There is a notion of duality due to Serre on the category of perfect connected commutative unipotent groups (cpu) over an algebraically closed field of positive characteristics k. For the objects in (cpu), there is also an analogue of the Fourier transform known as the Fourier-Deligne transform due to Deligne.

In this talk, we introduce the category of Tate objects Tate(cpu), whose objects are certain connected commutative unipotent group ind-schemes. We then extend the aforementioned notions of duality and Fourier-Deligne transform to the category of Tate objects. This talk is based on joint work with Tanmay Deshpande.