The problem of computing the cohomology of line bundles on flag varieties in arbitrary characteristic has remained an open question despite decades of study and many important results. A more recent perspective on this problem has instead focused on stability; more precisely, how does the cohomology of a fixed line bundle vary as the dimension of the ambient flag becomes arbitrarily large? In recent joint work with Raicu, we answer this: the cohomology is eventually modeled by a fixed polynomial functor, depending only on the weight of the line bundle. Moreover, for certain classes of weights, we can show that these stability functors exhibit remarkable periodicity properties determined by the characteristic of the underlying field. In this talk, we will see some of the background and tools that go into these results, including an unexpected connection to a long-open problem of Akin--Buchsbaum on "universal" categorifications of the Jacobi-Trudi identity.