Georges de Rham

Georges de Rham (French: [dəʁam]; 10 September 1903 – 9 October 1990) was a Swiss mathematician, known for his contributions to differential topology.

Georges de Rham was born on 10 September 1903 in Roche, a small village in the canton of Vaud in Switzerland. He was the fifth born of the six children in the family of Léon de Rham, a constructions engineer. Georges de Rham grew up in Roche but went to school in nearby Aigle, the main town of the district, travelling daily by train. By his own account, he was not an extraordinary student in school, where he mainly enjoyed painting and dreamed of becoming a painter. In 1919 he moved with his family to Lausanne in a rented apartment in Beaulieu Castle where he would live for the rest of his life. Georges de Rham started the Gymnasium in Lausanne with a focus on humanities, following his passion for literature and philosophy but learning little mathematics. On graduating from the Gymnasium in 1921 however, he decided not to continue with the Faculty of Letters in order to avoid Latin. He opted instead for the Faculty of Sciences of the University of Lausanne. At the faculty he started out studying biology, physics and chemistry and no mathematics initially. While trying to learn some mathematics by himself as a tool for physics, his interest was raised and by the third year he abandoned biology to focus decisively on mathematics.

At the University he was mainly influenced by two professors, Gustave Dumas and Dmitry Mirimanoff, who guided him in studying the works of Émile Borel, René-Louis Baire, Henri Lebesgue and Joseph Serret. After graduating in 1925, de Rham remained at the University of Lausanne as an assistant to Dumas. Starting work towards completing his doctorate, he read the works of Henri Poincaré on topology on the advice of Dumas. Although he found inspiration for a thesis subject in Poincaré, progress was slow as topology was a relatively new topic and access to the relevant literature was difficult in Lausanne. With the recommendation of Dumas, de Rham contacted Lebesgue and went to Paris for a few months in 1926 and, again, for a few months in 1928. Both trips were financed by his own savings and he spent his time in Paris taking classes and studying at the University of Paris and the Collège de France. Lebesgue provided de Rham with a lot of help in this period, both with his studies and supporting his first research publications. When he finished his thesis Lebesgue advised him to send it to Élie Cartan and, in 1931, De Rham received his doctorate from the University of Paris before a commission led by Cartan and including Paul Montel and Gaston Julia as examiners.

In 1932 de Rham returned to the University of Lausanne as an extraordinary professor. In 1936 he also became a professor at the University of Geneva and continued to hold both positions in parallel until his retirement in 1971.

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 H_k to H_n-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.