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Sophus Lie made major advances in the theory of continuous groups of transformations and differential equations. Lie groups and Lie algebras are named after him.
Lie was taught mathematics at school by Sylow and then attended Sylow's lectures on group theory at the University of Christiania from where he graduated in 1865 (not gaining a distinction). There followed a few years when he could not decide what career to follow.
A turning point came in 1868 when he read papers on geometry by Poncelet and Plücker. In 1869 Lie went to Berlin where he met Felix Klein. They met again in Paris and Lie started to work on transformation groups. He was to collaborate later with Klein in publishing several papers. This joint work had as one of its outcomes Klein's characterisation of geometry (1872) as properties invariant under a group action. While in Paris Lie discovered contact transformations. These transformations allowed a 1-1 correspondence between lines and spheres in such a way that tangent spheres correspond to intersecting lines.
Because of the French-German war of 1870 both Klein and Lie left France, Lie deciding to go to Italy. On the way however he was arrested as a German spy and his mathematics notes were assumed to be coded messages. Only after the intervention of Darboux was Lie released and he decided to return to Christiania. In 1871 Lie became an assistant at Christiania (which became Kristiania then Oslo in 1925) and obtained his doctorate.
Lie had started examining partial differential equations, hoping that he could find a theory which was analogous to Galois' theory of equations. He examined his contact transformations considering how they affected a process due to Jacobi of generating further solutions from a given one. This led to combining the transformations in a way that Lie called a group, but which is not a group with our definition, rather what is today called a Lie algebra.
At this point he left his original intention of examining partial differential equations and examined Lie algebras. Killing was to examine Lie algebras quite independently of Lie, and Cartan was to publish the classification of semisimple Lie algebras in 1900.
Lie collaborated for nine years with Engel after which Lie and Engel jointly published Theorie der Transformationsgruppen in three volumes in 1893. This was Lie's major work on continuous groups of transformations. Engel was a student of Klein's sent by him to study under Lie.
In 1886 Lie succeeded Klein in the chair of mathematics at Leipzig with Engel as his assistant. In 1892 the lifelong friendship between Lie and Klein broke down and the following year Lie publically attacked Klein saying
I am no pupil of Klein, nor is the opposite the case, although this might be closer to the truth.
Lie returned to Kristiania in 1898 to take up a post specially created for him but his health was already deteriorating and he died soon after taking up the post.
Lie's other contributions were to differential geometry although his main aim in mathematical research was always to further the theory of differential equations.
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