Lattice points counting and Diophantine approximable on manifolds

It has been known for at least about a century that there exist algebraic badly approximable points confined by algebraic relations. On the other hand, until very recently nothing was known regarding badly approximable points subject to generic smooth (non-algebraic) relations. In this talk I will describe a technique in metric Diophantine approximation that has recently enabled to shed some light on this problem. Time permitting I will mention some other applications of the technique.