Title: An "almost all versus no" dichotomy for orbits on homogeneous spaces

Abstract: Several years ago I proved (in an involved and somewhat mysterious way) the following theorem: suppose M is an analytic submanifold of R^n which contains a not very well approximable vector; then almost all its vectors are not very well approximable. Recently I found a simple argument establishing this and other more general results. Both old and new proofs use quantitative nondivergence on the space of lattices. No background is necessary to follow the talk, everything will be explained from scratch.