Singularities and perfectoid geometry (Q2023887)

This survey paper gives a quick overview of the author's work on the existence of big Cohen-Macaulay algebras in mixed characteristic via perfectoid theory, as well as its consequences (e.g., the solutions of several homological conjectures in commutative algebra). He briefly discusses the role of almost mathematics and perfectoid theory in the proof. Finally, the paper concludes with a summary of recent applications of these results to singularity theory in both equal and mixed characteristic.