Appears in collection : Algebraic Analysis in honor of Masaki Kashiwara's 70th birthday

We will briefly recall the general philosophy of non-commutative (and derived) algebraic geometry in order to establish a precise link between dg-derived category of singularities of Landau-Ginzburg models and vanishing cohomology, over an arbitrary henselian trait. We will then focus on a trace formula for dg-categories and a recent application to Bloch’s conductor conjecture. This second, and main part of the talk refers to work in progress, joint with B. Toën.