What a treat, a fortnightly links which is actually posted on Sunday (as I intend it every time).

• Alberto Canonaco, Mattia Ornaghi, Paolo Stellari: Localizations of the category of $\mathrm{A}_\infty$-categories and internal Homs is a preprint proving the fundamental result that it doesn't matter whether you use dg categories, strictly unital $\mathrm{A}_\infty$-categories or cohomologically unital $\mathrm{A}_\infty$-categories (as long as you do everything up to quasi-equivalence). As an application they give a proof for a claim of Kontsevich, that the dg category of strictly unital $\mathrm{A}_\infty$-functors between 2 dg categories is a model for the internal Hom. A proof of this claim was first given by Faonte, but they point out some issues in the original proof.

• Daniel Bragg: Derived equivalences of twisted supersingular K3 surfaces discusses a class of K3 surfaces which only exist in characteristic $p$, and are characterised by the rank $\rho$ of their Picard group being 22 (compared to the maximum of 20 in characteristic 0). The Brauer group of such surfaces behaves totally differently, instead of being isomorphic to $(\mathbb{Q}/\mathbb{Z})^{22-\rho}$ it is isomorphic to the underlying additive group of the field $k$. This makes the analogue of the twisted derived Torelli theorem in finite characteristic behave differently, but using crystalline cohomology one can prove its analogue.

My favourite result is probably theorem 4.4.6, which gives the number of twisted Fourier–Mukai partners, in terms of the Artin invariant (which takes on values on from 1 to 11), involving powers of the characteristic.

• Marco Antonio Armenta, Bernhard Keller: Derived invariance of the Tamarkin--Tsygan calculus of an algebra shows that the pair $(\operatorname{HH}^\bullet(A),\operatorname{HH}_\bullet(A))$ with its Tamarkan–Tsygan calculus is a derived invariant. This is a result I wanted to see for quite some time! Next up, proving it for arbitrary dg categories (provided it is true in that generality of course).

• Homological projective duality is a wonderful machine, and I'd love to understand it much better. Anyone up for a seminar in Bonn? Recently, 2 more foundational papers have appeared:

One can also watch the short talk the first author gave introducing this subject.