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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