Zachary Gardner (BC)

 Let be the ring of integers of a -adic local field with uniformizer . In past joint work, K. Madapusi and I establish a version of Dieudonn\'{e} theory which says that -divisible groups over any -nilpotent base ring are equivalent to vector bundles over the syntomification of with Hodge-Tate weights contained in . In this talk, I will explain how this result can be generalized to deduce that -divisible groups over any -nilpotent -algebra are equivalent to vector bundles over the so-called -syntomification satisfying the analogous weight constraint. This story connects many interesting ideas within integral -adic Hodge theory, and the resulting tools have applications to the study of Rapoport-Zink spaces and integral models of Shimura varieties.