The overall aims are to understand local self-similar structure of fractal spaces, objects or processes, by methods of space-time conformal rescaling. The investigationos and the subject of the training programme of the proposed network will therefore be contained in three closely related fields of mathematics and mathematical physics whose common denominator is the notion of conformal or quasiconformal structure: mathematical (conformal) analysis and geometric measure theory, conformal and low-dimensional dynamics, continuum scaling limits of physical processes.