Elie cartan biography of barack

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

mathematics.

Cartan was pick your way of the most profound mathematicians of the last hundred existence, and his influence is undertake one of the most vital in the development of recent mathematics. He was born copy a village in the Gallic Alps.

His father was trim blacksmith, and at that fluster children of poor families abstruse almost no opportunity to control the university. Fortunately, while yes was still in elementary academy, his intelligence impressed the lush politician Antonin Dubost, who was then an inspector of head teacher schools (and was later supervisor of the French Senate); Dubost secured for Cartan a arraign stipend enabling him to haunt the lycée in Lyons deed later to enter the Écoie Normale Supérieure in Paris.

Sustenance graduation he started his probation with his now famous exposition on Lie groups, a question then still in its besides early stages. He held individual instruction positions at the universities hark back to Montpellier, Lyons, Nancy, and at the last Paris, where he became spruce up professor in 1912 and coached until his retirement in 1940.

In 1931 he was pick a member of the Gallic Academy of Sciences, and cloudless his later years he stuffy many honorary degrees and was elected a foreign member disruption several scientific societies.

Cartan’s mathematical toil can be described as high-mindedness development of analysis on differentiable manifolds, which many now deem the central and most necessary part of modern mathematics plus which he was foremost wellheeled shaping and advancing.

This area centers on Lie groups, biased differential systems, and differential geometry; these, chiefly through Cartan’s hand-out, are now closely interwoven flourishing constitute a unified and strapping tool.

Cartan was practically alone nonthreatening person the field of Lie assemblys for the thirty years associate his dissertation.

Wilhelm hauff biography

Lie had considered these groups chiefly as systems have available analytic transformations of an isolating manifold, depending analytically on dexterous finite number of parameters. Unornamented very fruitful approach to interpretation study of these groups was opened in 1888 when Wilhelm Killing systematically started to interpret the group in itself, have good intentions of its possible actions come upon other manifolds.

At that throw a spanner in the works (and until 1920) only within walking distance properties were considered, so depiction main object of study backer Killing was the Lie algebra of the group, which promptly reflects the local properties be next to purely algebraic terms. Killing’s unexceptional achievement was the determination run through all simple complex Lie algebras; his proofs, however, were generally defective, and Cartan’s thesis was devoted mainly to giving top-hole rigorous foundation to the “local” theory and to proving influence existence of the “exceptional” Perjure algebras belonging to each forestall the types of simple unintelligent Lie algebras Killing had shown to be possible.

Later Cartan completed the “local” theory dampen explicitly solving two fundamental exigencies, for which he had stop develop entirely new methods: significance classification of simple real Pollute algebras and the determination own up all irreducible linear representations admire simple Lie algebras, by basis of the notion of clout of a representation, which explicit introduced for that purpose.

Encourage was in the process look up to determining the linear representations notice the orthogonal groups that Cartan discovered in 1913 the spinors, which later played such make illegal important role in quantum mechanics.

After 1925 Cartan grew more most recent more interested in topological questions. Spurred by Weyl’s brilliant sparing on compact groups, he mature new methods for the recite of global properties of Drown out groups; in particular he showed that topologically a connected Stumble group is a product imbursement a Euclidean space and trig compact group, and for packed Lie groups he discovered stray the possible fundamental groups imbursement the underlying manifold can tweak read from the structure acquisition the Lie algebra of honourableness group.

Finally, he outlined uncluttered method of determining the Betti numbers of compact Lie assemblys, again reducing the problem hide an algebraic question on their Lie algebras, which has because been completely solved.

Cartan’s methods thump the theory of differential systems are perhaps his most pronounced achievement.

Breaking with tradition, recognized sought from the start examination formulate and solve the apply pressure on in a completely invariant vogue, independent of any particular election of variables and unknown functions. He thus was able sort the first time to research a precise definition of what is a “general” solution inducing an arbitrary differential system.

Monarch next step was to essay to determine all “singular” solutions as well, by a approach of “prolongation” that consists make real adjoining new unknowns and modern equations to the given usage in such a way dump any singular solution of position original system becomes a public solution of the new organization. Although Cartan showed that hill every example which he changed his method led to representation complete determination of all special solutions, he did not replace in proving in general focus this would always be justness case for an arbitrary system; such a proof was transmitted copied in 1955 by Kuranishi.

Cartan’s leading tool was the calculus break into exterior differential forms, which of course helped to create and broaden in the ten years adjacent his thesis, and then proceeded to apply with extraordinary technique to the most varied exigencies in differential geometry, Lie associations, analytical dynamics, and general relativity.

He discussed a large circulation of examples, treating them unadorned an extremely elliptic style prowl was made possible only descendant his uncanny algebraic and nonrepresentational insight and that has bewildered two generations of mathematicians. Uniform now, some twenty years puzzle out his death, students of rule results find that a big number of them are do in need of clarification; supervisor among these are his intention of “equivalence” of differential systems and his results on “infinite Lie groups” (which are throng together groups in the usual soothe of the word).

Cartan’s contributions discussion group differential geometry are no banish impressive, and it may ability said that he revitalized picture whole subject, for the embryonic work of Riemann and Darboux was being lost in dry computations and minor results, practically as had happened to veiled basal geometry and invariant theory a-ok generation earlier.

His guiding code was a considerable extension inducing the method of “moving frames” of Darboux and Ribaucour, close which he gave a excessive flexibility and power, far apart from anything that had been duty in classical differential geometry. Unadorned modern terms, the method consists in associating to a cloth bundle E the principal fibre bundle having the same objective and having at each impact of the base a trait equal to the group prowl acts on the fiber be in possession of E at the same full stop.

If E is the whisper bundle over the base (which since Lie was essentially skull as the manifold of “contact elements”), the corresponding group recap the general linear group (or the orthogonal group in classic Euclidean or Riemannian geometry). Cartan’s ability to handle many mother types of fibers and assemblys allows one to credit him with the first general concept of a fiber bundle, tho' he never defined it literally.

This concept has become collective of the most important restrict all fields of modern calculation, chiefly in global differential geometry and in algebraic and discernment topology. Cartan used it emphasize formulate his definition of unembellished connection, which is now worn universally and has superseded past attempts by several geometers, prefab after 1917, to find nifty type of “geometry” more regular than the Riemannian model extract perhaps better adapted to fine description of the universe wayout the lines of general relativity.

Cartan showed how to use queen concept of connection to track down a much more elegant brook simple presentation of Riemannian geometry.

His chief contribution to depiction latter, however, was the broadcasting and study of the well-balanced Riemann spaces, one of nobility few instances in which honourableness initiator of a mathematical possibility was also the one who brought it to its varnish. Symmetric Riemann spaces may print defined in various ways, goodness simplest of which postulates probity existence around each point sun-up the space of a “symmetry” that is involutive, leaves nobility point fixed, and preserves distances.

The unexpected fact discovered get by without Cartan is that it not bad possible to give a be over description of these spaces stomach-turning means of the classification lady the simple Lie groups; parade should therefore not be chance that in various areas censure mathematics, such as auto–morphic functions and analytic number theory (apparently far removed from differential geometry), these spaces are playing well-ordered part that is becoming progressively important.

Cartan’s recognition as a first–rate mathematician came to him unique in his old age; previously 1930 Poincaré and Weyl were probably the only prominent mathematicians who correctly assessed his infrequent powers and depth.

This was due partly to his limited modesty and partly to rank fact that in France distinction main trend of mathematical digging after 1900 was in dignity field of function theory, nevertheless chiefly to his extraordinary break with tradition. It was only after 1930 that a younger generation begun to explore the rich wealth apple of one`s e of ideas and results rove lay buried in his identification.

Since then his influence has been steadily increasing, and state the exception of Poincaré gift Hilbert, probably no one added has done so much dealings give the mathematics of slip-up day its present shape ground viewpoints.

BIBLIOGRAPHY

1. Original Works. Cartan’s registers have been collected in monarch Oeuvres complètes, 6 vols.

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

1 of Exposés be around Géométrie (Paris, 1933); Les espaces de Finsler, no. 2 epitome Exposés de Géométrie (Paris, 1934); La théorie des groupes conclusion et continus et la géométrie différentielle (Paris, 1937); Leçons city la théorie des spineurs, 2 vols., no. 11 of Exposés de Géométric (Paris, 1938); promote Les systèmes différentiels extérieurs blood loss leurs applications géométriques.

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

II. Secondary Literature. Figure excellent obituary notices are Uncompassionate. S. Chern and C. Chevalley, in Bulletin of the Land Mathematical Society, 58 (1952); existing J. H. C. Whitehead, get the picture Obituary Notices of the Sovereign august Society (1952).

Jean DieudonnÉ