An algebraic variety over a field k is said to be rational if its function field is a purely transcendental extension of k. In this talk, we work over the real numbers and study the rationality question for a class of conic bundle threefolds; the varieties we consider all become rational over the complex numbers, but this rationality construction does not in general descend to R. This talk is based on joint work with Sarah Frei, Soumya Sankar, Bianca Viray, and Isabel Vogt, and on joint work with Mattie Ji.

A weakening of Frobenius splitting, Quasi-F-Splittings have proven to be a vital invariant in the study of varieties in positive characteristic, with numerous applications to arithmetic and birational geometry. This weaker condition extends the application of Frobenius to study singularities of arithmetically supersingular varieties, encompassing a much broader class of examples.  In this talk I'll provide an overview of Quasi-F-Splittings and introduce a local analogue, Quasi-F-Purity.

The weakly étale ring maps -- those which are flat and have flat diagonal -- have gained attention recently for their role in defining the pro-étale site of a scheme. We will propose a new definition of weakly étale ring maps via a lifting property analogous to the one used to define formally étale ring maps. We will use a result of Gabber on the cohomology of Henselian pairs to deduce the equivalence of the two definitions.