The classical Dieudonne theory classifies p-divisible groups over a perfect field k in terms of semi-linear algebra over W(k). In this talk, I will explain a conjectural generalization -- due to Drinfeld -- of Dieudonne's classification to a broader class of rings, whose spectra form a basis for the fpqc topology on p-nilpotent schemes. A key input in Drinfeld's approach is a certain square-zero extension of the ring of Witt vectors, known as the ring of sheared Witt vectors. I will explain some results on this ring due to Bhatt, Mathew, Zhang and myself. 
If time permits, I will also describe another application of sheared Witt vectors: the sheared prismatization. While the classical prismatization construction -- due to Drinfeld and Bhatt–Lurie -- provides a deformation of the category of nilpotent D-modules on smooth p-adic formal schemes, the sheared prismatization yields a deformation of the category of all D-modules.
The last part of my talk is based on a joint work with Bhatt, Kanaev, Mathew, and Zhang.