In this talk we report some recent joint work with Biao Ma.

We introduce a new type of Monge-Ampere type PDEs in complex geometry and relate their solvability to corresponding paired slope stability conditions through the bridge of analytical condition of sub-solution existence. This work generalizes previous works on J-equations and deformed Hermitian-Yang-Mills equations by G. Chen, Datar-Pingali, Song and many others.

In order to find more examples, we also explore a generalization of  the 1959 works of Gårding on hyperbolic polynomials, which has been widely used to study geometric PDEs under  the framework of Caffarelli-Nirenberg-Spruck theory. 

Generalizing earlir works of C.-M. Lin and some of our previous results, we introduce a class of polynomials that is named after Gårding. Gårding polynomials are characterized by their associated cones and defined in a similar fashion of Lorentzian polynomial of Branden-Huh.  Gårding  polynomials share similar algebraic properties of stable and Lorentzian polynomials and have potential applications in other fields of mathematics.