摘要: Hall algebras (first week). We will define and study the Hall algebras attached to an abelian category. These come in several flavors (Hopf algebra, Lie algebra, derived versions of these,...) and we will try to compare the various versions. All examples well understood so far are related to abelian categories of homological dimension one (ie curves and quivers); after reviewing these, we will try to give some examples in higher dimension (eg in dimension two).Canonical bases and moduli spaces (second week). We give, following the lead of Lusztig, a geometric counterpart to the Hall algebras defined during the first week. For this we consider, given an abelian category A, the category of constructible complexes over the moduli space (stack) of objects in A, and construct several functors on this category. This “geometric Hall algebra”(or rather,“geometric Hall category”) is related to the usual one via some trace map …