With the organisation of Noncommutative Shapes taking up quite a bit of my time, this post is 1 week late.
Dmitrii Pirozhkov: Categorical Torelli theorem for hypersurfaces shows how the residual category of a hypersurface together with its natural polarisation completely determines the hypersurface. The trick is to reduce it to the usual Torelli theorem in terms of the Jacobian ring, by looking at the Hochschild cohomology of this residual category. Really cool result, and a pleasant paper to read too!
Nick, if you're reading this: this is the type of content you should post on your blog. Go make a blog!
Marco Andreatta, Roberto Pignatelli: Fano's Last Fano discusses from a modern point-of-view a Fano 3-fold originally studied by Fano in 1949. It is wonderful to see the interaction of classical methods and modern methods, in order to show how Fano was looking at what we would now call 2–16 on Fanography.