Tuesday, April 29, 2014

Henri Poincaré - the Last Universalist of Mathematics

Henri Poincaré
(1854 – 1912)
On April 29, 1854, French mathematician, theoretical physicist, engineer, and a philosopher of science Henri Poincaré was born. He is often described as a polymath, and in mathematics as The Last Universalist since he excelled in all fields of the discipline as it existed during his lifetime.

Jules Henri Poincaré was born near France and excelled in every class from the very beginning. It is assumed that his mother had a conversation with his teacher, when he was about 13 years old. The teacher told her that "Henri will become a mathematician … I would say a great mathematician". However, when Poincare graduated in 1871 and only received the grade 'fair' in science. In mathematics, Poincare received zero points, it is assumed that he answered the wrong questions. Two years later, Poincare enrolled at the École Polytechnique and again, he excelled in every subject, but graduated only as the second in class due to his inability to draw. Poincare submitted a dissertation on partial differential equations and he was then put in charge of the course on differential and integral calculus at the University of Caen.

In 1880, the mathematician made use of non-Euclidean geometry for the first time and resolved a problem in the theory of differential equations to the competition for the grand prize in mathematics of the Academy of Sciences in Paris. He was put on the faculty of sciences at the University of Paris and later on, he succeeded G. Lippmann in the chair of mathematical physics and probability. Poincare switched institutes and universities a lot in the next years, and in 1904, he became professor of general astronomy at the École Polytechnique.

Poincare managed to make significant contributions to classical mechanics and even more important, he was able to publish a founding document in chaos theory. Poincare showed that general, the stability of n-body systems (like the solar system) cannot be demonstrated. In this context, he also proved his recurrence theorem. When working on the foundations of topology, Poincare became increasingly interested in what topological properties characterized a sphere. In 1900, he claimed that homology was sufficient to tell if a 3-manifold was a 3-sphere and four years later, he described a counterexample to this claim, a space now called the Poincaré homology sphere. The Poincaré sphere was the first example of a homology sphere Poincare now had to establish that the Poincaré sphere was different from the 3-sphere and he introduced a new topological invariant, the fundamental group. He was able to show, that the Poincaré sphere had a fundamental group of order 120, while the 3-sphere had a trivial fundamental group. In this way he was able to conclude that these two spaces were, indeed, different. Pointcare also wondered whether a 3-manifold with the homology of a 3-sphere and also trivial fundamental group had to be a 3-sphere. In November 2002, Russian mathematician Grigori Perelman published his outline of a solution of the Poincaré conjecture and four years later Perelman was awarded, but declined, the Fields Medal for his proof. In 2010, the Clay Mathematics Institute awarded Perelman the $1 million Millennium Prize in recognition of his proof, which he rejected as well.

Poincaré became the president of the French Academy in 1906 and was elected to the Académie Française in 1908. During his lifetime, Henri Poincaré published over five hundred scientific papers and over thirty books. He passed away on July 17, 1912 in Paris.

At yovisto, you may be interested in an entertaining video explaining the Poincare Conjecture and its importance for science.



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