Abstract: Bridgeland assigned to any triangulated category a complex manifold, whose elements are referred to as stability conditions. Homological mirror symmetry and string theory predict a parallel between Teichmüller theory and categories whereby the stability space plays the role of the Teichmüller space. However it is hard to extract global information for these spaces. Beilinson observed patterns in the structure of some triangulated categories called later exceptional collections. The interplay between the two notions in the title unveils novelties for both. An explicit determining of the entire stability spaces on wild Kronecker quivers in our latest joint work with L. Katzarkov is a subsequent result arising from this interplay. In this talk I will try to put the new result in a context, supporting what Bridgeland wrote: the whole idea of extracting geometry from homological algebra is, to me at least, a very attractive one.