VI.52 Camille Jordan

b. Lyon, France, 1838; d. Milan, Italy, 1922
Nominally an engineer until 1885; teacher of mathematics,
École Polytechnique and Collège de France (1873-1912)

Jordan was the leading group theorist of his generation. His immense Traité des Substitutions et des Équations Algébriques (1870), which brought together all his earlier results on PERMUTATION GROUPS [III.68] and provided a synthesis of GALOIS’S [VI.41] ideas, remained a cornerstone for group theorists for many years. Included in the Traité, in the chapter on what he calls linear substitutions (now written in matrix form as y = Ax), is the definition of what today is called the JORDAN NORMAL FORM [III.43] of a matrix, although in 1868 WEIERSTRASS [VI.44] had already defined an equivalent normal form.

Jordan is also known for his work in topology, especially for the theorem now known as the Jordan curve theorem. This states that a simple closed curve in the plane separates the plane into two disjoint regions, an inside and an outside, and it was given by him in his influential Cours d’Analyse (1887). Although the theorem appears obvious, the proof, as Jordan recognized, is difficult and the one he gave was incorrect. (The proof is relatively easy for smooth curves; the difficulties arise when dealing with nowhere-smooth curves, such as the Koch snowflake.) The first rigorous proof was given by Oswald Veblen in 1905. There is a stronger form of the theorem, known as the Jordan-Schönflies theorem, which states that in addition the two regions of the plane, the inside and the outside, are homeomorphic to the standard circle in the plane. Unlike the original theorem, this stronger form of the theorem cannot be generalized to higher dimensions, a famous counterexample being the Alexander horned sphere.