Motivic homotopy theory is the study of homotopy-theoretic ideas in the setting of algebraic geometry. The basic categories of interest are those of motivic spaces $\mathcal{H}(S)$ and motivic spectra $\mathcal{SH}(S)$ over a base scheme $S$. In recent work of Bachmann--Hoyois, these categories were equipped with norm monoidal structures, variants of monoidal structures richer than what is usually the richest for homotopy theory (i.e. $\mathbb{E}_\infty$). In this talk, I will discuss norm monoidal structures on various extensions of motivic homotopy theory where the spaces/spectra are equipped with (generalized) transfers. The construction of norms for motivic spaces with framed transfers will allow us to prove a norm monoidal enhancement of the motivic infinite loop space recognition principle of Elmanto--Hoyois--Khan--Sosnilo--Yakerson. We'll also discuss the interactions of norms with other flavors of transfer.
