Predicting, resolutions and the topological representation of operators on Banach spaces C(K)

We will describe three quite new classes of compact (nonmetrizable) Hausdorff spaces which are motivated by problems concerning Banach spaces and the algebras of operators on them:

Ks such that C(K)s have few operators; Ls such that C(L)s do not have support sets, and Ms which are not Radon-Nikodym but are continuous images of Radon-Nikodym compacta.

These classes are quite different and have diverse applications but the methods used in their constructions share some points: combinatorial predicting, topological representation of operators and the use topological resolutions.