Metric measure geometry
  In this book, we study Gromov's metric geometric theory on the space of metric measure spaces, based on the idea of concentration of measure phenomenon due to L\'evy and Milman. Although most of the details are omitted in the original article of Gromov, we present complete and detailed proofs for some main parts, in which we prove several claims that are not mentioned in any literature. We also discuss concentration with a lower bound of curvature, originally studied by Funano and the author. 
Takashi Shioya
math.MG