The study of the birational group of Fano hypersurfaces was undertaken by Fano in dimension three, and has lead to the proof, by Iskovskikh and Manin, that smooth quartic threefolds are not rational.
Their work eventually led to the notion of birational rigidity. In this talk I will overview the problem. I will start from Segre and Manin's theorem on cubic surfaces over nonclosed fields, and explain how similar ideas can be used to establish birational rigidity in higher dimensions. Speaker(s): Tommaso de Fernex (University of Utah)