Henri Poincaré can be said to have been the originator of algebraic topology and of the theory of analytic

functions of several complex variables.

Poincaré entered the École Polytechnique in 1873 and continued his studies at the École des Mines, as a student of Charles

Hermite, from which he received his doctorate in mathematics in 1879. He was appointed to a chair of mathematical physics at

the Sorbonne in 1881, a position he held until his death.

Before the age of 30 he developed the concept of automorphic functions which he used to solve second order linear differential

equations with algebraic coefficients. His Analysis situs , published in 1895, is an early systematic treatment of topology.

Poincaré can be said to have been the originator of algebraic topology and of the theory of analytic functions of several

complex variables. He also worked in algebraic geometry and made a major contribution to number theory with work on

Diophantine equations.

In applied mathematics he studied optics, electricity, telegraphy, capillarity, elasticity, thermodynamics, potential theory,

quantum theory, theory of relativity and cosmology.

He is often described as the last universalist in mathematics.

In the field of celestial mechanics he studied the three-body-problem, and theories of light and electromagnetic waves. He is

acknowledged as a co-discoverer, with Albert Einstein and Hendrik Lorentz, of the special theory of relativity.

His major works include Les Méthods nouvelle de la méchanique celeste in three volumes published between 1892 and

1899 and Lecons de mecanique celeste (1905). In the first of these he aimed to completely characterise all motions of

mechanical systems. He invoked an analogy with fluid flow. He also showed that previous series expansions used in studying

the 3-body problem were convergent, but not in general uniformly convergent, so putting in doubt the stability proofs of

Lagrange and Laplace.

He also wrote many popular scientific articles including Science and Hypothesis (1901), Science and Method (1908), and

The Value of Science (1904). A quote from Poincaré is particularly relevant to this collection on the history of mathematics. In

1908 he wrote

*The true method of foreseeing the future of mathematics is to
study its history and its actual state. *

The Poincaré conjecture is as one of the most baffling and challenging unsolved problems in algebraic topology. Homotopy

theory reduces topological questions to algebra by associating with topological spaces various groups which are algebraic

invariants. Poincaré introduced the fundamental group to distinguish different categories of two-dimensional surfaces. He was

able to show that any 2-dimensional surface having the same fundamental group as the two-dimensional surface of a sphere is

topologically equivalent to a sphere. He conjectured that the result held for 3-dimensional manifolds and this was later extended

to higher dimensions.

Surprisingly proofs are known for the equivalent of Poincaré's conjecture for all dimensions strictly greater than 3. No complete

classification scheme for 3-manifolds is known so there is no list of possible manifolds that can be checked to verify that they all

have different homotopy groups.

Poincaré was first to consider the possibility of chaos in a deterministic system, in his work on planetary orbits. Little interest

was shown in this work until the modern study of chaotic dynamics began in 1963.