Products of conics and products of Severi-Brauer surfaces in the Grothendieck ring

Let ${P_i}$ and ${Q_j}$ be two sets of Severi-Brauer surfaces (or two sets of conics) over a field $k$. The main goal of this talk is to explain the relationship between the classes of products $[\Pi P_i]$ and $[\Pi Q_j]$ in the Grothendieck ring and the subgroups $\langle P_i\rangle$ and $\langle Q_j\rangle$ in the Brauer group $\mathrm{Br}(k)$. Under some restrictions on the base field, the following conditions are equivalent: (1) $\langle P_i\rangle = \langle Q_j\rangle$ in $\mathrm{Br}(k)$ (2) $[\Pi P_i] = [\Pi Q_j]$ in $K_0(\mathrm{Var}_k)$ (3) $\Pi P_i$ and $\Pi Q_j$ are birational. In this talk, we will prove this statement. We will also show some properties of the Grothendieck subring generated by conics, in particular, it will be explicitly described by generators and relations. The talk is based on the works of J. Kollar "Conics in the Grothendieck ring" and A. Hogadi "Products of Brauer Severi surfaces".