Joint IAS/PU Arithmetic Geometry

Given a cohomology theory, it is sometimes possible to associate to each $X$ a stack $X'$, called the transmutation of $X$, in such a way that the cohomology of $X$ agrees with the coherent cohomology of $X'$. In practice, each cohomology theory has an associated six-functor formalism Shv, and transmutation has the feature Shv$(X)$ is equivalent to the derived category of quasi-coherent sheaves on $X'$. Transmutations were first constructed by Simpson for de Rham cohomology in characteristic zero, and have in recent years become available for some (but not all) six-functor formalisms of interest. The goal of this talk is to explain joint work with Scholze in which we construct an infinity topos of geometric objects which we call Gestalten, with the feature that *every* six-functor formalism gives rise to a transmutation functor into Gestalten.