From Analysis situs to the theory of periods
Apparaît également dans la collection : Colloque Scientifique International Poincaré 100
The talk will focus on the pairing between singular homology and de Rham cohomology: Combinatorics of cells of a triangulation on one side, differential forms on the other side. The two aspects of the subject were already present in Poincaré's work, but the fact that this pairing is perfect has been proved much later by de Rham. De Rham's comparison theorem is not the end but the beginning of the story. Indeed, the two sides compute cohomology with different coefficients: In particular, in the context of algebraic geometry, the de Rham side is sensitive to the definition field of the variety, while the singular or Betti side sees the cohomolgy with integral coefficients. This leads to the modern theory of periods, and rises arithmetic questions related to the Hodge conjecture.
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