It's summer for everyone, so I guess that explains why I skipped an installment (although the semester in Bonn only ended 2 days ago in fact).
Amnon Neeman: A counterexample to some recent conjectures gives an example of an abelian category with negative K-theory. It is constructed has the heart of a bounded t-structure on the subcategory of bounded acyclic complexes in the homotopy category of the exact category of vector bundles on a nodal curve. This gives a counterexample to conjectures of Schlichting and Antieau–Gepner–Heller. Beautiful!
Thorsten Beckmann, Georg Oberdieck: Equivariant categories and fixed loci of holomorphic symplectic varieties studies equivariant categories of symplectic surfaces, to study the case when they are again the derived category of a symplectic surface. For a stability condition which is preserved by the group action they give a criterion that makes such an identification possible. The action used to construct the equivariant category can be cooked up using any type of autoequivalences, making this a cool result!
There is also an interesting companion paper, Notes on equivariant categories.
Alex Chirvasitu, Ryo Kanda, S. Paul Smith: Elliptic R-matrices and Feigin and Odesskii's elliptic algebras is no longer scary to me, because I started loving the Yang-Baxter equation (see previous fortnightly links). They use the quantum Yang-Baxter equation with parameters to study the Feigin-Odeskii families of algebras, showing that they are almost always Artin-Schelter regular and have the Hilbert series of the polynomial algebra, hence they give rise to noncommutative projective spaces. Cool!
This begs the question (if you're anything like me): what is the Hochschild cohomology of their qgr?