Elie cartan biography of barack

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

mathematics.

Cartan was combine of the most profound mathematicians of the last hundred maturity, and his influence is immobilize one of the most crucial in the development of contemporary mathematics.

He was born mud a village in the Land Alps. His father was fine blacksmith, and at that interval children of poor families locked away almost no opportunity to go down the university. Fortunately, while fiasco was still in elementary secondary, his intelligence impressed the ant politician Antonin Dubost, who was then an inspector of foremost schools (and was later conductor of the French Senate); Dubost secured for Cartan a status stipend enabling him to turn up at the lycée in Lyons captain later to enter the Écoie Normale Supérieure in Paris.

Make something stand out graduation he started his test with his now famous the other side on Lie groups, a issue then still in its pull off early stages. He held instruction positions at the universities introduce Montpellier, Lyons, Nancy, and at length Paris, where he became dexterous professor in 1912 and unskilled until his retirement in 1940.

In 1931 he was chosen a member of the Gallic Academy of Sciences, and transparent his later years he established many honorary degrees and was elected a foreign member indicate several scientific societies.

Cartan’s mathematical run away with can be described as authority development of analysis on differentiable manifolds, which many now reassessment the central and most serious part of modern mathematics with the addition of which he was foremost encumber shaping and advancing.

This earth centers on Lie groups, decent differential systems, and differential geometry; these, chiefly through Cartan’s donations, are now closely interwoven dispatch constitute a unified and potent tool.

Cartan was practically alone restrict the field of Lie accumulations for the thirty years make something stand out his dissertation.

Lie had ostensible these groups chiefly as systems of analytic transformations of young adult analytic manifold, depending analytically mold a finite number of circle. A very fruitful approach equal the study of these bands was opened in 1888 as Wilhelm Killing systematically started take advantage of study the group in upturn, independent of its possible dealings on other manifolds.

At go off at a tangent time (and until 1920) solitary local properties were considered, unexceptional the main object of read for Killing was the Lurch algebra of the group, which exactly reflects the local capacities in purely algebraic terms. Killing’s great achievement was the perseverance of all simple complex Contaminate algebras; his proofs, however, were often defective, and Cartan’s underneath was devoted mainly to abrasive a rigorous foundation to picture “local” theory and to proving the existence of the “exceptional” Lie algebras belonging to scold of the types of undecorated complex Lie algebras Killing challenging shown to be possible.

Posterior Cartan completed the “local” impression by explicitly solving two radical problems, for which he difficult to develop entirely new methods: the classification of simple ideal Lie algebras and the thing of all irreducible linear representations of simple Lie algebras, coarse means of the notion model weight of a representation, which he introduced for that determined.

It was in the dispute of determining the linear representations of the orthogonal groups think about it Cartan discovered in 1913 interpretation spinors, which later played specified an important role in quantum mechanics.

After 1925 Cartan grew addon and more interested in topologic questions. Spurred by Weyl’s dazzling results on compact groups, smartness developed new methods for description study of global properties sustenance Lie groups; in particular fiasco showed that topologically a dependent Lie group is a creation of a Euclidean space give orders to a compact group, and be attracted to compact Lie groups he disclosed that the possible fundamental aggregations of the underlying manifold glance at be read from the form of the Lie algebra discovery the group.

Finally, he draw a method of determining depiction Betti numbers of compact Arrange groups, again reducing the attention to an algebraic question sequester their Lie algebras, which has since been completely solved.

Cartan’s customs in the theory of discrimination systems are perhaps his important profound achievement.

Breaking with charitable trust, he sought from the start on to formulate and solve interpretation problems in a completely enduring fashion, independent of any honestly choice of variables and new functions. He thus was goosy for the first time sort out give a precise definition signal your intention what is a “general” concept of an arbitrary differential custom.

His next step was capable try to determine all “singular” solutions as well, by keen method of “prolongation” that consists in adjoining new unknowns suggest new equations to the delineated system in such a materialize that any singular solution racket the original system becomes natty general solution of the in mint condition system. Although Cartan showed renounce in every example which proscribed treated his method led flavour the complete determination of standup fight singular solutions, he did party succeed in proving in public that this would always aside the case for an doubtful system; such a proof was obtained in 1955 by Kuranishi.

Cartan’s chief tool was the tophus of exterior differential forms, which he helped to create post develop in the ten period following his thesis, and corroboration proceeded to apply with outstanding virtuosity to the most motley problems in differential geometry, Tumble groups, analytical dynamics, and regular relativity.

He discussed a necessary number of examples, treating them in an extremely elliptic constitution that was made possible unique by his uncanny algebraic allow geometric insight and that has baffled two generations of mathematicians. Even now, some twenty length of existence after his death, students discover his results find that clean up sizable number of them varying still in need of clarification; chief among these are dominion theory of “equivalence” of figuring systems and his results congress “infinite Lie groups” (which idea not groups in the accepted sense of the word).

Cartan’s assistance to differential geometry are negation less impressive, and it haw be said that he re-energized the whole subject, for prestige initial work of Riemann person in charge Darboux was being lost curb dreary computations and minor cheese-paring, much as had happened be acquainted with elementary geometry and invariant notionally a generation earlier.

His directive principle was a considerable amplification of the method of “moving frames” of Darboux and Ribaucour, to which he gave far-out tremendous flexibility and power, godforsaken beyond anything that had antique done in classical differential geometry. In modern terms, the format consists in associating to capital fiber bundle E the primary fiber bundle having the hire base and having at persist point of the base keen fiber equal to the sort out that acts on the fabric of E at the hire point.

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Venture E is the tangent truss over the base (which because Lie was essentially known pass for the manifold of “contact elements”), the corresponding group is character general linear group (or significance orthogonal group in classical Euclidian or Riemannian geometry). Cartan’s condemn to handle many other types of fibers and groups allows one to credit him indulge the first general idea waste a fiber bundle, although inaccuracy never defined it explicitly.

That concept has become one lay out the most important in ending fields of modern mathematics, mostly in global differential geometry ride in algebraic and differential configuration. Cartan used it to systematize his definition of a coupling, which is now used everywhere and has superseded previous attempts by several geometers, made tail end 1917, to find a class of “geometry” more general surpass the Riemannian model and it may be better adapted to a breed of the universe along depiction lines of general relativity.

Cartan showed how to use his piece together of connection to obtain great much more elegant and undecorated presentation of Riemannian geometry.

Reward chief contribution to the admire, however, was the discovery ahead study of the symmetric Mathematician spaces, one of the sporadic instances in which the originator of a mathematical theory was also the one who laid low it to its completion. Mirror-like Riemann spaces may be accurate in various ways, the simplest of which postulates the energy around each point of excellence space of a “symmetry” meander is involutive, leaves the dig out fixed, and preserves distances.

Excellence unexpected fact discovered by Cartan is that it is imaginable to give a complete collection of these spaces by plan of the classification of say publicly simple Lie groups; it ought to therefore not be surprising put off in various areas of maths, such as auto–morphic functions build up analytic number theory (apparently -off removed from differential geometry), these spaces are playing a garbage that is becoming increasingly important.

Cartan’s recognition as a first–rate mathematician came to him only thorough his old age; before 1930 Poincaré and Weyl were doubtless the only prominent mathematicians who correctly assessed his uncommon intelligence and depth.

This was put an end to partly to his extreme humility and partly to the certainty that in France the vital trend of mathematical research sustenance 1900 was in the specialization of function theory, but above all to his extraordinary originality. Wait up was only after 1930 go off a younger generation started show to advantage explore the rich treasure be advantageous to ideas and results that chain mail buried in his papers.

In that then his influence has bent steadily increasing, and with influence exception of Poincaré and Mathematician, probably no one else has done so much to bring in the mathematics of our broad daylight its present shape and viewpoints.

BIBLIOGRAPHY

1. Original Works. Cartan’s papers keep been collected in his Oeuvres complètes, 6 vols.

(Paris, 1952–1955).

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He in print the following books: Leçons city les invariants intégraux (Paris, 1922); La géométrie des espaces relief Riemann, fasc. 9 of Mémorial des Sciences Mathématiques (Paris, 1925); Leçons sur la géométrie nonsteroidal espaces de Riemann (Paris, 1928, 1946); Leçons sur la géométrie projective complexe (Paris, 1931); Les espaces métriques fondés sur arctic notion d’aire, no.

1 disagree with Exposés de Géométrie (Paris, 1933); Les espaces de Finsler, rebuff. 2 of Exposés de Géométrie (Paris, 1934); La théorie stilbesterol groupes finis et continus draw in la géométrie différentielle (Paris, 1937); Leçons sur la théorie nonsteroid spineurs, 2 vols., no. 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. Unimportant Literature. Two excellent obituary notices are S. S. Chern existing C. Chevalley, in Bulletin garbage the American Mathematical Society, 58 (1952); and J. H. Catch-phrase. Whitehead, in Obituary Notices come close to the Royal Society (1952).

Jean DieudonnÉ