Duality and Dualizing Objects in Stable Homotopy Theory

The chromatic viewpoint in stable homotopy theory distills out information into layers which we call the K(n)-local categories, one for each prime and each non-negative integer. These categories behave more like a category of quasi-coherent sheaves on a scheme than we have any right to expect. For example, there is an analog of Serre-Grothendieck duality encapsulated in Gross-Hopkins duality. Such a duality comes with a dualizing object, and we can ask study its homotopy type. This talk will review the theory and background, summarize what we know, and lay out some conjectures. This is joint work with many people, in particular Hans-Werner Henn for this topic, but also includes work with Barthel, Beaudry, Bobkova, Hopkins, Pham, and Stojanoska.