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The subject of homotopy theory studies the properties of shapes which are invariant under continuous change. Despite its geometric origins, one of the great discoveries of XX century mathematics is that such phenomena are closely related to arithmetic, the study of phenomena arising from the integers. Very informally, this means that the deep properties of prime numbers are reflected in the structure of space and the shapes which can inhibit it. The goal of this Hakubi project is the study of prismatic cohomology, which is a recently discovered invariant of varieties in mixed characteristic which led to large advances in the field of arithmetic geometry. As first shown by Hahn-Raksit-Wilson, prismatic cohomology naturally arises from the even filtration, which informally means that it is essentially encoded by the structure of stable homotopy theory itself. This surprising discovery connects prismatic cohomology to a wide range of mathematics, such as the theory of motives or chromatic homotopy theory and allows one to apply it to the study of localizing invariants of ring spectra.