Abstract. Differential graded categories enhance our understanding of triangulated categories appearing in algebra and geometry. In this survey, we review their foundations and report on recent work by Drinfeld, Dugger-Shipley,..., Toën and Toën-Vaquié. 1.

This paper contains some applications of Bridgeland-Douglas stability conditions on triangulated categories, and Joyce’s work on counting invariants of semistable objects, to the study of birational geometry. We introduce the notion of motivic Gopakumar-Vafa invariants as counting invariants of D2-branes, and show that they are invariant under birational transformations between Calabi-Yau 3-folds. The result is similar to the fact that birational Calabi-Yau 3-folds have the same betti numbers or Hodge numbers. 1

Abstract. We present a simple description of moduli spaces of torsion-free D-modules (D-bundles) on general smooth complex curves, generalizing the identification of the space of ideals in the Weyl algebra with Calogero-Moser quiver varieties. Namely, we show that the moduli of D-bundles form twisted cotangent bundles to moduli of torsion sheaves on X, answering a question of Ginzburg. The corresponding (untwisted) cotangent bundles are identified with moduli of perverse vector bundles on T ∗ X, which contain as open subsets the moduli of framed torsion-free sheaves (the Hilbert schemes T ∗ X [n] in the rank one case). The proof is based on the description of the derived category of D-modules on X by a noncommutative version of the Beilinson transform on P 1. 1.

The following thesis plays a central role in deformation theory: (∗) If X is a moduli space over a field k of characteristic zero, then a formal neighborhood of any point x ∈ X is controlled by a differential graded Lie algebra. This idea was developed in unpublished work of Deligne, Drinfeld, and Feigin, and has powerfully influenced

2.1. Compatible autoequivalences 2 2.2. Application of Theorem 2.3 to structure of Mσn

With an appendix by Michel Van den Bergh at Diepenbeek Abstract. We define and investigate deformed n-Calabi–Yau completions of homologically smooth di¤erential graded ( dg) categories. Important examples are: deformed preprojective algebras of connected non-Dynkin quivers, Ginzburg dg algebras associated to quivers with potentials and dg categories associated to the category of coherent sheaves on the canonical bundle of a smooth variety. We show that deformed Calabi–Yau completions do have the Calabi–Yau property and that their construction is compatible with derived equivalences and with localizations. In particular, Ginzburg dg algebras have the Calabi–Yau property. We show that deformed 3-Calabi–Yau completions of algebras of global dimension at most 2 are quasi-isomorphic to Ginzburg dg algebras and apply this to the study of cluster-tilted algebras and to the construction of derived equivalences associated to mutations of quivers with potentials. In the appendix, Michel Van den Bergh uses non-commutative di¤erential geometry to give an alternative proof of the fact that Ginzburg dg algebras have the Calabi–Yau property.