French mathematician Joseph Liouville was an accomplished teacher and a gifted researcher. His work in mathematical physics influenced the study of electrodynamics, heat flow, and addressed problems in astronomy. In addition, his purely mathematical contributions included the integration of certain algebraic functions, transcendental numbers, and examination of boundary values in differential equations. During his incredibly productive career, he taught almost continuously and published over 400 scientific papers.
Liouville was the son of a captain in Napoleon's army, causing him to live with his uncle until his father's return from the Napoleonic wars. After the wars, Liouville attended school, eventually attending the Collège St. Louis in Paris. It was there that he first studied high-level mathematics and began writing papers (though none were published). In 1825 Liouville entered the Ecole Polytechnique, taking classes from André Ampère (1775-1836), Dominique Arago (1786-1853), and Simeon Poisson (1781-1840).
In 1831 Liouville took his first academic position. For the next several years, he taught an average of 35-40 hours weekly at several different schools in the Paris area. In spite of this, he continued his research, but sometimes at the expense of his students. In 1836, unhappy with the low quality of mathematics journals in France, he founded a journal devoted to pure and applied mathematics, which became known as the Journal de Liouville.
Liouville developed an international reputation through his publications in August Crelle's journal and, based in part on this, he was elected to the Astronomy Section of the Acadèmie des Sciences in 1839. The following year, he was elected to the Bureau des Longitudes to fill a vacancy left by the death of Poisson. This was a significant event because it secured Liouville's career, allowing him to help other mathematicians with theirs. For the next 20 years he worked extensively toward helping younger mathematicians develop and promote their own ideas and work.
During this time, Liouville suffered a painful indignity when he was passed over for a chair in the Collège de France. He was so upset at the appointment of a professional rival over him that he immediately resigned. Expressing his outrage, he stated, "I am profoundly humiliated as a person and as a geometer by the events that took place yesterday at the Collége de France. From this moment on it is impossible for me to lecture at this institution."
Liouville, encouraged by Arago, also made a successful bid for political office. However, his failure to win reelection seemed to have a profound negative impact on him. Prior to this defeat, his friends noted that he never failed to stand up for his beliefs, while afterwards he simply became bitter and was beset by melancholic thoughts that interrupted even his mathematical notes and research.
Liouville reached his most productive years in 1856 and 1857, prior to becoming bogged down with excessive teaching duties again. After this time, his papers, while still numerous, were less detailed and not as polished, often leaving proofs to others rather than completing them himself.
Liouville's most significant contribution was developed in the 1830s, when he and Jacques Sturm (1803-1855) worked on boundary value problems in differential equations. By helping to solve these problems, Liouville contributed immeasurably to mathematical physics and to the solution of integral equations. His work on some aspects of Hamiltonian dynamics produced results that were—and continue to be—of major importance in statistical mechanics.
In spite of his personal and professional disappointments, Liouville won esteem during his life and remains highly regarded as a mathematician. Any single one of his major contributions would have been sufficient to secure his reputation as a mathematician of the first rank; his ability to contribute so much in so many areas marks him indelibly as a genius.
P. ANDREW KARAM