This develops the applications to Kahler geometry mentioned in lecture 2. One application is a uniqueness theorem for twisted Kahler-Einstein equations; another one is Moser-trudinger inequalities.

   This lecture is based on: arXiv:math/0608385 Positivity of direct image bundles and convexity on the space of Kahler metrics and arXiv:1103.0923 A Brunn-Minkowski type inequality for Fano manifolds and the Bando-Mabuchi uniqueness theorem.