Orbit Closures

Let G be a connected linear algebraic group, let V be a finite dimensional algebraic G-module, and let O_1 and O_2 be two G-orbits in V. The talk is aimed at a discussion of the constructive ways of finding out whether or not O_1 lies in the Zariski closure of O_2. This yields the constructive ways of finding out whether given two points of V lie in the same orbit or not. Several classical problems in algebra and algebraic geometry are reduced to this problem.