We construct explicit GL-equivariant minimal free resolutions of certain (truncations of) modules of relative invariants over Veronese subrings in arbitrary characteristic. The free modules in the resolution correspond to certain skew Schur modules associated to "ribbon" or "skew-hook" diagrams, and the differentials at each step are surprisingly uniform. We then utilize the uniformity of these resolutions to explicitly compute information about tensor products, Hom, and Tor between these modules and show that they also have rather simple descriptions in terms of ribbon skew-Schur modules. I will emphasize the hidden role of symmetric function theory in detecting the answer to this question and guiding our intuition to build new tools to prove our results in arbitrary characteristic. This is joint work with Mike Perlman, Sasha Pevzner, Vic Reiner, and Keller VandeBogert.

Building on a result of Swanson, Cutkosky--Herzog--Trung and Kodiyalam described the surprisingly predictable asymptotic behavior of Castelnuovo--Mumford regularity for powers of ideals on a projective space P^n: given an ideal I, there exist integers d and e such that for large enough n the regularity of I^n is exactly dn+e. Through a medley of examples we will see why asking the same question about an ideal I in the total coordinate ring S of a smooth projective toric variety X is interesting. After that I will summarize the ideas and methods we used to bound the region reg(I^n) as a subset of Pic(X) by proving that it contains a translate of reg(S) and is contained in a translate of Nef(X), with each bound translating by a fixed vector as n increases. Along the way will see some surprising behavior for multigraded regularity of modules. This is joint work with Juliette Bruce and Lauren Cranton Heller.