When a property holds for almost every point in Euclidean space, it is
natural to ask whether it continues to hold generically on some submanifold
of Euclidean space. So, for example, Khintchine’s theorem gives a condition
under which almost all points in Euclidean space are rationally approximable
at some given rate. Does this condition still guarantee the same for almost all
points on an embedded surface? I will discuss this, and other similar questions
arising in Diophantine approximation.