• Roland Abuaf, Compact hyperkähler categories is an updated preprint defining the notion of a hyperkähler category. Given how hard it is to construct compact hyperkähler varieties, it is an interesting idea to construct categories with similar categories. Roland gives several new examples using noncommutative crepant resolutions.

What is new in the most recent update is a counterexample to a conjecture of Kuznetsov, that says that if $\mathcal{A}$ is a Calabi–Yau subcategory of some $\mathbf{D}^{\mathrm{b}}(X)$, then $\mathcal{A}\simeq\mathbf{D}^{\mathrm{b}}(X')$ where $X\to X'$ is a birational morphism.

• Andrea Maiorana, Moduli of semistable sheaves as quiver moduli is a really nice write-up of the moduli theory of semistable sheaves from the point of view of quiver moduli. It gives an overview of the different aspects of the theory, discussing all the different facets.

• Matthew Ballard, Blake Farman, Kernels for noncommutative projective schemes gives the Artin–Zhang analogue of the result that dg functors between derived categories of schemes are given by Fourier–Mukai transforms: the derived category of the product~$X\times_kY$ where kernels live in the commutative setting is now replaced by the derived category of $\mathop{\mathrm{qgr}}A^{\mathrm{op}}\otimes_kB$. Ample detail is given to the homological conditions that are required in the proof.