Jules Henri Poincare

Before looking briefly at the many contributions that Poincare made to mathematics and to other sciences, we should say a little about his way of thinking and working. He is considered as one of the great geniuses of all time and there are two very significant sources which study his thought processes. One is a lecture which Poincare gave to l’Institut General Psychologique in Paris in 1908 entitled “Mathematical invention” in which he looked at his own thought processes which led to his major mathematical discoveries. The other is the book by Toulouse who was the director of the Psychology Laboratory of l’Ecole des Hautes Etudes in Paris.

Toulouse explains that Poincare kept very precise working hours. He undertook mathematical research for four hours a day, between 10 am and noon, then again from 5 pm to 7 pm. He would read articles in journals later in the evening. An interesting aspect of Poincare’s work is that he tended to develop his results from first principles. For many mathematicians there is a building process with more and more being built on top of the previous work. This was not the way that Poincare worked and not only his research, but also his lectures and books, were all developed carefully from basics.

He worked in algebraic geometry making fundamental contributions in papers written in 1910-1911. He examined algebraic curves on an algebraic surface F(x, y, z) = 0 and developed methods which enabled him to give easy proofs of deep results due to Emile Picard and Severi. He gave the first correct proof of a result stated by Castelnuovo, Enriques and Severi, these authors having suggested a false method of proof. His first major contribution to number theory was made in 1901 with work: “...on the Diophantine problem of finding the points with rational coordinates on a curve f(x,y) = 0, where the coefficient of f is rational numbers”.

In applied mathematics he studied optics, electricity, telegraphy, capillarity, elasticity, thermodynamics, potential theory, quantum theory, theory of relativity and cosmology. In the field of celestial mechanics he studied the three-body-problem, and the theories of light and of electromagnetic waves. He is acknowledged as a co-discoverer, with Albert Einstein and Hendrick Lorentz, of the special theory of relativity. We should note that, despite his great influence on the mathematics of his time, Poincare never founded his own school since he did not have any students. Although his contemporaries used his results, they seldom adopted his techniques.

Poincare achieved the highest honours for his contributions of true genius. He was elected to the Academie des Sciences in 1887 and in 1906 was elected President of the Academy. Due to his research Poincare was the only member elected to every one of the five sections of the Academy, namely the geometry, mechanics, physics, geography and navigation sections.

(Adapted from the Internet sites)

6.3 Read the text to learn about the outstanding Russian mathematician A.Kolmogorov.

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