Metric Diophantine approximation: well approximable theory on manifolds

The well known theorems of Khintchine and Jarnik in metric Diophantine approximation provide a comprehensive description of the measure theoretic properties of real numbers approximable by rational numbers with a given error. Various generalisations of these fundamental results have been obtained for other settings, in particular, for curves and more generally manifolds. In this talk I will explain my recent Jarnik type results for a parabola in homogeneous settings and for a planar curve in `functional' inhomogeneous settings. These represents the first comprehensive study of its kind in the theory of Diophantine approximation on manifolds.