Keywords: Moduli of sheaves Donaldson-Thomas invariants Virtual classes, localisation Semiorthogonal decompositions Grothendieck rings of varieties

**Refined invariants in moduli theory**. Moduli spaces of stable sheaves on complex algebraic 3-folds*Y*have a rich geometric structure which can be used to define enumerative invariants (numbers) that virtually count the number of points in the moduli space; in some cases (when Y is Calabi-Yau), these numbers can be refined to more sophisticated invariants, e.g. in the K-theoretic or motivic setting. These are called**Donaldson-Thomas invariants**. The computation of these refined invariants can be performed in various ways, depending on the nature of the refinement: for instance one might use virtual torus localisation for the K-theoretic invariants, Hall algebras and the theory of quiver representations for the motivic invariants.**Geometry of Hilbert and Quot schemes**and their virtual structures, such as existence of virtual fundamental classes, but also their very structure as algebraic schemes.**Semiorthogonal decompositions**. A semiorthogonal decomposition of a triangulated category*T*allows one to "decompose"*T*in smaller pieces, hopefully easier to study. The deformation theory of semiorthogonal decompositions has still many interesting aspects to be explored.**Rings of motives**, such as the Grothendieck ring of varieties, are interesting and mysterious. It is sometimes possible to compute the generating function of the motives of a given family moduli spaces and observe its behaviour, good or bad. It is a challenge to find a family of moduli spaces for which such generating function has a nice intrinsic expression.