A brilliant student, Jordan followed the usual career of French mathematicians from Cauchy to Poincaré, and at seventeen he entered the École Polytechnique.
Career
Achievements
Membership
Academy of Sciences
1881
Academy of Sciences, Paris, France
Jordan was elected a member of the Academy of Sciences in 1881.
Awards
Légion d'Honneur
1890
On July 12, 1890, Jordan became an officer of the Légion d'Honneur.
A brilliant student, Jordan followed the usual career of French mathematicians from Cauchy to Poincaré, and at seventeen he entered the École Polytechnique.
Camille Jordan was a French mathematician. He is known for his work on substitution groups and the theory of equations, which first brought a full understanding of the importance of the theories of the eminent mathematician Évariste Galois. He also served as a professor at the Ecole Polytechnique and College de France.
Background
Camille Jordan was born on January 5, 1838, in Lyons, France, in a well-to-do family. One of his granduncles (also named Camille Jordan) was a fairly well-known politician who took part in many events from the French Revolution in 1789 to the beginning of the Bourbon restoration; a cousin, Alexis Jordan, is known in botany as the discoverer of “smaller species” which still bear his name (“jordanons”). Jordan’s father Esprit-Alexandre Jordan, an engineer, was a graduate of the École Polytechnique; his mother, Joséphine Puvis de Chavannes, was a sister of the painter Pierre Puvis de Chavannes.
Education
A brilliant student, Jordan followed the usual career of French mathematicians from Cauchy to Poincaré, and at seventeen he entered the École Polytechnique. His academical advisors were Victor Puiseux and Joseph Alfred Serret.
Jordan was an engineer by profession, and it left him ample time for mathematical research, and most of his 120 papers were written before he retired as an engineer. From 1873 until his retirement in 1912 he taught simultaneously at the Ecole Polytechnique and the College de France.
Jordan’s place in the tradition of French mathematics is exactly halfway between Hermite and Poincare. Like them he was a “universal” mathematician who published papers in practically all branches of the mathematics of his time. In one of his first papers, devoted to questions of “analysis situs” (as combinatorial topology was then called), he investigated symmetries in polyhedrons from an exclusively combinatorial point of view, which was then an entirely new approach. In analysis his conception of what a rigorous proof should be was far more exacting than that of most of his contemporaries; and his Cours d'analyse, which was first published in the early 1880s and had very widespread influence, set standards which remained unsurpassed for many years.
Jordan took an active part in the movement which started modern analysis in the last decades of the nineteenth century: independently of Peano, he introduced a notion of exterior measure for arbitrary sets in a plane or in n-dimensional space. The concept of a function of bounded variation originated with him; and he proved that such a function is the difference of two increasing functions, a result which enabled him to extend the definition of the length of a curve and to generalize the known criteria of convergence of Fourier series. His most famous contribution to topology was to realize that the decomposition of a plane into two regions by a simple closed curve was susceptible of mathematical proof and to imagine such a proof for the first time.
Although these contributions would have been enough to rank Jordan very high among his mathematical contemporaries, it is chiefly as an algebraist that he reached celebrity when he was barely thirty; and during the next forty years, he was universally regarded as the undisputed master of group theory.
When Jordan started his mathematical career, Galois' profound ideas and results (which had remained unknown to most mathematicians until 1845) were still very poorly understood, despite the efforts of A. Serret and Liouville to popularize them; and before 1860 Kronecker was probably the only first-rate mathematician who realized the power of these ideas and who succeeded in using them in his own algebraic research. Jordan was the first to embark on systematic development of the theory of finite groups and of its applications in the directions opened by Galois. Chief among his first results were the concept of composition series and the first part of the famous Jordan-Hôlder theorem, proving the invariance of the system of indexes of consecutive groups in any composition series (up to their ordering).
He also was the first to investigate the structure of the general linear group and of the “classical” groups over a prime finite field, and he very ingeniously applied his results to a great range of problems; in particular, he was able to determine the structure of the Galois group of equations having as roots the parameters of some well-known geometric configurations (the twenty-seven lines on a cubic surface, the twenty-eight double tangents to a quartic, the sixteen double points of a Kummer surface, and so on).
Another problem for which Jordan’s knowledge of these classical groups was the key to the solution, and to which he devoted a considerable amount of effort from the beginning of his career, was the general study of solvable finite groups. From all we know today (in particular about p-groups, a field which was started, in the generation following Jordan, with the Sylow theorems) it seems hopeless to expect a complete classification of all solvable groups which would characterize each of them, for instance, by a system of numerical invariants. Perhaps Jordan realized this; at any rate, he contented himself with setting up the machinery that would automatically yield all solvable groups of a given order n. This in itself was no mean undertaking; and the solution imagined by Jordan was a gigantic recursive scheme, giving the solvable groups of order n when one supposes that the solvable groups of which the orders are the exponents of the prime factors of n are all known. This may have no more than a theoretical value; but in the process of developing his method, Jordan was led to many important new concepts, such as the minimal normal subgroups of a group and the orthogonal groups over a field of characteristic 2 (which he called “hypoabelian” groups).
In 1870 Jordan gathered all his results on permutation groups for the previous ten years in a huge volume, Traité des substitutions, which for thirty years was to remain the bible of all specialists in group theory. His fame had spread beyond France, and foreign students were eager to attend his lectures; in particular Felix Klein and Sophus Lie came to Paris in 1870 to study with Jordan, who at that time was developing his researches in an entirely new direction: the determination of all groups of movements in three-dimensional space. This may well have been the source from which Lie conceived his theory of “continuous groups” and Klein the idea of “discontinuous groups” (both types had been en-countered by Jordan in his classification).
The most profound results obtained by Jordan in algebra are his “finiteness theorems,” which he proved during the twelve years following the publication of the Traité.
Achievements
Camille Jordan is remembered for a number of results in mathematics, which include the Jordan curve theorem, a topological result required in complex analysis; the Jordan normal form and the Jordan matrix in linear algebra; the Jordan measure in mathematical analysis; the Jordan-Hölder theorem on composition series in group theory; and the Jordan's theorem on finite linear groups.
Jordan's work did much to bring Galois theory into the mainstream. He is also remembered for his investigation of the Mathieu groups, the first examples of sporadic groups. His Traité des substitutions won for Jordan the 1870 prix Poncelet.
On July 12, 1890, he became an officer of the Légion d'Honneur.
The asteroid 25593 Camillejordan and Institut Camille Jordan are named in his honour.
