Haonan Zhang - Institute of Science and Technology Austria (IST Austria) In a celebrated paper in 1973, Lieb proved what we now call Lieb's concavity theorem and resolved a conjecture of Wigner-Yanase-Dyson. In this talk we will explain how to easily derive the concavity/convexity of certain trace functionals from some fundamental results: mainly this 1973 concavity theorem of Lieb and its complementary convexity result of Ando in 1979. Along the way we settle a conjecture of Carlen-Frank-Lieb. A weaker form of this conjecture was made earlier by Audenaert-Datta when studying the data processing inequalities for alpha-z Rényi relative entropies. Time permitting, we also revisit another 1973 concavity result of Lieb. The talk is based on arXiv:1811.01205.