Invariant theory of theta-representations

数学的主要任务之一是描述某些对象直到某种等价关系。通常，这种关系是由代数群作用给出的。那么等价类就是轨道，而轨道闭包对应于我们对象的简并。因此，描述代数作用的轨道，以及判断一个轨道是否位于另一个轨道的闭包中，是一个重要而有趣的问题。然而，这只在极少数情况下是可能的。其中一个例子是E.B.Vinberg在《70-S》中介绍的题元表示法。这个项目的目的是研究theta表示和相关的对象，更好地了解它们的轨道结构，对称不变量，找出轨道闭包是否具有奇点的正规性或合理性等良好性质。

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.