Marius Sophus Lie ( / l iː / LEE ; Norwegian: [liː] ; 17 December 1842 – 18 February 1899) was a Norwegian mathematician. He helped develop the theory of smooth, repeating patterns and used it to study geometry and math problems involving rates of change. He also made important contributions to the study of mathematical symbols and rules.

Life and career

Marius Sophus Lie was born on December 17, 1842, in the small town of Nordfjordeid. He was the youngest of six children born to Johann Herman Lie, a Lutheran pastor, and his wife, who was from a well-known family in Trondheim.

Lie received his primary education on the southeastern coast of Moss before attending high school in Oslo, which was then called Christiania. After finishing high school, his goal to join the military was stopped when the army refused to accept him because of poor eyesight. He then enrolled at the Royal Frederick University.

In 1869, Lie published his first mathematical work, Repräsentation der Imaginären der Plangeometrie (Representations of the Imaginaries in Plane Geometry), in the Academy of Sciences in Christiania and in Crelle's Journal. That same year, he received a scholarship and traveled to Berlin, where he stayed from September 1870 to February 1870. In Berlin, he met Felix Klein and became close friends. After leaving Berlin, Lie traveled to Paris, where Klein joined him two months later. In Paris, they met Camille Jordan and Gaston Darboux. However, on July 19, 1870, the Franco-Prussian War began, and Klein, who was Prussian, had to leave France quickly. Lie was arrested in Fontainebleau and suspected of being a German spy, which made him famous in Norway. He was released after one month, thanks to Darboux's help.

Lie earned his PhD from the Royal Frederick University (now in Oslo) in 1871 with a thesis titled Over en Classe geometriske Transformationer (On a Class of Geometric Transformations). Darboux later called this work "one of the most beautiful discoveries of modern geometry." The following year, the Norwegian Parliament created a special professorship for Lie. That same year, Lie visited Klein, who was working on the Erlangen program in Erlangen.

In 1872, Lie spent eight months with Peter Ludwig Mejdell Sylow, editing and publishing the mathematical works of their countryman, Niels Henrik Abel.

At the end of 1872, Lie proposed to Anna Birch, who was 18 years old at the time. They married in 1874 and had three children: Marie (born 1877), Dagny (born 1880), and Herman (born 1884).

From 1876, Lie co-edited the journal Archiv for Mathematik og Naturvidenskab with the physician Jacob Worm-Müller and the biologist Georg Ossian Sars.

In 1884, Friedrich Engel came to Christiania to help Lie, with support from Klein and Adolph Mayer, who were professors at Leipzig. Engel assisted Lie in writing his most important work, Theorie der Transformationsgruppen, which was published in Leipzig in three volumes from 1888 to 1893. Later, Engel also helped compile Lie's collected works.

In 1886, Lie became a professor at Leipzig, replacing Klein, who had moved to Göttingen. In November 1889, Lie experienced a mental breakdown and was hospitalized until June 1890. He returned to his job but later suffered from anemia, which worsened over time. In May 1898, he resigned and returned to Norway in September of the same year. He died in 1899 at age 56 from pernicious anemia, a disease caused by the body not absorbing enough vitamin B12.

Lie was honored with several titles, including Honorary Member of the London Mathematical Society in 1878, Corresponding Member of the French Academy of Sciences in 1892, Honorary Member of the Manchester Literary and Philosophical Society in 1894, Foreign Member of the Royal Society of London in 1895, and Foreign Associate of the National Academy of Sciences of the United States of America in 1895.

• 1888 copy of Theorie der Transformationsgruppen, volume I
• Title page to Theorie der Transformationsgruppen
• Preface to Theorie der Transformationsgruppen

Legacy

Sophus Lie's most important contribution was discovering that groups of continuous transformations (now called Lie groups) could be better understood by "linearizing" them. This process involves studying the related generating vector fields, which are also called infinitesimal generators. These generators follow a simplified version of the group's rules, now known as the commutator bracket, and form a structure called a Lie algebra.

Hermann Weyl used Lie's work on group theory in his papers from 1922 and 1923. Today, Lie groups are used in quantum mechanics. However, the study of Lie groups now is very different from what Sophus Lie originally researched. Compared to other 19th-century mathematicians, Lie's work is now the least well-known.

Sophus Lie worked hard to create the Abel Prize. Inspired by the Nansen Fund, which honored Fridtjof Nansen, and because the Nobel Prize did not have a mathematics award, he gathered support to create an award for important work in pure mathematics.

Lie mentored many doctoral students who later became successful mathematicians. Élie Cartan became one of the greatest mathematicians of the 20th century. Kazimierz Żorawski's work was found to be important in many areas. Hans Frederick Blichfeldt contributed to several areas of mathematics.

Books

• Lie, Sophus (1888), Theorie der Transformationsgruppen I (in German), published in Leipzig by B. G. Teubner. Co-authored with Friedrich Engel. An English translation exists, edited and translated by Joël Merker, with a foreword. Details: ISBN 978-3-662-46210-2 and arXiv:1003.3202.
• Lie, Sophus (1890), Theorie der Transformationsgruppen II (in German), published in Leipzig by B. G. Teubner. Co-authored with Friedrich Engel.
• Lie, Sophus (1891), Vorlesungen über differentialgleichungen mit bekannten infinitesimalen transformationen (in German), published in Leipzig by B. G. Teubner. Co-authored with Georg Scheffers.
• Lie, Sophus (1893), Vorlesungen über continuierliche Gruppen (in German), published in Leipzig by B. G. Teubner. Co-authored with Georg Scheffers.
• Lie, Sophus (1893), Theorie der Transformationsgruppen III (in German), published in Leipzig by B. G. Teubner. Co-authored with Friedrich Engel.
• Lie, Sophus (1896), Geometrie der Berührungstransformationen (in German), published in Leipzig by B. G. Teubner. Co-authored with Georg Scheffers.
• Lie, Sophus, Engel, Friedrich; Heegaard, Poul (eds.), Gesammelte Abhandlungen, published in Leipzig by Teubner. Seven volumes published between 1922 and 1960.