I have made major contributions to the study of the dynamical behavior of flows on homogeneous spaces of Lie groups and applications of the theory to problems in Diophantine approximation. One of the problems concerns the set of values of quadratic forms, evaluated at points with integer coordinates. Conditions for the set to be dense, and estimates for the number of solutions with values in prescribed intervals, have now been understood, through the work of various authors including myself, in the case when the number of variables is at least 3.