On Unirationality of hypersurfaces

There are several notions that are used to characterize algebraic varieties in terms of their similarity to a projective space. Two of these notions are rationality and unirationality. The problem of determining whether a variety is rational or unirational is in general very hard and unirationality in particular is poorly understood. In this talk, I will start by reviewing basic properties of rational/unirational varieties and recent progress in the study of these notions. I will then focus on the case of hypersurfaces and will discuss recent joint work with Eric Riedl on unirationality of hypersurfaces of low degree and evidence for non-unirationality of Fano hypersurfaces of large degree.