2018-05-11 15:00-16:00 Room 1114, Sciences Building No. 1

Abstract: In algebraic geometry, one says that an algebraic variety $X$ defined over a field $K$ is rational if it is very close to being the affine space $K^n$. This means that one can parametrize the points of $X$ by $K^n$, in an almost one-to-one fashion. Deciding whether an algebraic variety, given by polynomial equations, is rational is in general very difficult. I will start with very classical and elementary examples (like the rational parametrization of the circle) and move on to spectacular results obtained recently on the behavior of rationality in a family: given a family $(X_t)$ of algebraic varieties, is the set of $t$ for which $X_t$ is rational open, closed?