2.10. Historical interlude: Sophus Lie 27 Lie and Klein traveled to Paris together, and there Lie produced the famous contact transformation, which mapped straight lines into spheres. An application of this expertise to the Earth sphere, however, did not serve him well. After the outbreak of the Franco-Prussian war, Lie could not find a better way to return to Norway than by first hiking to Italy. With his peculiar hiking habits, such as taking off his clothes in the rain and putting them into his backpack, he was not able to flee very far. The French quickly apprehended him and found papers filled with mysterious symbols. Lie’s efforts to explain the meaning of his mathematical notation did not dispel the authorities’ suspicion that he was a German spy. A short stay in prison afforded him some quiet time to complete his studies, and upon return to Norway, Lie successfully defended his doctoral dissertation. Unable to find a job in Norway, Lie resolved to go to Sweden, but Norwegian patriots intervened, and the Norwegian National Assembly voted by a large majority to establish a personal extraordinary professorship for Lie at the University of Christiania. Although the salary offered was less than extraordinary, he stayed. Lie’s research on sphere mapping and his lively exchanges with Klein led both of them to think of more general connections between group theory and geometry. In 1872 Klein presented his famous Er- langen Program, in which he suggested unifying specific geometries under a general framework of projective geometry and using group theory to organize all geometric knowledge. Lie and Klein clearly ar- ticulated the notion of a transformation group, the continuous analog of a permutation group, with promising applications to geometry and differential equations, but they lacked a general theory of the sub- ject. The Erlangen Program implied one aspect of this project — the group classification problem — but Lie had no intention of attacking this bastion at the time. As he later wrote to Klein, “[I]n your essay the problem of determining all groups is not posited, probably on the grounds that at the time such a problem seemed to you absurd or impossible, as it did to me” [22, pp. 41–42]. By the end of 1873, Lie’s pessimism gave way to a much brighter outlook. After dipping into the theory of first order differential equa- tions, developed by Jacobi and his followers, and making considerable

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