Smooth compactifications in derived noncommutative algebraic geometry

We will formulate several results on the notion of a smooth compactification of differential graded categories, the questions of its existence and its construction. Existence of a smooth compactification is closely related with the notion of homotopy finiteness of DG category (the analogue of finite domination for topological spaces).
We will formulate the result on the existence of a smooth compactification for the derived category of coherent sheaves on any quasi-projective (possibly singular) scheme. We will also discuss an example of a homotopically finite DG category which does not have a smooth compactification (a counterexample to a conjecture of Kontsevich). The later result can be interpreted as a lack of resolution of singularities in derived noncommutative geometry.

^* Zoom Meeting ID: 979 5396 3790