Seminar

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In positive characteristic, the F-signature can be viewed as a quantitative measure of F-regularity – an important class of singularities central to the celebrated theory of tight closure pioneered by Hochster and Huneke, and closely related to

Kawamata Log Terminal (KLT) singularities via standard reduction techniques from characteristic zero. The definition can also be extended to divisor (or hypersurface) pairs, and the resulting F-signature functions given by scaling enjoy a number of nice properties such as convexity. In this talk, I will give an overview of some of the positive characteristic theory, and detail recent progress in developing an analogue in the mixed characteristic setting. Based in part on joint work with Hanlin Cai, Seungsu Lee, Linquan Ma, and Karl Schwede, this involves leveraging the perfectoidization functor of Bhatt-Scholze to define the perfectoid signature. In addition, I will mention joint work in progress with a subset of the same authors to extend this definition to pairs and show properties of the resulting perfectoid signature functions given by scaling.