I'll describe two results that fit a recent theme of research in commutative algebra and algebraic geometry: pushing known results from the graded to the multigraded setting. One involves using graded ultraproducts to characterize which multigraded algebras satisfy Stillman's conjecture and prove the existence of various bounds, and the other using homological techniques to get a bound on the regularity of curves. I'll emphasize some future directions that these tools may help to address.