Prismatic cohomology is an exciting new tool recently introduced by Bhatt and Scholze. Properly, this provides a badly missing tool in p-adic Hodge theory, namely a systematic way to relate the myriad zoo of cohomology theories, which come out of a genuine mathematical object called a prism. The crux of the story is that in contrast to projective complex manifolds, where there are essentially two primary cohomology theories of interest, namely singular and de Rham which are classically naturally isomorphic, the p-adic version of the story is much more complicated. To approximate the singular cohomology, or at times replace it, one needs to consider not two but many cohomology theories, e.g., ́etale, crystalline, etc. The relationships between them are suitably arithmetically enhanced versions of the complex picture, and were explored in foundations laid out by Fontane, Faltings, and many others over a long period of time. The prismatic theory is built on previous work, certain enhancements to the classic p-adiccohomology theories, and builds on and extends Scholze's revolutionary theory of perfectoid spaces.

Already this theory has paid immediate dividends in the form of strong comparison theorems as well as many conceptual improvements to the integral versions of these cohomology theories. As a taste of the more tangible applications, [Bhac] applied these ideas to settle a long standing question about absolute integral closures; an applications which at first does not appear to depend at all on prismatic cohomology. Improvements to the theory are already being developed, as well as many lecture series and courses.

In these lectures, we will describe the motivations behind this exciting theory, starting with the setting of projective complex manifolds, and then will spend some time giving a description of the various p-adic cohomology theories which are being unified. We will not assume a background in p-adic cohomologies. We will then describe the foundations of the prismatic theory which by comparison to the now more historical versions of p-adic Hodge theory are in some ways quite simple and direct. The astute reader will note that in some meaningful ways, this theories is more complicated. In particular, the main references make clear that to provide rigorous general proofs, one must pass to derived algebraic geometry. We will do our best to avoid these techniques when they are not necessary for the story. Time permitting, we will also outline the triumphs of the theory.