Stability conditions on abelian categories

The first goal of this talk is to give the definition of stability condition for an abelian category, for which one needs to introduce the notions of stability functions and stability of an object for a stability function.

After these first definitions, one wants to discuss the existence of the Harder-Narasimhan filtration. This question is positive for a stability condition on a noetherian category with a stability function having discrete real image. The presentation will be complemented with examples, mentioning the case of curves and the slope stability. The main reference is Section 4 of Macrì-Schmidt's notes.