Georges de Rham
Encyclopedia
Georges de Rham 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 manifold
s 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 to great advantage be reformulated 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.
Switzerland
Switzerland name of one of the Swiss cantons. ; ; ; or ), in its full name the Swiss Confederation , is a federal republic consisting of 26 cantons, with Bern as the seat of the federal authorities. The country is situated in Western Europe,Or Central Europe depending on the definition....
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
, known for his contributions to differential topology
Differential topology
In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.- Description :...
.
He studied at the University of Lausanne
University of Lausanne
The University of Lausanne in Lausanne, Switzerland was founded in 1537 as a school of theology, before being made a university in 1890. Today about 12,000 students and 2200 researchers study and work at the university...
and then in Paris
University of Paris
The University of Paris was a university located in Paris, France and one of the earliest to be established in Europe. It was founded in the mid 12th century, and officially recognized as a university probably between 1160 and 1250...
for a doctorate, becoming a lecturer in Lausanne in 1931; where he held positions until retirement in 1971; he held positions in Geneva
University of Geneva
The University of Geneva is a public research university located in Geneva, Switzerland.It was founded in 1559 by John Calvin, as a theological seminary and law school. It remained focused on theology until the 17th century, when it became a center for Enlightenment scholarship. In 1873, it...
in parallel.
In 1931 he proved de Rham's theorem, identifying the de Rham cohomology
De Rham cohomology
In mathematics, de Rham cohomology is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes...
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
Henri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
and Élie Cartan
Élie Cartan
Élie Joseph Cartan was an influential French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications...
. The first proof of the general Stokes' theorem
Stokes' theorem
In differential geometry, Stokes' theorem is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. Lord Kelvin first discovered the result and communicated it to George Stokes in July 1850...
, for example, is attributed to Poincaré, in 1899. At the time there was no cohomology theory, one could reasonably say: for manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
s the homology theory
Homology theory
In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. It can be broadly defined as the study of homology theories on topological spaces.-The general idea:...
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
Differential form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...
terms an n-form that is never zero (and two being equivalent if related by a positive scalar field). The duality can to great advantage be reformulated in terms of the Hodge dual
Hodge dual
In mathematics, the Hodge star operator or Hodge dual is a significant linear map introduced in general by W. V. D. Hodge. It is defined on the exterior algebra of a finite-dimensional oriented inner product space.-Dimensions and algebra:...
—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
Hodge theory
In mathematics, Hodge theory, named after W. V. D. Hodge, is one aspect of the study of the algebraic topology of a smooth manifold M. More specifically, it works out the consequences for the cohomology groups of M, with real coefficients, of the partial differential equation theory of generalised...
and sheaf
Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
theory.
De Rham also worked on the torsion invariants of smooth manifolds.
Further reading
- Bott, Raoul. "Georges de Rham 1901–1990," Notices of the American Mathematical Society 38 2 (Feb. 1991), 114–115.
- Eckmann, Beno. "Georges de Rham 1903–1990," Elemente der Mathematik 47 3 (1992), 118–122.
External links
- Barile, Margherita. "Georges de Rham." Biographical sketch at The First Century of the International Commission on Mathematical Education.