Although Henri Poincaré (1854-1912) died before the outbreak of the First World War, he founded subjects of intense interest to present-day mathematicians. Poincaré was a pioneer in hyperbolic geometry, which in the 1970's and 1980's became important in the study of 3-manifolds. He originated many of the central concepts of algebraic topology, a subject which only came to full flower in the mid-twentieth century. He invented qualitative methods for the study of solutions of differential equations, and applied them to the motions of the planets. His work in this area prefigured another subject of much current interest, dynamical systems and chaos. In addition he did work of considerable importance in mathematical physics, and could be considered a co-discoverer (with Einstein and Lorenz) of the special theory of relativity.
Poincaré formulated an important problem that motivated a great deal of work in twentieth-century topology. The Poincaré conjecture asserts that every simply connected closed 3-manifold is homeomorphic to the 3-sphere. The conjecture was later generalized to higher dimensions: the generalized Poincaré conjecture asserts that any n-dimensional manifold homotopy equivalent to the n-sphere is homeomorphic to it. Ironically the higher-dimensional versions proved easier to tackle. Stephen Smale proved the generalized Poincaré conjecture for n ≥ 5 in 1961, and Michael Freedman disposed of the case n = 4 in 1982. But the original case n = 3 remained open until the twenty-first century, when Grigori Perelman implemented a program of Richard Hamilton to prove the conjecture in a series of papers that appeared in 2002-3. (His papers were never submitted to a formal journal, but the proof was verified to the satisfaction of the mathematical community in 2006.)
Poincaré was born in 1854 in Nancy, France. As a a child he had poor muscular coordination and suffered a bout of diphtheria. He soon showed a gift for written composition. In 1873-76 he attended the École Polytéchnique in Paris, where he won top honors in mathematics. He continued his studies at the École National Supérieure des Mines as a student of Hermite, and received a doctorate from the University of Paris in 1879 after writing a thesis (under the supervision of Darboux) on differential equations. His first appointment was at the University of Caen, but in 1881 he joined the University of Paris, where he spent the rest of his career. He wrote prolifically (though not as prolifically as Euler or Erdös), publishing almost 500 papers.
Poincaré made his reputation in the theory of automorphic functions, which are functions invariant under linear fractional transformations. He recognized the relationship of this problem to non-Euclidean (or more specifically hyperbolic) geometry. Later he made the first major use of geometric (or "qualitative") methods in the investigation of solutions of differential equations when he tackled the n-body problem. The problem, which was the subject of an international competition sponsored by King Oscar II of Sweden, was to determine the stability of the Solar System (assuming Newton's law of gravitation). Poincaré's essay won the prize in 1889, even though he only partially solved the problem; what Poincaré found was that (to use modern terminology) mathematical chaos was lurking in Newton's equations for three or more bodies. Poincaré summarized the results of his investigation in his three-volume work Les Méthodes nouvelles de la mécanique céleste, published from 1892-99.
Poincaré's use of qualitative methods led him into the study of "analysis situs", or what is now called topology. Beginning in 1892, he began a systematic development of this subject. He originated many of the key concepts of algebraic topology, including the fundamental group and the basic ideas of homology theory, in a series of papers that culminated with his statement in 1904 of the Poincaré conjecture.