A variety is "unirational" if it admits a dominant rational map from projective space. For moduli spaces this amounts to an explicit “recipe” for writing down a general member of the universal family. In characteristic zero, tensor forms obstruct unirationality -- famously employed by Harris--Mumford (1982) to prove that M_g is not unirational for g > 22. In positive characteristic, unirationality behaves much wilder due to the existence of inseparable maps. Consequently, we know the (non)-unirationality of few moduli spaces in positive characteristic. I will exhibit new techniques to obstruct unirationality in positive characteristic inspired by methods used to prove hyperbolicity in complex geometry. As applications, I will present a counterexample to a 1977 conjecture of Shioda regarding the unirationality of general type surfaces and prove that many Hilbert modular varieties over positive characteristics are not unirational.

For the discretisation of symmetric, divergence-conforming stress tensors in continuum mechanics, we prove inf-sup stability bounds which are uniform in polynomial degree and mesh size for the Hu–Zhang finite element in two dimensions. This is achieved via an explicit construction of a bounded right inverse of the divergence operator, with the crucial component being the construction of bounded Poincaré operators for the stress elasticity complex which are polynomial-preserving, in the Bernstein–Gelfand–Gelfand framework of the finite element exterior calculus. We also construct hp-bounded projection operators satisfying a commuting diagram property and hp-stable Hodge decompositions. Numerical examples are provided.