Alberto Merici (University of Milan) - Log prismatic cohomology, motivic spectra and comparison theorems

We prove that (logarithmic, Nygaard completed) prismatic and (logarithmic) syntomic cohomology are representable in the category of logarithmic motives. As an application, we immediately obtain Gysin maps for prismatic and syntomic cohomology, and we precisely identify their cofibers. In the second part of the talk we develop a descent technique that we call saturated descent, inspired by the work of Niziol on log K-theory. Using this, we prove crystalline comparison theorems for log prismatic cohomology, log Segal conjectures and log analogues of the Breuil-Kisin prismatic cohomology, from which we get Gysin maps for the A_inf cohomology.