Fermat's Last Theorem: Joseph Liouville

Saturday, September 26, 2009

Joseph Liouville

Joseph Liouville was born on March 24, 1809 in Saint-Omer, France. His father was an officer in Napoleon's army so for his earliest years, he lived with his uncle. Only when Napoleon was defeated did his father return and the family live together in Toul, France.

In Paris, Liouville studied mathematics at the College St. Louis. Already at this time, he showed interest and talent in advanced mathematical topics.

In 1825, when he was 16, he entered the Ecole Polytechnique. There, he attended lectures by Andre Marie Ampere and Dominique Francois Jean Arago. While he did not attend any lectures by Augustin Louis Cauchy, he was greatly influenced by him. Among his examiners when he graduated were Gaspard de Pony and Simeon Denis Poisson.

He graduated in 1827 and entered the Ecole des Ponts et Chaussees with the intention of becoming an engineer. Engineering projects in those days were physically demanding and Liouville found his health severely affected. He took some time off by returning to Toul. He got married to Marie-Louise Balland and decided to resign from the Ecole des Ponts et Chaussees which he did in 1830.

In 1831, he accepted an academic position at the Ecole Polytechnique. He was assistant to Claude Louis Mathieu and the role carried with it responsibilities of 35-40 hours of lectures per week. In doing this, Liouville developed a reputation for focusing on advanced topics and for being difficult to follow.

Many times, Liouville attempted to improve his position at the Ecole Polytechnique without success. He was also frustrated by the quality of math journals in France at the time. In 1836, he started his own math journal, Journal de Mathematiques Pures et Appliquees which was also known as the Journal de Liouville. By this time, he had developed an international reputation based on his papers which he published in Crelle's Journal. The journal was a success and published many significant papers by French mathematicians.

By 1838, he became Professor of Analysis and Mechanics at the Ecole Polytechnique. Many honors followed. In 1840, he was elected to the Academie des Sciences in Astronomy and he was also elected to the Bureau des Longitudes.

Liouville had become close friends with Arago who become the head of the Republican Party in France. Liouville was encouraged to run for office. In 1848, Liouville was elected to the Constituting Assembly. Unfortunately, his political career did not last long for he was voted out of office the following year. This had a big impact on his spirits as noted by one of his biographers:

The political defeat changed Liouville's personality. In earlier letters, he was often depressed because of illness, and could vent his anger towards his enemies such as Libri, but he always fought for what he believed was right. After the election in 1849, he resigned and became bitter, even towards his old friends. When he sat down at his desk, he did not only work, ... he also pondered his ill fate. ... his mathematical notes were interrupted with quotes from poets and philosophers...

In 1850, the mathematical chair at the College de France opened. Up for consideration were Liouville and Cauchy. After a heated contest, the position went to Liouville in 1851.

Liouville's mathematical output was phenomenal writing over 400 mathematical papers. Over 200 of papers were on number theory. Other papers covered a range of topics including mathematical physics, astronomy, as well as pure mathematics.

He introduced the fractional calculus as part of his analysis of electromagnetism. He was also to the first to prove the existence of transcendental numbers, numbers that is not algebraic (that is, it cannot be a solution to an equation of a nonconstant polynomial with rational coefficients). He did very significant work on the boundary value problems with differential equations in what is today called Sturm-Liouville Theory. He did important work in statistical mechanics and measure theory.

Perhaps, his most important impact in mathematics was his discovery of the memoir by Evariste Galois. In 1843, he announced to the Paris Academy that he had discovered very brilliant insights by Galois. Galois's memoir was then published in 1846 which would introduce group theory and place Galois among the most celebrated mathematicians in the history of the subject.

Liouville died on September 8, 1882. Many historians consider him the greatest mathematician of his day. The Liouville Crater on the Moon is named in his honor.

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