We give an overview of some recent developments in algebraic geometry in characteristic p. First, we introduce a good substitute of dg Lie algebras in this setting, which leads to a classification of formal moduli problems. Next, we use these new Lie algebras to construct a Galois correspondence for purely inseparable field extensions, generalising a result of Jacobson at height one. Finally, we prove that ordinary Calabi-Yau varieties in characteristic p are unobstructed and admit canonical lifts, generalising results of Serre-Tate, Deligne-Nygaard, Ward, and Achinger-Zdanowic. This talk is based on separate joint works with Mathew, Taelman, and Waldron.

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