The Rochberg garden

In 1996, it was published the seminal work of Rochberg “Higher order estimates in complex interpolation theory” (Rochberg, 1996). Among many other things, the paper contains a new method to construct new Banach spaces having an intriguing behaviour: they are simultaneously interpolation spaces and twisted sums of increasing complexity. The fundamental idea of Rochberg is to consider for each $z\in S$ the space formed by the arrays of the truncated sequence of the Taylor coefficients of the elements of the Calderón space. What was probably unforeseen is that the Rochberg constructions would lead to a deep theory connecting Interpolation theory, Homology, Operator Theory and the Geometry of Banach spaces. This work aims to synthetically present such connections, an up-to-date account of the theory and a list of significative open problems.