Title: A survey of recent progress in hessian equations in Kahler geometry.

Abstract: We report some of the recent works on fully non-linear Hessian PDEs on Kahler manifolds. We propose the Gårding/ideal Gårding theory as an extension of the classical hyperbolic polynomial theory. We discuss the corresponding Liouville type theorem and $L^\infty$ theory using entropy control. Applications to semi-stable J equation will also be discussed. This talk contains joint work with Biao Ma and Jinyang Wu.