Aleksandr Pukhlikov (University of Liverpool)
pukh@liverpool.ac.uk The classical rationality problem investigates whether a given algebraic variety can be parametrised by rational functions. In other words, whether it is possible to introduce a global system of coordinates on a dense open subset. In this talk, we will show how the classical rationality problem in the higher-dimensional contexts generalizes to the problem of description of rationally connected structures on a given rationally connected fiber space. The key concept in higher-dimensional situation is that of birational rigidity. We will discuss some examples and open problems.