Pseudogroups, Moving Frames and Differential Invariants

Abstract:
I will report on my ongoing joint work with Peter Olver on developing systematic and constructive algorithms for analyzing the structure of continuous pseudogroups and identifying various invariants for their action.

In our approach we employ moving frames, which for general pseudogroup actions are defined as equivariant mappings from the space of jets of submanifolds into the pseudogroup jet bundle. The existence of a moving frame requires local freeness of the action in a suitable sense and, as in the finite dimensional case, moving frames can be used to systematically produce complete sets of differential invariants and invariant coframes for the pseudogroup action and to effectively analyze their algebraic structure.

In this talk I will focus on the method of moving frames combined with techniques from commutative algebra to discuss Tresse-Kumpera type existence results for differential invariants of a pseudogroup action and to describe methods for analyzing the algebraic structure of such invariants.

Our constructions are equally applicable to finite dimensional Lie group actions and provide a slight generalization of the classical moving frame methods in this case.