Abstract: It will be the continuation, on a more technical level, of the first lecture. Topics to be (very briefly) covered include: (1) recurrence to compact sets and Diophantine approximation on manifolds; (2) orbit closures, the Oppenheim conjecture and the Littlewood conjecture; (3) classification of ergodic invariant measures and equidistribution; (4) quantitative Oppenheim conjecture and counting of integral points on homogeneous varieties.