Duality in tt-geometry via proxy-smallness

Proxy-smallness is a finiteness property for objects in (tensor-) triangulated categories, and I will describe how one can exploit this notion to prove duality statements in tensor-triangular geometry. 

I will explain how to recover a form of Grothendieck duality on a canonical subcategory building on work of Balmer--Dell'Ambrogio--Sanders, and how this setting provides a suitable framework to generalise the concept of Gorenstein duality for ring spectra in the sense of Dwyer--Greenlees--Iyengar. 

This is joint work with Tom Peirce.