Forcing "NSω₁ is ω₁-dense" From Large Cardinals

We answer a question of Woodin by showing that assuming an inaccessible cardinal κ which is a limit of <κ-supercompact cardinals exists, there is a stationary set preserving forcing P so that VP``NSω₁ is ω₁-dense". We also introduce a new forcing axiom QM, show it is consistent assuming a supercompact limit of supercompact cardinals and prove that it implies Qₘₐₓ-(*). Consequently, QM implies ``NSω₁ is ω₁-dense".