Georges de Rham : biography
Georges de Rham ( 10 September 1903 – 9 October 1990) was a Swiss mathematician, known for his contributions to differential topology.
He studied at the University of Lausanne and then in Paris for a doctorate, becoming a lecturer in Lausanne in 1931; where he held positions until retirement in 1971; he held positions in Geneva in parallel.
In 1931 he proved de Rham’s theorem, identifying the de Rham cohomology groups as topological invariants. This proof can be considered as sought-after, since the result was implicit in the points of view of Henri Poincaré and Élie Cartan. The first proof of the general Stokes’ theorem, for example, is attributed to Poincaré, in 1899. At the time there was no cohomology theory, one could reasonably say: for manifolds the homology theory was known to be self-dual with the switch of dimension to codimension (that is, from Hk to Hn-k, where n is the dimension). That is true, anyway, for orientable manifolds, an orientation being in differential form terms an n-form that is never zero (and two being equivalent if related by a positive scalar field). The duality can be reformulated, to great advantage, in terms of the Hodge dual—intuitively, ‘divide into’ an orientation form—as it was in the years succeeding the theorem. Separating out the homological and differential form sides allowed the coexistence of ‘integrand’ and ‘domains of integration’, as cochains and chains, with clarity. De Rham himself developed a theory of homological currents, that showed how this fitted with the generalised function concept.
The influence of de Rham’s theorem was particularly great during the development of Hodge theory and sheaf theory.
De Rham also worked on the torsion invariants of smooth manifolds.