I will describe a Linearization Conjecture that identifies the spectral dualizing module of a p-adic Lie group in terms of a representation sphere built from the Lie algebra. We can prove this when the action is restricted to certain small finite subgroups. These results are enough to determine Spanier-Whitehead duals of some chromatically interesting spectra. This is joint work in progress with Beaudry, Goerss, and Hopkins.

I will talk about homotopical dualities of Brown-Comenetz type through the lens of algebra and algebraic geometry. In particular, I will discuss details of self-duality in the prominent examples of K-theory and topological modular forms.