Global results related to scalar curvature and isoperimetry

I will first survey some recent progress on global problems related to scalar curvature and area/volume, focusing in particular on scale breaking phenomena in such problems. I will then discuss the role of the Hawking mass in the resolution of this scale-breaking issue for the stable CMC uniqueness problem in asymptotically Schwarzschild manifolds (joint work with M. Eichmair) and possibly mention some features of the isoperimetric problem in asymptotically Schwarzschild-anti-de Sitter manifolds (joint work with M. Eichmair, Y. Shi, J. Zhu).