• Jianxun Hu, Hua-Zhong Ke, Changzheng Li, Tuo Yang: Gamma conjecture I for del Pezzo surfaces discusses conjecture $\mathcal{O}$ and the first Gamma conjecture for del Pezzo surfaces. These conjectures describe properties of quantum cohomology, as predicted by mirror symmetry for Fano varieties. Recall that quantum cohomology $\mathrm{QH}^\bullet(X)$ is a deformation of the cohomology ring $\mathrm{H}^\bullet(X,\mathbb{C})$, and conjecture $\mathcal{O}$ predicts the distribution of the eigenvalues of quantum multiplication with the class $\mathrm{c}_1(X)$.

They give a conceptual proof for del Pezzo surfaces, whereas I had been planning to discuss the purely computational proof which is nothing but a case-by-case analysis of the 10 different deformation families of del Pezzo surfaces. Maybe I'll still discuss this approach at some point, but the conceptual proof is much more interesting of course :).

• Amnon Neeman: Metrics on triangulated categories explains how to interpret some recent results of Neeman and Krause in terms of the classical (but in the triangulated category unknown) notion of metrics on categories. Besides being a very interesting and pleasant to read preprint, it features some interesting quotes, such as the following advice on how to write a paper:

All we have shown so far is that there is no law barring a mathematician from making a string of ridiculous definitions. To persuade the reader that this formalism has some value we need to use it to prove a theorem.
• Sarah Scherotzke, Nicolò Sibilla, Mattia Talpo: Gluing semi-orthogonal decompositions discusses the interaction of homotopy limits of dg categories with semiorthogonal decompositions for the categories appearing in the diagram. They use this to construct a semiorthogonal decomposition for root stacks, without a condition on the divisor. Cool!