The goal of this talk is to present some results on the reachable space for the heat equation, based on several works : SE, Kévin Le Balc’h & Marius Tucsnak; SE & Adrien Tendani-Soler. As I will explain, this question is in fact closely related to the possibility to extend the heat semigroup on some spaces of holomorphic functions on an appropriate square or rhombus. Note that this property is also the one essentially used in the work by Alexander Strohmaier and Alden Waters in their study of the reachable space of the heat equation thanks to the so-called Wick rotation. In particular, our work should --This is still an ongoing work ;-)-- allow to describe almost optimally the reachable space for the heat equation in the presence of lower order terms and semi-linear terms.