A. Marius Sophus Lie: A Mathematician’s Life

Born: Nordfjordeide, Norway, 17 December 1842
Died: Christiania (Oslo), Norway, 18 February 1899.

Sophus Lie: Derived from a painting by Erik Werenskiold (Engel and Heegaard Vol. 2 [1912]).

Marius Sophus Lie  was the youngest of six children of a Lutheran pastor, Johann Herman Lie.

Lie first attended school in Moss (Kristianiaford); then, from 1857 to 1859, he attended Nissen's Private Latin School in Christiania. Lie mastered the classes without any difficulties.

He studied at Christiania University from 1859 to 1865, mainly mathematics and sciences.

During his studies at Christiania, he had no preference for mathematics.

After his examination in 1865, he gave private lessons, became somewhat interested in astronomy, and tried to learn mechanics; but he could not decide what to do.

Lie himself said that the road to mathematics for him was long and difficult.

The situation changed when, in 1868, he hit upon Poncelet's and Plücker's writings. Later, he called himself a student of Plücker.

Lie’s first publication brought him a scholarship for studying abroad.

In 1869 Lie went to Berlin, where he met Felix Klein, with whom he later cooperated in publishing several papers. He spent the winter of 1869-1870 in Berlin where he met Kummer and Weierstraß.

In the summer of 1870, Lie and, later, Klein traveled to Paris via Göttingen to meet Darboux and Jordan.

Jordan acquainted Lie and Klein with the notion of a group introduced into algebra by Galois in 1832.

In 1870, Lie discovered contact transformations.

At the outbreak of the Franco-Prussian war in July of 1870, Klein left Paris; Lie, a Norwegian, stayed. In August, he decided to hike to Italy, but on his way he was arrested as a German spy near Fontainebleau. His mathematical notes were suspected to be military secrets in code—a letter from Klein seemed suspicious. After being locked in prison for a month, he was freed through Darboux's intervention. Just before the Germans blockaded Paris, he escaped to Italy. From there, he returned to Germany, where he again met Klein in Düsseldorf.

In 1871, he became an assistant tutor at the University of Cristiania (Oslo). In the same year, he submitted for his doctor’s degree a memoir in which he advanced the theory of tangential transformations.

During the period 1871–1872, he developed the integration theory of partial differential equations, now found in many textbooks, although rarely under his name.

Appointed extraordinary professor on 1 July 1872, he began his researches on continuous transformation groups in 1873.

In 1874, Lie married Anna Birch from Tvedestrand. They had two daughters and a son. The marriage was very happy, and Lie was very fond of his family.

In 1873, Lie turned from the invariants of contact transformations to the principles of the theory of transformation groups. Together with Sylow, he assumed the editorship of Niels Abel's work.

Lie was quite isolated in Christiania at that time. He had no students who were interested in his research. He was disappointed that his works did not receive more attention abroad. Except for Klein, Mayer, and later, Picard, nobody paid attention to his work. Lie’s results on the integration theory of partial differential equations were found by Adolph Mayer at that time, with whom he had conducted a lively correspondence. In a letter to Mayer he writes, “If I only knew how to get the mathematicians interested in transformation groups and their applications to differential equations. I am certain, absolutely certain in my case, that in the future these theories will be recognized as fundamental. I want to form thus an impression now, since for one thing, I could then achieve ten times as much.” His main interest turned to transformation groups, his most celebrated creation; although in 1876, he returned to differential geometry. In the same year, he joined G.O. Sars and Worm Müller in founding the Archiv för mathematik og naturvidenskab.

In 1884, Klein and Mayer induced Engel, who had just received his Ph.D., to visit Lie in order to learn about transformation groups and to help him write a comprehensive book on the subject.

Engel stayed 9 months with Lie. Thanks to Engel’s activity, the work was accomplished, its three parts being published between 1888 and 1893. Engel and Lie developed a warm and lifelong friendship. Engel helped Lie by giving his rather intuitive geometrical ideas a more precise mathematical form. Lie often felt it a burden to prepare his ideas for publication. After 9 years of collaboration with Engel, Lie published , 3 vols. (1893). This work contains the results of his investigations of the general theory of finite continuous groups of transformations. It was followed by Geometrie der Berührungstransformationen (1896).

In 1886, he succeeded Klein on the chair of mathematics at the University of Leipzig, with Engel as his assistant. At Leipzig, he found interested students, among them Scheffers, Zorawski, and Kowalewski.

With Scheffers, Lie published textbooks on transformation groups and on differential equations, and a fragmentary geometry of contact transformations. Afterward, his student Kowalewski, wrote many books about Lie’s work. At this time, it was quite unusual for young French mathematicians to go to Germany for studying. But the Ecole Normale Supérieure in Paris sent some of their best students to Lie; and he was very proud of this. Lie did not plan to stay in Leipzig forever; he had in mind a period of 6 to 8 years. So he did not resign from his professorship in Christiania, but was granted an extraordinary leave of absence.

Life in Leipzig was not that easy for Lie. His teaching duties were much heavier than at home, the language caused him some problems, and he became tired of supervising weak and dependent graduate students. As time passed, he also ran into trouble with some of his colleagues. In the last years of his life, Lie turned to foundations of geometry, which at that time meant the Helmholtz space problem.

In 1898, he returned to Cristiania to accept a special chair of mathematics created for him, but his health was already broken. Lie, who was described as on open-hearted man of gigantic stature and excellent physical health, was struck by what was then called neurasthenia. This was the result of his rushing work and the overload of mental action. Treatment in a mental hospital led to his recovery, and in 1890, he could resume his work. His character, however, had changed greatly. He became increasingly sensitive, irascible, suspicious, and misanthropic, despite the many tokens of recognition that were heaped upon him. He died of pernicious anemia in February 1899. His papers were edited, with excellent annotations, by Engel and Heegaard.

An analysis of Lie’s work is given in the (1900). His collected works are contained in Gesammelte Abhandlungen, 7 vols (1922–37). Two other standard works are his Differentialgleichungen (1891) and Vorlesungen über continuierliche Gruppen (1893).

In 1890, Lie himself wrote on his work in a letter to his friend Motzfeldt, “…my life’s work will stand through all times and, in the years to come, be more and more appreciated—no doubt about it.” It seems he was absolutely right!

Created by Mathematica  (September 15, 2003)