Seminar talk, 19 October 2011

Speaker: Pavel Bibikov

Title: On differential invariants of G-manifolds

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.