After a short description of the perfectoid fields introduced by Peter Scholze, I will explain my construction with Laurent Fargues of a « curve » X_F associated to a perfectoid field F. This curve is defined over the field Q_p of p-adic numbers. It is not algebraic but is « complete » meaning that one can define the degree of a vector bundle and that Harder-Narasimhan theorem holds. There is a complete classification of these vector bundles. When F is of positive characteristic, it leads to an analogue of the Narasimhan-Sheshadri theorem, i.e. there is a natural equivalence between semi-stable vector bundles of slope 0 and p-adic representations of the absolute Galois group of F. When F is of characteristic 0, it gives rise to a new way of understanding p-adic Hodge theory.
This lecture is part of the discussion meeting "The Fargues-Fontaine Curve: Insights from Perfectoid spaces and p-adic Hodge Theory".