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  "body": "\n## Summary\n\nA perfectoid space is an analytic space over a non-archimedean field of residue characteristic $p$.\nIt is so ramified that the Frobenius map is surjective modulo $p$.\nPeter Scholze introduced perfectoid spaces in 2012.\nTheir key feature is tilting: perfectoid spaces in mixed characteristic correspond to perfectoid spaces in characteristic $p$.\nThis lets one transfer problems between the two worlds.\nThey have become a basic tool of $p$-adic geometry and the arithmetic of [[Galois representations|JQEYEQJY2G]].\n\n## Explanations\n\n### L1 human\n\nThe $p$-adic numbers and the field of Laurent series over $\\F_p$ look very different, since one has characteristic $0$ and the other characteristic $p$.\nAdjoin all $p$-power roots of $p$ to the first, all $p$-power roots of the variable to the second, and complete both.\nThe two resulting fields have equivalent Galois theories.\nPerfectoid spaces extend this surprising match from fields to geometry.\nA hard question in characteristic $0$ can then be moved to characteristic $p$, where the Frobenius map gives extra tools.\n",
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