(Measured) groupoids: from the beginning to some recent constructions

As The Ugly Duckling among algebraic structures, groupoids may be less fascinating than other categories, such as groups. Nevertheless, in some situations in Mathematics, it turns out that groupoids fit better as a model to describe objects. In this talk we will adopt a bird’s eye view on this topic, with particular interest on the role played by groupoids in Measured Group Theory, namely in the dynamical study of measure preserving actions. In the last part, we will see how bounded cohomology, a classical notion introduced in the seventies and fruitfully exploited in the last decades in Rigidity Theory and in Geometric Group Theory, has been recently defined for measured groupoids. Time permitting, some motivations for the interest in such an extension will be presented. This is a joint work with A. Savini.