in this survey, we give an overview over some aspects of the set of tilting objects in an $m-$cluster category, with focus on those properties which are valid for all $m geq 1$. we focus on the following three combinatorial aspects: modeling the set of tilting objects using arcs in certain polygons, the generalized assicahedra of fomin and reading, and colored quiver mutation.

In this survey, we give an overview over some aspects of the set of tilting objects in an $m-$cluster category, with focus on those properties which are valid for all $m geq 1$. We focus on the following three combinatorial aspects: modeling the set of tilting objects using arcs in certain polygons, the generalized assicahedra of Fomin and Reading, and colored quiver mutation.

We show that Gorenstein singularities that are cones over singular Fano varieties provided by so-called flag quivers are smoothable in codimension three. Moreover, we give a precise characterization about the smoothability in codimension three of the Fano variety itself.

We use Moyal-type formulas to construct a Hopf algebra quantization of the necklace Lie bialgebra associated with a quiver.

Quivers of finite mutation type have been classified recently in [4]. Main examples of these quivers are the quivers associated with triangulations of surfaces as introduced in [5]. They are also closely related to the representation theory of algebras [1]. In this paper, we study structural properties of finite mutation type quivers. We determine a class of subquivers, which we call basic quiv...