Andrei Caldararu, Kevin Costello, Junwu Tu: Categorical enumerative invariants, I: String vertices and Andrei Caldararu, Junwu Tu: Categorical enumerative invariants, II: Givental formula is a double-header, giving a general definition of categorical invariants, which should recover various enumerative invariants in different fields. I say should, because the actual comparison is only known in some cases (see e.g. Andrei Caldararu, Junwu Tu: Computing a categorical Gromov-Witten invariant).
This is a beautiful combination of tools and introduction of new ideas, and I'd love to spend more time on mastering these. Anyone up for a reading group of some sorts?
Alexander Efimov: Wall finiteness obstruction for DG categories and for algebras over colored DG operads are the slides of a talk by Sasha, in which he discusses the following amazing and unexpected result: every phantom category embeds into a proper dg category with a full exceptional collection. So phantom subcategories are even more elusive than I (and I guess others) expected; as they can lurk in the shadows of categories where you never expected them to live.
I could only see part of the talk, but the proof is inspired by a beautiful parallel between dg categories and topological spaces, and develops the analog of Wall's finiteness obstruction in this setting. This result disproves several conjectures (discussed in the slides). Sasha has a nice track record in disproving conjectures, and showing how phantoms are even spookier than we thought is a very interesting addition to this!