PICARD, (CHARLES-) MILE

born , July 24, 1856, Paris, Fr. died Dec. 11, 1941, Paris French mathematician whose theories did much to advance research into analysis, algebraic geometry, and mechanics. Picard became a lecturer at the University of Paris in 1878 and a professor at the University of Toulouse the following year. From 1881 to 1898 he held various posts with the University of Toulouse and the cole Normale Suprieure, Paris, and in 1898 he was appointed professor at the University of Paris. In 1917 he was elected permanent secretary for the mathematical sciences in the Academy of Sciences, Paris. Picard worked on quadratic forms, on Fuchsian and Abelian functions, and on the allied theories of discontinuous and continuous groups of transformation; he also discovered hyperfuchsian and hyperabelian functions. His work led to a study of the algebraic manifold now known as the Picard variety, which plays a fundamental role in algebraic geometry. In 1879 he proved the theorem known by his name, that an integral function of the complex variable takes every finite value, with one possible exception. His theorem became the starting point for many important studies in the theory of complex functions. Picard introduced a new means of proving the existence of solutions to differential equations: the method of successive approximations. His method proved so useful that for ordinary differential equations it replaced the Cauchy-Lipschitz method. He also created a theory of linear differential equations, later extended by his pupil Ernest Vessiot, analogous to the Galois theory of algebraic equations. Inspired by Niels H. Abel of Norway and Bernhard Riemann of Germany, Picard's study of the integrals attached to algebraic surfaces and the related topological questions developed into an important part of algebraic geometry, with varied applications to topology and function theory. His research was expounded in the treatise that he published with Georges Simart, Thorie des fonctions algbriques de deux variables indpendantes, 2 vol. (1897, 1906; Theory of Algebraic Functions of Two Independent Variables). His studies of harmonic vibrations, coupled with the contributions of Hermann Schwarz of Germany and Henri Poincar of France, marked the beginning of the theory of integral equations.