MSW-bangle functions are generic bases for marked surfaces

This is a report on a joint project with Daniel Labardini and Jon Wilson. We extend our previous result, joint with Daniel Labardini and Jan Schröer from unpunctured surfaces to punctured surfaces with non-empty boundaries. Let T be a tagged triangulation of such a marked surface and A(T) the corresponding Jacobian algebra for the Labardini potential. A(T) is finite-dimensional and tame, however it is only gentle if the surface has no punctures. More precisely, A(T) is skewed-gentle if T is of signature 0, otherwise there is no explicit classification of the indecomposable representations known. Moreover, the correspondence between curves and indecomposable representations is complicated by the presence of kinks. We sketch briefly, how to deal with these difficulties.