This is a scheduled post, I've been on a vacation for the past week, and I will be for the next. So I might've missed some really cool preprints and other links.
Anthony Blanc, Marco Robalo, Bertrand Töen and Gabriele Vezzosi: Motivic realizations of matrix factorizations and vanishing cycles is a very exciting paper. First it generalises the correspondence between singularity categories and Landau–Ginzburg models, then it constructs a functor from noncommutative motives to (Morel–Voevodsky) commutative motives, and finally it gives a deep comparison result between the inertia invariant part of the vanishing cohomology of a morphism and the realization of the category of a Landau–Ginzburg pair. All very complicated, and I would love to understand more about this. But for that I will have to immerse myself a little more in the recent work of the authors.
Bertrand Toën and Gabriele Vezzosi, The l-adic trace formula for dg-categories and Bloch's conductor conjecture is a research announcement of results that build on the previous fortnightly link. I had missed it when it came out a few weeks ago, but it deserves to be mentioned. Bertrand Toën lectured about this in Stuttgart, and it looks really exciting to see how noncommutative geometry can be used to prove arithmetic results.
Goncalo Tabuada, A note on secondary K-theory II is a nice continuation of an earlier paper by Tabuada. The secondary K-theory of the title is also known as the Grothendieck ring of pretriangulated dg categories à la Bondal–Larsen–Lunts. The goal of the paper is to study the map from the derived Brauer group of a ring (or scheme) to the secondary K-theory, and show that in many cases it is injective.