UCLA

This thesis develops an emerging relationship between the number theoretic notion of prisms and the equivariant algebra of Tambara functors. We develop some of the theory of Weyl-invariant Tambara functors and present Witt vectors from this perspective. A new interpretation of divided power structures in terms of Tambara thickenings is also given. We reprove classical deformation-theoretic results and more recent work on presentations for the Witt vectors of perfect algebras. We then turn our attention to prisms, and show that there is an explicit recipe for constructing a Tambara functor from an oriented prism. Earlier results relating quotients of perfect prisms to Witt vectors are reproved from this perspective. We also offer a new Tambara-theoretic characterization of perfectoid rings, and introduce a candidate theory of global prisms with associated Q/Z-Tambara functors.