Explore the Schatten properties of commutators in quantum tori during this 43-minute lecture by Kai Zeng from Université de Bourgogne-Franche-Comté at the Institut des Hautes Etudes Scientifiques (IHES). Delve into the characterization of Schatten properties for the commutator [$\mathfrak{R}_i,M_x$] on quantum tori $\mathbb{T}_{\theta}^d$, where $\mathfrak{R}_j$ represents Riesz transforms and $M_x$ is the pointwise multiplication operator. Discover how these properties relate to Besov spaces $B_{p,q}^{\alpha} ({\mathbb T}_{\theta}^d)$ on quantum tori. Examine the extension of this characterization to cases involving arbitrary Calderon-Zygmund operators. Gain insights into quantum differentiability in strictly noncommutative settings through these new results.