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Singularities are measured in different ways in characteristic zero, positive characteristic, and mixed characteristic. However, the classes of singularities usually form analogous groups with similar properties, with an example of such a group being klt, strongly F-regular and BCM-regular.  In this talk we shall focus on a mixed characteristic counterpart of F-injective and Du Bois singularities.  We will see that the three have a unified description and satisfy many expected properties.

We recall the definition of parasolid closure, which is a closure operation for all noetherian rings, the parasolid algebras (which are the 'allowed covers') and the test ideals deriving from this approach. In an open discussion, we would like to understand how these test ideals (and the concept of a pararegular ring) is related to similar notions defined with the help of big Cohen-Macaulay algebras, resolutions and alterations.

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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.

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In this talk, I will discuss two methods: the 0th local cohomology and the bigraded algebra, that are used to extend the classical multiplicities such as the Hilbert-Samuel multiplicity, the Hilber-Kunz multiplicity, the mixed multiplicity from 0-dimensional ideals to general ideals.