Title: An overview of the stable polynomial theory

Abstract: In studying fully-nonlinear PDE and geometric analysis, elementary symmetric functions of $n$-variables play crucial roles. These simple polynomials have rich algebraic, geometric and analytical structures. They belong to a more general class, stable polynomials that that are widely applied in analysis, control theory, probability, dynamics and combinatoric studies.

In this talk, we will review theories of real stable theory from classical era to works of Borcea-Branden in early 2000s. We will go over definitions, key properties and structural results. We loosely follow the survey of D. Wagner.

In future talks, we will discuss the Lorentzian polynomial theory of Branden-Huh that is a modern generalization with great success in many fields.

Our goal is to introduce a different type of generalized stable polynomial theory that is closed related to items above, enjoys similar advantages and may be applied in PDE theory and many other fields.