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Passcode: The three digit integer that is the cube of the sum of its digits

Up to a finite covering, a sequence of nested subvarieties of an affine algebraic variety just looks like a flag of vector spaces (Noether); understanding this «Ìýup toÌý» is a primary motivation for a fine study of finite coverings.Ìý

The aim of this talk is to give a bird-eye view of some fundamental questions about them, which took root in Algebraic Geometry (descent problems etc.), then motivated major trends in Commutative Algebra (F-singularities etc.), and recently found complete solutions using p-adic methods (perfectoids). Rather than going into detail of the latter, the emphasis will be on synthesizing, from the geometric viewpoint, a rather scattered theme.Ìý

This is based on joint work with Luisa Fiorot ()