Invariant theory of theta-representations

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One of the main tasks of mathematics is to describe certain objects up to a certain equivalence relation. Often this relation is given by an algebraic group action. Then the equivalence classes are the orbits and orbits closures correspond to degenerations of our objects. Thus, describing orbits of algebraic actions, as well as deciding whether one orbit lies in the closure of another, is an important and interesting problem. However, this is possible only in a very few cases. One of this instances is provided by the theta-representations introduced by E.B.Vinberg in the 70-s. The aim of this project is to study theta-representations and related objects, get a better understanding of their orbit structure, symmetric invariants, find out whether orbits closures have such nice properties as normality or rationality of singularities.