In this talk, I will introduce the category of synthetic $E_\infty$-rings. This category is distinct from, though monadic and comonadic over, the category of $E_\infty$-rings in even, MU-synthetic spectra, and is closely related to the even filtration. Under the relationship between p-complete synthetic and cellular motivic spectra, synthetic $E_\infty$-rings are expected to correspond to the normed motivic spectra of Bachmann--Hoyois. I will also discuss how synthetic $E_\infty$-rings can be used to study the even filtration on THH.
This is joint work in progress with Devalapurkar, Hahn and Raksit. It takes inspiration from work in the motivic setting by Bachmann--Burklund and joint work of the speaker with Burklund.
