I was invited to give a talk at the 2019 edition of Rutgers MAMLS taking place Nov 1-3, 2019 at Rutgers University, USA.
Sealed Trees and the Perfect Subtree Property for Weakly Compact Cardinals
Abstract: We investigate the consistency strength of the statement: $\kappa$ is weakly compact and there is no tree on $\kappa$ with exactly $\kappa^{+}$ many branches. We show that this property fails strongly (there is a sealed tree) if there is no inner model with a Woodin cardinal. On the other hand, we show that this property as well as the related Perfect Subtree Property for $\kappa$, implies the consistency of $\operatorname{AD}_{\mathbb{R}}$. This is joint work with Yair Hayut.