L'exposé de cette semaine fera partie de la conférence GAIL : https://gail.sciencesconf.org/
In this talk I will first give a basic overview of K\"ahler and Sasaki structures and their special (constant scalar curvature and more generally extremal) metrics. Then I will describe the so-called join construction of two Sasaki manifolds (introduced by Boyer and Galicki) and highlight some questions relating to this construction and the existence of special metrics in the Sasaki cone. Motivated by these questions, I will exhibit some concrete examples that are part of work in progress with Charles P. Boyer as well as past work with Charles P. Boyer, Hongnian Huang, and Eveline Legendre.