Perfectoid Algebras and Singularities in Mixed Characteristic

In this lecture series, we give an introduction to recent applications of p-adic methods to commutative algebra. We start by proving a p-adic version of Kunz's theorem characterizing regular local rings via perfectoid algebras. We then focus on the direct summand theorem and the existence of big Cohen-Macaulay algebras, and we will sketch a proof of them via Andre's flatness lemma. Finally, we use perfectoid big Cohen-Macaulay algebras to define and study singularities in mixed characteristic, and we will discuss some recent work studying various notions of mixed characteristic test ideals via the p-adic Riemann-Hilbert functor. 

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