Mathematics

The course is an introduction to some of the newest approaches to non-archimedean analytic geometry including:- Huber's adic spaces;- Raynaud's formal schemes and blow-ups;- Clausen-Scholze's analytic spaces.We will focus on specific examples arising from algebraic geometry, Scholze's tilting equivalence of perfectoid spaces and the Fargues-Fontaine curve.We will also show how to define (motivic) homotopy equivalences in this setting, with the aim of defining a relative de Rham cohomology for adic spaces over $\mathbb{Q}_p$  and a relative rigid cohomology for schemes over  $\mathbb{F}_p$.