Seminar talk, 19 October 2011

Speaker: Pavel Bibikov

Title: Classification of linear actions of algebraic groups on the space of homogeneous forms

Abstract:
Consider an action of a group $G$ on a manifold $M$. This action prolongs to action on the space of functions on $M$. The talk will discuss a new method of construction the field of differential invariants of this action (and, as a consequence, classification of $G$-orbits of functions on $M$).

As an example, there will be discussed solutions of the problems of classification $\mathrm{GL}_3(\mathbb{C})$- and $\mathrm{SO}_3(\mathbb{C})$-orbits of ternary forms. Then there will be considered a solution of more general problem of classification of $G$-orbit of homogeneous form in many variables. Surprisingly, this classification does not depend on the degree of the form, on the number of variables, and even on (in a sense) the group $G$.

In conclusion, there will be explained how to generalized these results to classification of $G$-orbit of functions on manifolds.