Lie, (Marius) Sophus (1842–1899)

Norwegian mathematician who provided the foundations for the science of topology in transformation groups known as the Lie groups. He was one of the first mathematicians to emphasize the importance of the notion of groups in geometry.

Lie was born near Bergen and studied at Christiania (now Oslo) and abroad at Berlin and Paris. He was professor at Christiania 1873–86 and 1898–99 and at Leipzig, Germany, 1886–98.

Lie's first great discovery, made in 1870, was that of his contact transformation, which mapped straight lines with spheres and principal tangent curves into curvature lines. In his theory of tangential transformations occurs the particular transformation that makes a sphere correspond to a straight line. By 1873 Lie had begun to investigate transformation groups. In this work on group theory he chose a new space element, the contact element, which is an incidence pair of point and line or of point and hyperplane. This led him to the discovery of Lie groups, one of the basic notions of which is that of infinitesimal transformation.

The Lie integration theorem, which he developed, made it possible to classify partial differential equations in such a way as to make most of the classical methods of solving such equations reducible to a single principle.