Forcing "$\mathrm{NS}_{\omega_1}$ is $\omega_1$-dense" From Large Cardinals

Abstract

We answer a question of Woodin by showing that assuming an inaccessible cardinal $\kappa$ which is a limit of ${<}\kappa$-supercompact cardinals exists, there is a stationary set preserving forcing $\mathbb{P}$ so that $V^{\mathbb P}\models``\mathrm{NS}_{\omega_1}\text{ is }\omega_1\text{-dense}"$. We also introduce a new forcing axiom $\mathrm{QM}$, show it is consistent assuming a supercompact limit of supercompact cardinals and prove that it implies $\mathbb{Q}_{\mathrm{max}}\text{-}(*)$. Consequently, $\mathrm{QM}$ implies ``$\mathrm{NS}_{\omega_1}$ is $\omega_1$-dense".