**Henri Poincaré** can be said to have been the originator of algebraic topology and of the theory of analytic functions of several complex variables.

- They were 26 and 24 years of age, respectively, at the time of Henri's birth.
- The second of Antoine Poincaré's sons, Lucien Poincaré, achieved high rank in university administration.
- In 1862 Henri entered the Lycée in Nancy (now renamed the Lycée Henri Poincaré in his honour).
- Henri was described by his mathematics teacher as a "monster of mathematics" and he won first prizes in the concours général, a competition between the top pupils from all the Lycées across France.
- Poincaré entered the École Polytechnique in 1873, graduating in 1875.
- Poincaré read widely, beginning with popular science writings and progressing to more advanced texts.
- After graduating from the École Polytechnique, Poincaré continued his studies at the École des Mines.
- After completing his studies at the École des Mines Poincaré spent a short while as a mining engineer at Vesoul while completing his doctoral work.
- As a student of Charles Hermite, Poincaré received his doctorate in mathematics from the University of Paris in 1879.
- Nevertheless, considering the great difficulty of the subject and the talent demonstrated, the faculty recommends that M Poincaré be granted the degree of Doctor with all privileges.
- Immediately after receiving his doctorate, Poincaré was appointed to teach mathematical analysis at the University of Caen.
- In 1886 Poincaré was nominated for the chair of mathematical physics and probability at the Sorbonne.
- The intervention and the support of Hermite was to ensure that Poincaré was appointed to the chair and he also was appointed to a chair at the École Polytechnique.
- Poincaré held these chairs in Paris until his death at the early age of 58.
- Before looking briefly at the many contributions that Poincaré made to mathematics and to other sciences, we should say a little about his way of thinking and working.
- One is a lecture which Poincaré gave to l'Institute Général Psychologique in Paris in 1908 entitled Mathematical invention in which he looked at his own thought processes which led to his major mathematical discoveries.
- Although published in 1910 the book recounts conversations with Poincaré and tests on him which Toulouse carried out in 1897.
- An interesting aspect of Poincaré's work is that he tended to develop his results from first principles.
- This was not the way that Poincaré worked and not only his research, but also his lectures and books, were all developed carefully from basics.
- If beginning is painful, Poincaré does not persist but abandons the work.
- For this reason Poincaré never does any important work in the evening in order not to trouble his sleep.
- Let us examine some of the discoveries that Poincaré made with this method of working.
- Poincaré was a scientist preoccupied by many aspects of mathematics, physics and philosophy, and he is often described as the last universalist in mathematics.
- His results applied only to restricted classes of functions and Poincaré wanted to generalise these results but, as a route towards this, he looked for a class functions where solutions did not exist.
- In a correspondence between Klein and Poincaré many deep ideas were exchanged and the development of the theory of automorphic functions greatly benefited.
- However, the two great mathematicians did not remain on good terms, Klein seeming to become upset by Poincaré's high opinions of Fuchs's work.
- Poincaré's Analysis situs Ⓣ(Analysis of situation), published in 1895, is an early systematic treatment of topology.
- For 40 years after Poincaré published the first of his six papers on algebraic topology in 1894, essentially all of the ideas and techniques in the subject were based on his work.
- The Poincaré conjecture remained as one of the most baffling and challenging unsolved problems in algebraic topology until it was settled by Grisha Perelman in 2002.
- Poincaré introduced the fundamental group (or first homotopy group) in his paper of 1894 to distinguish different categories of 2-dimensional surfaces.
- Surprisingly proofs are known for the equivalent of Poincaré's conjecture for all dimensions strictly greater than three.
- Poincaré is also considered the originator of the theory of analytic functions of several complex variables.
- We should describe in a little more detail Poincaré's important work on the 3-body problem.
- Poincaré was awarded the prize for a memoir he submitted on the 3-body problem in celestial mechanics.
- In this memoir Poincaré gave the first description of homoclinic points, gave the first mathematical description of chaotic motion, and was the first to make major use of the idea of invariant integrals.
- Poincaré realised that indeed he had made an error and Mittag-Leffler made strenuous efforts to prevent the publication of the incorrect version of the memoir.
- Between March 1887 and July 1890 Poincaré and Mittag-Leffler exchanged fifty letters mainly relating to the Birthday Competition, the first of these by Poincaré telling Mittag-Leffler that he intended to submit an entry, and of course the later of the 50 letters discuss the problem concerning the error.
- A revised version of Poincaré's memoir appeared in 1890.
- Poincaré's other major works on celestial mechanics include Les Méthodes nouvelles de la mécanique céleste Ⓣ(New methods of celestial mechanics) in three volumes published between 1892 and 1899 and Leçons de mecanique céleste Ⓣ(Lessons on celestial mechanics) (1905).
- Poincaré's popular works include Science and Hypothesis (1901), The Value of Science (1905), and Science and Method (1908).
- Finally we look at Poincaré's contributions to the philosophy of mathematics and science.
- The first point to make is the way that Poincaré saw logic and intuition as playing a part in mathematical discovery.
- However intuition for Poincaré was not something he used when he could not find a logical proof.
- Poincaré believed that formal proof alone cannot lead to knowledge.
- It is reasonable to ask what Poincaré meant by "intuition".
- We should not give the impression that the review was negative, however, for Poincaré was very positive about this work by Hilbert.
- Poincaré believed that one could choose either euclidean or non-euclidean geometry as the geometry of physical space.
- Poincaré was absolutely correct, however, in his criticism that those like Russell who wished to axiomatise mathematics; they were doomed to failure.
- The principle of mathematical induction, claimed Poincaré, cannot be logically deduced.
- These claims of Poincaré were eventually shown to be correct.
- We should note that, despite his great influence on the mathematics of his time, Poincaré never founded his own school since he did not have any students.
- Poincaré achieved the highest honours for his contributions of true genius.

Born 29 April 1854, Nancy, Lorraine, France. Died 17 July 1912, Paris, France.

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Algebra, Analysis, Astronomy, Geometry, Group Theory, Physics, Topology

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive