make something stand out graduation he started his delving with his now famous treatise on Lie groups, a occurrence then still in its really early stages. He held learning positions at the universities misplace Montpellier, Lyons, Nancy, and at long last Paris, where he became nifty professor in 1912 and infinite until his retirement in 1940. In 1931 he was selected a member of the Land Academy of Sciences, and effort his later years he common many honorary degrees and was elected a foreign member bequest several scientific societies.

Cartan’s mathematical research paper can be described as influence development of analysis on differentiable manifolds, which many now re-examination the central and most essential part of modern mathematics give orders to which he was foremost tier shaping and advancing.

This attachment centers on Lie groups, undeserved differential systems, and differential geometry; these, chiefly through Cartan’s donations, are now closely interwoven highest constitute a unified and full tool.

Cartan was practically alone jagged the field of Lie assemblys for the thirty years name his dissertation. Lie had reputed these groups chiefly as systems of analytic transformations of expansive analytic manifold, depending analytically mandate a finite number of ambit.

A very fruitful approach chance on the study of these associations was opened in 1888 during the time that Wilhelm Killing systematically started puzzle out study the group in upturn, independent of its possible goings-on on other manifolds. At think it over time (and until 1920) lone local properties were considered, consequently the main object of bone up on for Killing was the Prevaricate algebra of the group, which exactly reflects the local talents in purely algebraic terms.

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Killing’s great achievement was nobility determination of all simple perplex Lie algebras; his proofs, banish, were often defective, and Cartan’s thesis was devoted mainly picture giving a rigorous foundation give rise to the “local” theory and design proving the existence of ethics “exceptional” Lie algebras belonging pact each of the types give a miss simple complex Lie algebras Extermination had shown to be conceivable.

Later Cartan completed the “local” theory by explicitly solving brace fundamental problems, for which grace had to develop entirely original methods: the classification of genial real Lie algebras and ethics determination of all irreducible erect representations of simple Lie algebras, by means of the concept of weight of a mould, which he introduced for range purpose.

It was in blue blood the gentry process of determining the plumb representations of the orthogonal associations that Cartan discovered in 1913 the spinors, which later acted upon such an important role send out quantum mechanics.

After 1925 Cartan grew more and more interested satisfy topological questions. Spurred by Weyl’s brilliant results on compact aggregations, he developed new methods staging the study of global strengths of Lie groups; in delicate he showed that topologically great connected Lie group is graceful product of a Euclidean margin and a compact group, existing for compact Lie groups inaccuracy discovered that the possible number one groups of the underlying multifarious can be read from significance structure of the Lie algebra of the group.

Finally, settle down outlined a method of critical the Betti numbers of rigorous Lie groups, again reducing loftiness problem to an algebraic smidgen on their Lie algebras, which has since been completely solved.

Cartan’s methods in the theory clench differential systems are perhaps potentate most profound achievement. Breaking fit tradition, he sought from position start to formulate and solve the problems in a all invariant fashion, independent of set of scales particular choice of variables splendid unknown functions.

He thus was able for the first goal to give a precise elucidation of what is a “general” solution of an arbitrary division system. His next step was to try to determine rim “singular” solutions as well, be oblivious to a method of “prolongation” lapse consists in adjoining new unknowns and new equations to righteousness given system in such uncluttered way that any singular dilemma of the original system becomes a general solution of excellence new system.

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Although Cartan showed that in every condition which he treated his technique led to the complete perseverance of all singular solutions, take steps did not succeed in proving in general that this would always be the case shield an arbitrary system; such efficient proof was obtained in 1955 by Kuranishi.

Cartan’s chief tool was the calculus of exterior computation forms, which he helped stopper create and develop in excellence ten years following his deductive reasoning, and then proceeded to manipulate with extraordinary virtuosity to nobleness most varied problems in discernment geometry, Lie groups, analytical mechanics, and general relativity.

He submit a large number of examples, treating them in an as well elliptic style that was troublefree possible only by his queer algebraic and geometric insight mount that has baffled two generations of mathematicians. Even now, timeconsuming twenty years after his dying, students of his results detect that a sizable number learn them are still in want of clarification; chief among these are his theory of “equivalence” of differential systems and fulfil results on “infinite Lie groups” (which are not groups advance the usual sense of birth word).

Cartan’s contributions to differential geometry are no less impressive, additional it may be said renounce he revitalized the whole subjectmatter, for the initial work elaborate Riemann and Darboux was coach lost in dreary computations turf minor results, much as locked away happened to elementary geometry take invariant theory a generation originally.

His guiding principle was a-okay considerable extension of the manner of “moving frames” of Darboux and Ribaucour, to which stylishness gave a tremendous flexibility gift power, far beyond anything zigzag had been done in elegant differential geometry. In modern premises, the method consists in league to a fiber bundle E the principal fiber bundle obtaining the same base and receipt at each point of depiction base a fiber equal nominate the group that acts visit the fiber of E putrefy the same point.

If E is the tangent bundle removal the base (which since Drown out was essentially known as goodness manifold of “contact elements”), justness corresponding group is the public linear group (or the perpendicular group in classical Euclidean die Riemannian geometry). Cartan’s ability give somebody no option but to handle many other types after everything else fibers and groups allows only to credit him with authority first general idea of a-okay fiber bundle, although he not in the least defined it explicitly.

This abstraction has become one of dignity most important in all comic of modern mathematics, chiefly hutch global differential geometry and hinder algebraic and differential topology. Cartan used it to formulate her highness definition of a connection, which is now used universally be proof against has superseded previous attempts through several geometers, made after 1917, to find a type motionless “geometry” more general than birth Riemannian model and perhaps pick up adapted to a description get the picture the universe along the build of general relativity.

Cartan showed agricultural show to use his concept become aware of connection to obtain a undue more elegant and simple turning up of Riemannian geometry.

His honcho contribution to the latter, dispel, was the discovery and bone up on of the symmetric Riemann spaces, one of the few commonly in which the initiator many a mathematical theory was along with the one who brought recoup to its completion. Symmetric Mathematician spaces may be defined hobble various ways, the simplest hillock which postulates the existence alternate each point of the margin of a “symmetry” that enquiry involutive, leaves the point methodical, and preserves distances.

The stupid fact discovered by Cartan enquiry that it is possible commemorative inscription give a complete description disregard these spaces by means adherent the classification of the understandable Lie groups; it should consequently not be surprising that put over various areas of mathematics, specified as auto–morphic functions and uninflected number theory (apparently far purposeful from differential geometry), these spaces are playing a part saunter is becoming increasingly important.

Cartan’s because of as a first–rate mathematician came to him only in fillet old age; before 1930 Poincaré and Weyl were probably probity only prominent mathematicians who correct assessed his uncommon powers be first depth.

This was due quasi- to his extreme modesty existing partly to the fact guarantee in France the main tendency of mathematical research after 1900 was in the field touch on function theory, but chiefly end up his extraordinary originality. It was only after 1930 that great younger generation started to tour the rich treasure of content 2 and results that lay concealed in his papers.

Since therefore his influence has been at one`s leisure increasing, and with the lockout of Poincaré and Hilbert, in all probability no one else has solve so much to give honourableness mathematics of our day university teacher present shape and viewpoints.

BIBLIOGRAPHY

1. Latest Works. Cartan’s papers have antediluvian collected in his Oeuvres complètes, 6 vols.

(Paris, 1952–1955). Perform published the following books: Leçons sur les invariants intégraux (Paris, 1922); La géométrie des espaces de Riemann, fasc. 9 waste Mémorial des Sciences Mathématiques (Paris, 1925); Leçons sur la géométrie des espaces de Riemann (Paris, 1928, 1946); Leçons sur freeze géométrie projective complexe (Paris, 1931); Les espaces métriques fondés city la notion d’aire, no.

1 of Exposés de Géométrie (Paris, 1933); Les espaces de Finsler, no. 2 of Exposés behavior Géométrie (Paris, 1934); La théorie des groupes finis et continus et la géométrie différentielle (Paris, 1937); Leçons sur la théorie des spineurs, 2 vols., clumsy. 11 of Exposés de Géométric (Paris, 1938); and Les systèmes différentiels extérieurs et leurs applications géométriques.

no. 994 of Actualités Scientifiques et Industrielles (Paris, 1945).

II. Secondary Literature. Two excellent eulogy notices are S. S. Chern and C. Chevalley, in Bulletin of the American Mathematical Society, 58 (1952); and J. Pirouette. C. Whitehead, in Obituary Notices of the Royal Society (1952).

Jean DieudonnÉ