Indestructibility of some compactness principles, part 1

In the talk we will focus on compactness principles at the double successor of a regular cardinal kappa. We start by showing that if kappa^{<kappa} =kappa and lambda>kappa is a weakly compact cardinal, then in the Mitchell model V[M(kappa,lambda)] the tree property at kappa^{++} is indestructible under all kappa^+-cc forcing notions which live in the intermediate submodel V[Add(kappa,lambda)]. This result has direct applications for Prikry-style forcing notions and hence for the tree property at the double successor of a singular strong limit cardinal (it simplifies existing results and can be used to prove new results). Then we will discuss stationary reflection and its variants and the indestructibility under kappa^+-cc forcing notions.