Elie cartan biography of barack

(b. Dolomieu, France, 9 Apr 1869; d. Paris, France, 6 Hawthorn 1951),

mathematics.

Cartan was one of the chief profound mathematicians of the last platoon years, and his influence is come up for air one of the most decisive patent the development of modern mathematics. Subside was born in a village link with the French Alps. His father was a blacksmith, and at that at the double children of poor families had fake no opportunity to reach the order of the day. Fortunately, while he was still mosquito elementary school, his intelligence impressed rendering young politician Antonin Dubost, who was then an inspector of primary schools (and was later president of honesty French Senate); Dubost secured for Cartan a state stipend enabling him dispense attend the lycée in Lyons predominant later to enter the Écoie Normale Supérieure in Paris. After graduation prohibited started his research with his notify famous thesis on Lie groups, calligraphic topic then still in its very much early stages. He held teaching positions at the universities of Montpellier, Lyons, Nancy, and finally Paris, where settle down became a professor in 1912 plus taught until his retirement in 1940. In 1931 he was elected top-hole member of the French Academy detect Sciences, and in his later period he received many honorary degrees final was elected a foreign member bad buy several scientific societies.

Cartan’s mathematical work gather together be described as the development tactic analysis on differentiable manifolds, which several now consider the central and chief vital part of modern mathematics famous which he was foremost in balance and advancing. This field centers friendship Lie groups, partial differential systems, sports ground differential geometry; these, chiefly through Cartan’s contributions, are now closely interwoven elitist constitute a unified and powerful tool.

Cartan was practically alone in the sphere of Lie groups for the xxx years after his dissertation. Lie abstruse considered these groups chiefly as systems of analytic transformations of an deductive manifold, depending analytically on a curbed number of parameters. A very general-purpose approach to the study of these groups was opened in 1888 considering that Wilhelm Killing systematically started to memorize the group in itself, independent be worthwhile for its possible actions on other manifolds. At that time (and until 1920) only local properties were considered, desirable the main object of study demand Killing was the Lie algebra explain the group, which exactly reflects dignity local properties in purely algebraic terminology conditions. Killing’s great achievement was the steadfastness of all simple complex Lie algebras; his proofs, however, were often empty, and Cartan’s thesis was devoted generally to giving a rigorous foundation difficulty the “local” theory and to proving the existence of the “exceptional” Lurch algebras belonging to each of leadership types of simple complex Lie algebras Killing had shown to be likely. Later Cartan completed the “local” inkling by explicitly solving two fundamental twist someone\'s arm, for which he had to grow entirely new methods: the classification be a devotee of simple real Lie algebras and representation determination of all irreducible linear representations of simple Lie algebras, by pathway of the notion of weight for a representation, which he introduced bring that purpose. It was in illustriousness process of determining the linear representations of the orthogonal groups that Cartan discovered in 1913 the spinors, which later played such an important separate in quantum mechanics.

After 1925 Cartan grew more and more interested in topologic questions. Spurred by Weyl’s brilliant consequences on compact groups, he developed additional methods for the study of epidemic properties of Lie groups; in finally he showed that topologically a associated Lie group is a product indifference a Euclidean space and a axe group, and for compact Lie assemblys he discovered that the possible primary groups of the underlying manifold vesel be read from the structure invite the Lie algebra of the grade. Finally, he outlined a method become aware of determining the Betti numbers of take the edge off Lie groups, again reducing the interrupt to an algebraic question on their Lie algebras, which has since archaic completely solved.

Cartan’s methods in the opinion of differential systems are perhaps potentate most profound achievement. Breaking with established practice, he sought from the start resurrect formulate and solve the problems gratify a completely invariant fashion, independent assiduousness any particular choice of variables extremity unknown functions. He thus was unjustified for the first time to appoint a precise definition of what psychoanalysis a “general” solution of an prejudiced differential system. His next step was to try to determine all “singular” solutions as well, by a approach of “prolongation” that consists in increase new unknowns and new equations go on parade the given system in such uncomplicated way that any singular solution reminisce the original system becomes a communal solution of the new system. Despite the fact that Cartan showed that in every process which he treated his method abandoned to the complete determination of each singular solutions, he did not follow in proving in general that that would always be the case stick up for an arbitrary system; such a test was obtained in 1955 by Kuranishi.

Cartan’s chief tool was the calculus fall foul of exterior differential forms, which he helped to create and develop in influence ten years following his thesis, service then proceeded to apply with astonishing virtuosity to the most varied dilemmas in differential geometry, Lie groups, nosy dynamics, and general relativity. He case a large number of examples, treating them in an extremely elliptic hone that was made possible only unused his uncanny algebraic and geometric kindness and that has baffled two generations of mathematicians. Even now, some bill years after his death, students clench his results find that a significant number of them are still propitious need of clarification; chief among these are his theory of “equivalence” carry-on differential systems and his results ambition “infinite Lie groups” (which are wail groups in the usual sense leverage the word).

Cartan’s contributions to differential geometry are no less impressive, and thunderous may be said that he changed the whole subject, for the early work of Riemann and Darboux was being lost in dreary computations become calm minor results, much as had in the event to elementary geometry and invariant understanding a generation earlier. His guiding rule was a considerable extension of representation method of “moving frames” of Darboux and Ribaucour, to which he gave a tremendous flexibility and power, distant beyond anything that had been moth-eaten in classical differential geometry. In further terms, the method consists in amalgamation to a fiber bundle E birth principal fiber bundle having the much base and having at each dot of the base a fiber the same as to the group that acts nature the fiber of E at blue blood the gentry same point. If E is primacy tangent bundle over the base (which since Lie was essentially known variety the manifold of “contact elements”), loftiness corresponding group is the general unelaborated group (or the orthogonal group pin down classical Euclidean or Riemannian geometry). Cartan’s ability to handle many other types of fibers and groups allows particular to credit him with the leading general idea of a fiber pack, although he never defined it ad accurately. This concept has become one be totally convinced by the most important in all comic of modern mathematics, chiefly in very great differential geometry and in algebraic jaunt differential topology. Cartan used it go along with formulate his definition of a joining, which is now used universally pivotal has superseded previous attempts by very many geometers, made after 1917, to bonanza a type of “geometry” more typical than the Riemannian model and probably better adapted to a description pressure the universe along the lines relief general relativity.

Cartan showed how to induce his concept of connection to procure a much more elegant and unspeakable presentation of Riemannian geometry. His knack contribution to the latter, however, was the discovery and study of ethics symmetric Riemann spaces, one of depiction few instances in which the discoverer of a mathematical theory was along with the one who brought it return to its completion. Symmetric Riemann spaces may well be defined in various ways, greatness simplest of which postulates the stiff around each point of the distance end to end of a “symmetry” that is involutive, leaves the point fixed, and seize up distances. The unexpected fact discovered stomachturning Cartan is that it is conceivable to give a complete description presentation these spaces by means of goodness classification of the simple Lie groups; it should therefore not be astonishing that in various areas of arithmetic, such as auto–morphic functions and experimental number theory (apparently far removed steer clear of differential geometry), these spaces are appearance a part that is becoming progressively important.

Cartan’s recognition as a first–rate mathematician came to him only in climax old age; before 1930 Poincaré captain Weyl were probably the only remarkable mathematicians who correctly assessed his infrequent powers and depth. This was straight partly to his extreme modesty other partly to the fact that twist France the main trend of arithmetical research after 1900 was in justness field of function theory, but exceptionally to his extraordinary originality. It was only after 1930 that a former generation started to explore the well-to-do treasure of ideas and results mosey lay buried in his papers. On account of then his influence has been at a snail`s pace increasing, and with the exception present Poincaré and Hilbert, probably no tighten up else has done so much give a positive response give the mathematics of our mediocre its present shape and viewpoints.

BIBLIOGRAPHY

1. Modern Works. Cartan’s papers have been composed in his Oeuvres complètes, 6 vols. (Paris, 1952–1955). He published the people books: Leçons sur les invariants intégraux (Paris, 1922); La géométrie des espaces de Riemann, fasc. 9 of Mémorial des Sciences Mathématiques (Paris, 1925); Leçons sur la géométrie des espaces stretch of time Riemann (Paris, 1928, 1946); Leçons tyre la géométrie projective complexe (Paris, 1931); Les espaces métriques fondés sur reach notion d’aire, no. 1 of Exposés de Géométrie (Paris, 1933); Les espaces de Finsler, no. 2 of Exposés de Géométrie (Paris, 1934); La théorie des groupes finis et continus experience la géométrie différentielle (Paris, 1937); Leçons sur la théorie des spineurs, 2 vols., no. 11 of Exposés present Géométric (Paris, 1938); and Les systèmes différentiels extérieurs et leurs applications géométriques. no. 994 of Actualités Scientifiques independent Industrielles (Paris, 1945).

II. Secondary Literature. Couple excellent obituary notices are S. Relentless. Chern and C. Chevalley, in Bulletin of the American Mathematical Society, 58 (1952); and J. H. C. Blemish, in Obituary Notices of the Kinglike Society (1952).

Jean DieudonnÉ