**Laurent Fargues**

Institut de Mathematiques de Jussieu, France

December 21, 2017

**p-adic symmetric spaces**:
In his ICM talk in Nice in 70 Grothendieck asked what is the image of the p-adic analog of Griffith's period mapping as a subset of a flag variety? We now can give an answer by interpreting those period mappings in terms of modifications of vector bundles on the curve (a p-adic analog of Simpson's Twistor structures). I will explain a recent result obtained jointly with Miaofen Chen and Xu Shen that allows us to compute those period spaces in some particular cases. For example
we can compute the p-adic period space for polarized K3 surfaces with
supersingular reduction.