We discuss an analogue of Morita theory for ring spectra, a thickening of the category of rings inspired by stable homotopy theory. This follows work by Rickard and Keller on Morita theory for derived categories. We also discuss two results for derived equivalences of DGAs which show they differ from derived equivalences of rings.

We study the derived categories of twisted supersingular K3 surfaces. prove a crystalline Torelli theorem for surfaces, characterizing Fourier-Mukai equivalences in terms crystals introduced [2] . This is positive characteristic analog Hodge-theoretic Orlov [23] and its extension to surfaces by Huybrechts Stellari [9] , [10] give applications various questions concerning partners, extending res...