I'm really interested in how the chromatic story plays out equivariantly. For example, what are the "right" equivariant/global versions of MU, BP, Morava K-theories, and Lubin-Tate theories, and what can they tell us about the structure of equivariant/global homotopy categories? Additionally, I'm interested in computing Balmer spectra and determining Nullstellensatzian objects (in the sense of Burklund-Schlank-Yuan). 

Lately I've been thinking a lot about equivariant algebra. Upcoming work with Ben Spitz and Jason Schuchardt classifies the Nullstellensatzian Tambara functors, and I've been wondering about how this might be used to port over definitions and results from basic Galois theory to the Tambara functor setting. More generally, it seems that there are lots and lots of elementary results in algebra that are ripe to be ported over to Tambara/Green functors.