We will explain the the definition of a perfectoid field and the tilting construction. We will then state the Fontaine-Wintenberger theorem and prove it in the algebraically closed case.

We will survey them. The relation to analysis lies in the fact that stratifications are related to differential equations, and to categories in the fact that the fundamental groups in question are those Tannakian fundamental groups of those categories. Else one should honestly say that the deep questions which will be mentioned are rather about the connection between Étale topology  and various crystalline sites.