Élie Cartan

Élie Joseph Cartan, ForMemRS was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems, differential geometry. He made significant contributions to general relativity and indirectly to quantum mechanics, he is regarded as one of the greatest mathematicians of the twentieth century. Cartan's recognition as a first–rate mathematician came to him only in his old age; this was due to his extreme modesty and to the fact that in France the main trend of mathematical research after 1900 was in the field of function theory, but chiefly to his extraordinary originality. It was only after 1930 that a younger generation started to explore the rich treasure of ideas and results that lay buried in his papers. Since his influence has been increasing, with the exception of Poincaré and Hilbert no one else has done so much to give the mathematics of 20th century its shape and viewpoints. Élie Cartan was born 9 April 1869 in the village of Dolomieu, Isère to Joseph Cartan and Anne Cottaz.

Joseph Cartan was the village blacksmith. Élie had an elder sister Jeanne-Marie. Élie Cartan was the best student in the school. One of his teachers, M. Dupuis, recalled "Élie Cartan was a shy student, but an unusual light of great intellect was shining in his eyes, this was combined with an excellent memory". Antonin Dubost the representative of Isère, visited the school and was impressed by Cartan's unusual abilities, he recommended Cartan to participate in a contest for a scholarship in a lycée. Cartan prepared for the contest under the supervision of M. Dupuis and passed at the age of ten years, he spent five years at the College of Vienne and two years at the Lycée of Grenoble. In 1887 he moved to the Lycée Janson de Sailly in Paris to study sciences for two years. Cartan enrolled in the École Normale Supérieure in 1888, he attended there lectures by Charles Hermite, Jules Tannery, Gaston Darboux, Paul Appell, Émile Picard, Edouard Goursat, Henri Poincaré whose lectures were what Cartan thought most of.

After graduation from the École Normale Superieure in 1891, Cartan was drafted into the French army, where he served one year and attained the rank of sergeant. For next two years Cartan returned to ENS and, following the advice of his classmate Arthur Tresse who studied under Sophus Lie in the years 1888–1889, worked on the subject of classification of simple Lie groups, started by Wilhelm Killing. In 1892 Lie came to Paris, at the invitation of Darboux and Tannery, met Cartan for the first time. Cartan defended his dissertation, The structure of finite continuous groups of transformations in 1894 in the Faculty of Sciences in the Sorbonne. Between 1894 and 1896 Cartan was a lecturer at the University of Montpellier. In 1903, while in Lyons, Cartan married Marie-Louise Bianconi. In 1904, Cartan's first son, Henri Cartan, who became an influential mathematician, was born. In 1909 Cartan moved his family to Paris and worked as a lecturer in the Faculty of Sciences in the Sorbonne. In 1912 Cartan became Professor there, based on the reference he received from Poincaré.

He remained in Sorbonne until his retirement in 1940 and spent the last years of his life teaching mathematics at the École Normale Supérieure for girls. In 1921 he became a foreign member of the Polish Academy of Learning and in 1937 a foreign member of the Royal Netherlands Academy of Arts and Sciences. In 1938 he participated in the International Committee composed to organise the International Congresses for the Unity of Science, he died in 1951 in Paris after a long illness. In the Travaux, Cartan breaks down his work into 15 areas. Using modern terminology, they are: Lie theory 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, 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, spinorsCartan's mathematical work can be described as the development of analysis on differentiable manifolds, which many now conside

Henri Cartan

Henri Paul Cartan was a French mathematician with substantial contributions in algebraic topology. He was the brother of composer Jean Cartan. Cartan studied at the Lycée Hoche in Versailles at the École Normale Supérieure in Paris, receiving his doctorate in mathematics, he taught at the University of Strasbourg from November 1931 until the outbreak of the Second World War, after which he held academic positions at a number of other French universities, spending the bulk of his working life in Paris. Cartan is known for work in algebraic topology, in particular on cohomology operations, the method of "killing homotopy groups", group cohomology, his seminar in Paris in the years after 1945 covered ground on several complex variables, sheaf theory, spectral sequences and homological algebra, in a way that influenced Jean-Pierre Serre, Armand Borel, Alexander Grothendieck and Frank Adams, amongst others of the leading lights of the younger generation. The number of his official students was small, but includes Adrien Douady, Roger Godement, Max Karoubi, Jean-Louis Koszul, Jean-Pierre Serre and René Thom.

Cartan was a founding member of the Bourbaki group and one of its most active participants. His book with Samuel Eilenberg Homological Algebra was an important text, treating the subject with a moderate level of abstraction with the help of category theory. Cartan used his influence to help obtain the release of some dissident mathematicians, including Leonid Plyushch and Jose Luis Massera. For his humanitarian efforts, he received the Pagels Award from the New York Academy of Sciences; the Cartan model in algebra is named after Cartan. Cartan died on 13 August 2008 at the age of 104, his funeral took place the following Wednesday on 20 August in Drome. Cartan received numerous honours and awards including the Wolf Prize in 1980, he was an Invited Speaker at the ICM in 1932 in Zurich and a Plenary Speaker at the ICM in 1950 in Cambridge, Massachusetts and in 1958 in Edinburgh. From 1974 until his death he had been a member of the French Academy of Sciences, he was a foreign member of the Finnish Academy of Science and Letters, Royal Danish Academy of Sciences and Letters, Royal Society of London, Russian Academy of Sciences, Royal Swedish Academy of Sciences, United States National Academy of Sciences, Polish Academy of Sciences and other academies and societies.

O'Connor, John J.. "Décès du mathématicien français Henri Cartan", Agence France-Presse, 2008-08-18, archived from the original on 2008-08-22 Chang, Kenneth, "Henri Cartan, French Mathematician, Is Dead at 104", The New York Times, p. A17, retrieved 2008-08-25 Cartan, Henri. Homological Algebra. Princeton Mathematical Series. 19. Princeton University Press. ISBN 978-0-691-04991-5. Rehmeyer, Julie, "Founder of the Secret Society of Mathematicians", Science News Henri Cartan on IMDb Jackson, Allyn, "Interview with Henri Cartan", Notices of the American Mathematical Society, 46: 782–8 Illusie, Luc. Notices of the American Mathematical Society, Sept. 2010, vol. 57, issue 8 Henri Cartan at l'Académie des Sciences Biographical sketch and bibliography by the Société Mathématique de France on the occasion of Cartan's 100th birthday. Cerf, Jean, "Trois quarts de siècle avec Henri Cartan", Gazette des Mathématiciens: 7–8, archived from the original on 2008-09-09 Samuel, Pierre, "Souvenirs personnels sur H. Cartan", Gazette des Mathématiciens: 13–15, archived from the original on 2008-09-09 "100th Birthday of Henri Cartan", European Mathematical Society Newsletter: 20–21, September 2004 Papers by Henri Cartan as member of the'Association européenne des enseignants' and the'Mouvement fédéraliste européen' are at the Historical Archives of the EU in Florence

Algebraic topology

Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though most classify up to homotopy equivalence. Although algebraic topology uses algebra to study topological problems, using topology to solve algebraic problems is sometimes possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. Below are some of the main areas studied in algebraic topology: In mathematics, homotopy groups are used in algebraic topology to classify topological spaces; the first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space. In algebraic topology and abstract algebra, homology is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group.

In homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains and coboundaries. Cohomology can be viewed as a method of assigning algebraic invariants to a topological space that has a more refined algebraic structure than does homology. Cohomology arises from the algebraic dualization of the construction of homology. In less abstract language, cochains in the fundamental sense should assign'quantities' to the chains of homology theory. A manifold is a topological space. Examples include the plane, the sphere, the torus, which can all be realized in three dimensions, but the Klein bottle and real projective plane which cannot be realized in three dimensions, but can be realized in four dimensions. Results in algebraic topology focus on global, non-differentiable aspects of manifolds. Knot theory is the study of mathematical knots. While inspired by knots that appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone.

In precise mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, R 3. Two mathematical knots are equivalent if one can be transformed into the other via a deformation of R 3 upon itself. A simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments and their n-dimensional counterparts. Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory; the purely combinatorial counterpart to a simplicial complex is an abstract simplicial complex. A CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory; this class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial nature that allows for computation. An older name for the subject was combinatorial topology, implying an emphasis on how a space X was constructed from simpler ones.

In the 1920s and 1930s, there was growing emphasis on investigating topological spaces by finding correspondences from them to algebraic groups, which led to the change of name to algebraic topology. The combinatorial topology name is still sometimes used to emphasize an algorithmic approach based on decomposition of spaces. In the algebraic approach, one finds a correspondence between spaces and groups that respects the relation of homeomorphism of spaces; this allows one to recast statements about topological spaces into statements about groups, which have a great deal of manageable structure making these statement easier to prove. Two major ways in which this can be done are through fundamental groups, or more homotopy theory, through homology and cohomology groups; the fundamental groups give us basic information about the structure of a topological space, but they are nonabelian and can be difficult to work with. The fundamental group of a simplicial complex does have a finite presentation.

Homology and cohomology groups, on the other hand, are abelian and in many important cases finitely generated. Finitely generated abelian groups are classified and are easy to work with. In general, all constructions of algebraic topology are functorial. Fundamental groups and homology and cohomology groups are not only invariants of the underlying topological space, in the sense that two topological spaces which are homeomorphic have the same associated groups, but their associated morphisms correspond — a continuous mapping of spaces induces a group homomorphism on the associated groups, these homomorphisms can be used to show non-existence of mappings. One of the first mathematicians to work with different types of cohomology was Georges de Rham. One can use the differential structure of smooth manifolds via de Rham cohomology, or Čech or sheaf co