The Hermitian geometry of the Chern connection

We survey some analytic problems concerning the geometry of the Chern con- nection of Hermitian manifolds: the existence of metrics with constant Chern-scalar curvature; the generalizations of the K¨ahler-Einstein condition to the non-K¨ahler setting; the convergence of the normalized Chern-Ricci flow on compact complex surfaces; the asymptotic behaviour of Monge-Amp`ere volumes of Hermitian metrics in the ddc-class. The talk is based on joint collaborations with Simone Calamai, Mauricio Corrˆea, Vincent Guedj, Chinh Lu, Francesco Pediconi, Cristiano Spotti, Valentino Tosatti.